§1. Introduction

Let \(\pi\) be a unitary cuspidal automorphic representation of \({\mathop{\mathrm{GL}}}_n\) over \(\mathbb{Q}\). The standard \(L\)-functions \(L(\pi,s)\) (\(s \in \mathbb{C}\)) were introduced by Godement–Jacquet (Godement and Jacquet 1972), and later interpreted more broadly by Jacquet–Piatetski-Shapiro–Shalika as the \({\mathop{\mathrm{GL}}}_n \times {\mathop{\mathrm{GL}}}_1\) case of Rankin–Selberg convolutions for \({\mathop{\mathrm{GL}}}_n \times {\mathop{\mathrm{GL}}}_m\); we refer to §2.14.3, §2.14.5 and references for further details, and also the summaries of (Hervé Jacquet 1979), (Rudnick and Sarnak 1996, sec. 2) and (Cogdell 2008). Conjectures of Langlands (R. P. Langlands 1970) predict that “all \(L\)-functions” arise as shifted products of such standard \(L\)-functions.

As a matter of convention, we exclude archimedean factors from the definition, so that \(L(\pi,s)\) is defined for \(s \in \mathbb{C}\) with \(\Re(s) > 1\) (H. Jacquet and Shalika 1981, I, Thm 5.3) by an Euler product \[L(\pi,s) = \prod_p L(\pi_p,s),\] and in general by meromorphic continuation. In the case \(n = 1\), these are the Dirichlet \(L\)-functions, which include the Riemann zeta function as a special case. We focus here on the case \(n \geqslant 2\), in which \(L(\pi,s)\) is known to be entire.

§1.1. The subconvexity problem in the \(t\)-aspect

For each natural number \(n\), let \(\delta_n\) denote the supremum of all real numbers \(\delta\) with the property that for each unitary cuspidal automorphic representation \(\pi\) of \({\mathop{\mathrm{GL}}}_n\) over \(\mathbb{Q}\), there exists \(C_0 \geqslant 0\) so that for all \(t \in \mathbb{R}\), \[\label{eq:lpi-tfrac12-+}\tag{1.1} |L(\pi,\tfrac{1}{2} + i t)| \leqslant C_0 (1 + |t|)^{(1 - \delta) n/4}.\] The convexity bound (see Lemma 2.14) implies that \(\delta_n \geqslant 0\), while the generalized Lindelöf hypothesis predicts that \(\delta_n = 1\). The \(t\)-aspect subconvexity problem for \({\mathop{\mathrm{GL}}}_n\) (over \(\mathbb{Q}\)) is to show that \(\delta_n > 0\). For an overview of the history on this problem, we refer to (Friedlander 1995), (Iwaniec and Sarnak 2000, sec. 2), (Philippe Michel and Venkatesh 2006), (Philippe Michel 2007, sec. 4, §5) and (Munshi 2018a). We survey below some of the existing literature, which has primarily addressed the case \(n \leqslant 3\). Our motivating result is the following.

We note that when \(n=2\) or \(3\), our estimate \((1.2)\) is not competitive with the existing numerical records. On the other hand, we have prioritized the clarity and uniformity of our arguments over their numerical optimality.

Summary of existing literature

It will be useful to introduce the modified quantity \(\delta_n^{(*)}\) (resp. \(\delta_{n}^{(**)}\)) defined similarly, but restricting the quantification to \(\pi\) that are unramified at every finite place (resp. that correspond to Hecke–Maass cusp forms on \({\mathop{\mathrm{SL}}}_n(\mathbb{Z})\)). Clearly \(\delta_n^{(**)} \geqslant\delta_n^{(*)} \geqslant\delta_n\). Many papers establish lower bounds for \(\delta_n^{(*)}\) or \(\delta_n^{(**)}\) using methods that are expected to extend to \(\delta_n\), although such extensions have not consistently been carried out.

It has been known that \(\delta_1 > 0\) since the classical work of Weyl, Hardy–Littlewood and Landau (Weyl 1916; Landau 1924), who in fact established the stronger bound \(\delta_1 \geqslant 1/3\). The current numerical record, \(\delta_1^{(*)} \geqslant 65/171 = 0.380\dotsb\), follows from work of Bombieri–Iwaniec (Bombieri and Iwaniec 1986) and Bourgain (Bourgain 2017).

It was shown first by Good (Good 1982) and Meurman (Meurman 1990) that \(\delta_2^{(*)} > 0\), and in fact that \(\delta_2^{(*)} \geqslant 1/3\). Recently Aggarwal (Aggarwal 2020) showed that \(\delta_2 \geqslant 1/3\) (see also Booker–Milinovich–Ng (Booker, Milinovich, and Ng 2019)).

It was shown first by Munshi (Munshi 2015), following earlier work of X. Li (Xiaoqing Li 2011), that \(\delta_3^{(**)} > 0\), and in fact that \(\delta_3^{(**)} \geqslant 1/24\). This has been improved to \(\delta_3^{(**)} \geqslant 1/10\) by Aggarwal (Aggarwal 2021) and to \(\delta_3^{(**)} \geqslant 1/6\) by Aggarwal–Leung–Munshi (Aggarwal, Leung, and Munshi 2022).

Sporadic cases of \(t\)-aspect subconvex bounds in higher rank have also been given, e.g., for Rankin–Selberg convolutions on \({\mathop{\mathrm{GL}}}_2 \times {\mathop{\mathrm{GL}}}_2\) (J. Bernstein and Reznikov 2010; Blomer, Jana, and Nelson 2023) or \({\mathop{\mathrm{GL}}}_3 \times {\mathop{\mathrm{GL}}}_2\) (Munshi 2018b).

Soundararajan (Soundararajan 2010, Example 3) established the following nontrivial bound, weaker than the assertion that \(\delta_n > 0\): for each \(n\) and \(\pi\) as above, and each \(\varepsilon> 0\), there exists \(C_0\) so that \[\label{eq:lpi-tfrac12-+-1}\tag{1.3} |L(\pi, \tfrac{1}{2} + i t)| \leqslant C_0 \frac{(1 + |t|)^{n/4} }{(\log (3 + |t|))^{1 - \varepsilon}}.\]

§1.2. Subconvex bounds assuming uniform parameter growth

We will establish Theorem 1.1 in a more general form.

The archimedean factor attached to \(L(\pi,s)\) may be written \[\label{eqn:L-infinity}\tag{1.4} L_\infty(\pi,s) = \prod_{j=1}^{n} \Gamma_\mathbb{R}(s + t_j), \quad \Gamma_{\mathbb{R}}(s) := \pi^{-s/2} \Gamma(s/2)\] for some complex numbers \(t_j\), which we call the archimedean \(L\)-function parameters of \(\pi\). The functional equation for \(L(\pi,s)\) involves a natural number, called the finite conductor of \(\pi\), which we denote by \(C(\pi_{\mathop{\mathrm{fin}}})\). The analytic conductor of \(\pi\) (Iwaniec and Sarnak 2000) is then defined to be the product \(C(\pi) := C(\pi_{\mathop{\mathrm{fin}}}) C(\pi_\infty)\), where \(C(\pi_\infty)\), the archimedean conductor, is the product \(\prod_{j=1}^n (3 + |t_j|)\) or some mild variant thereof.

Write \(\mathcal{A}(n)\) for the set of unitary cuspidal automorphic representations of \({\mathop{\mathrm{GL}}}_n\) over \(\mathbb{Q}\). The convexity bound (Lemma 2.14) asserts that for each natural number \(n\) and \(\varepsilon> 0\), there exists \(C_0 \geqslant 0\) so that for each \(\pi \in \mathcal{A}(n)\), \[|L(\pi,\tfrac{1}{2})| \leqslant C_0 C(\pi)^{1/4+\varepsilon}.\]

We say that a family \(\mathcal{F} \in \mathcal{A}(n)\) satisfies a subconvex bound if there exists \(\delta > 0\) and \(C_0 \geqslant 0\) so that for all \(\pi \in \mathcal{F}\), \[|L(\pi, \tfrac{1}{2} )| \leqslant C_0 C(\pi)^{(1 - \delta)/4}.\] The generalized Lindelöf hypothesis predicts that every family (or equivalently, the “universal family” \(\mathcal{F} = \mathcal{A}(n)\)) satisfies a subconvex bound with \(\delta = 1\). It is known by the classical work of Weyl (Weyl 1916), Hardy–Littlewood, Landau (Landau 1924), Burgess (Burgess 1957) and Heath-Brown (Heath-Brown 1978) that for \(n = 1\), every family satisfies a subconvex bound. The more recent work of Michel–Venkatesh (Philippe Michel and Venkatesh 2010) established the same conclusion when \(n=2\) (indeed, for the analogous problem over any fixed number field).

Soundararajan–Thorner (Soundararajan and Thorner 2019) recently established the general logarithmic improvement \[\label{eq:lpi-tfrac12-leq-2}\tag{1.5} |L(\pi,\tfrac{1}{2})| \leqslant C_0 \frac{C(\pi)^{1/4}}{(\log C(\pi))^{1/10^7 n^3}}.\] Their work made unconditional an earlier result of Soundararajan (Soundararajan 2010, Thm 1), which featured the stronger logarithmic improvement \((\log C(\pi))^{1-\varepsilon}\), as in \((1.3)\), but under the assumption of (a weakened form of) the generalized Ramanujan conjecture.

We focus in this paper on the spectral aspect, in which one regards the archimedean conductor \(C(\pi_\infty)\) as varying substantially and the finite conductor \(C(\pi_{\mathop{\mathrm{fin}}})\) as either fixed or varying mildly. We say that \(\mathcal{F} \subseteq \mathcal{A}(n)\) satisfies a spectral aspect subconvex bound, with polynomial dependence upon the level, if there exists \(\delta > 0\), \(C_0 \geqslant 0\) and \(A \geqslant 0\) so that for all \(\pi \in \mathcal{F}\), \[\label{eq:lpi-tfrac12-leq-1}\tag{1.6} |L(\pi,\tfrac{1}{2})| \leqslant C_0 C(\pi_\infty)^{(1-\delta)/4} C(\pi_{\mathop{\mathrm{fin}}})^{A}.\]

The following definition has played an important (although often implicit) role in the literature on the subject, and also in this paper. We say that \(\mathcal{F} \subseteq \mathcal{A}(n)\) has uniform parameter growth if there exists \(c > 0\) so that for each \(\pi \in \mathcal{F}\), the archimedean \(L\)-function parameters \(t_1,\dotsc,t_n\) of \(\pi\) satisfy \[\label{eq:c-leq-fract_j1}\tag{1.7} \frac{\min \{|t_1|, \dotsc, |t_n|\} }{1 + \max \{|t_1|,\dotsc,|t_n|\}} \geqslant c.\] Note that the LHS of \((1.7)\) is obviously bounded by \(1\), so this condition forces all of the \(t_j\) to be of roughly the same size. For instance, a result of Jutila–Motohashi (Jutila and Motohashi 2005) may be formulated as follows: for each \(\delta < 1/3\), there exists \(C_0 \geqslant 0\) so that for each \(\pi \in \mathcal{A}(2)\) of “full level” (i.e., \(C(\pi_{\mathop{\mathrm{fin}}}) = 1\)), we have \[|L(\pi,\tfrac{1}{2})| \leqslant C_0 (1 + \max \{|t_1|, |t_2|\})^{(1 - \delta)/2},\] where \(\{t_1, t_2\}\) denote the archimedean \(L\)-function parameters of \(\pi\). This result implies that a family \(\mathcal{F} \subseteq \mathcal{A}(2)\) with uniform parameter growth and full level satisfies a spectral aspect subconvex bound (with polynomial dependence upon the level, tautologically). However, specialized to the “conductor dropping” case in which one of the two parameters \(t_1\) or \(t_2\) is much smaller than the other (e.g., \(t_2 = \operatorname{O}(1)\) while \(t_1 \rightarrow \infty\)), this estimate does not improve upon the convexity bound. That case was not addressed until the work of Michel–Venkatesh noted above, following an important breakthrough of Duke–Friedlander–Iwaniec (Duke, Friedlander, and Iwaniec 2002) (see also (Blomer, Harcos, and Michel 2007; P. Michel 2004; Blomer and Khan 2019)).

For \(n \geqslant 3\), the first general examples of families \(\mathcal{F} \subseteq \mathcal{A}(n)\) proven to satisfy subconvex bounds, beyond the “\(t\)-aspect” setting discussed in §1.1 or special families arising from functorial lifts, were given by Blomer–Buttcane (Blomer and Buttcane 2020a, 2020b). To describe their result, let us say that \(\mathcal{F} \subseteq \mathcal{A}(n)\) avoids the walls if there exists \(c > 0\) so that for each \(\pi \in \mathcal{F}\), \[\frac{ \max_{j \neq k}|t_j - t_k|}{1 + \max \{|t_1|,\dotsc,|t_n|\}} \geqslant c.\] Blomer–Buttcane’s results may be formulated as follows: if \(\mathcal{F} \subseteq \mathcal{A}(3)\) has uniform parameter growth, avoids the walls, and each \(\pi \in \mathcal{F}\) has trivial central character, then \(\mathcal{F}\) satisfies a spectral aspect subconvex bound, with polynomial dependence upon the level. A preprint of Kumar–Mallesham–Singh (Kumar, Mallesham, and Singh 2020) establishes a similar conclusion, but with a modified form of the wall-avoidance condition.

We will show the following:

For the sake of illustration, we record the straightforward deduction from the latter result to the former:

It is similarly straightforward to see that Theorem 1.2 generalizes Theorem 1.1:

§1.3. Discussion of the proof

Overview

The closest result in the literature to Theorem 1.2 is the main result of (P. D. Nelson 2023). We summarize that result informally:

Theorem 1.2 may be understood as a specialization of a hypothetical generalization of Theorem 1.6: if we

• generalize Theorem 1.6 by dropping

  • the anisotropy condition on \(H\),

  • the cuspidality condition on the automorphic representations, and

  • the temperedness condition on their local components,

• specialize that generalization to the “least anisotropic” example \((G,H) = ({\mathop{\mathrm{GL}}}_{n+1}, {\mathop{\mathrm{GL}}}_n)\) and \(F = \mathbb{Q}\), and then

• specialize further by taking for \(\sigma\) the “least cuspidal” automorphic representation, consisting of Eisenstein series with trivial parameters, so that the Rankin–Selberg convolution is given by \[L(\pi \times \sigma, \tfrac{1}{2}) = L(\pi, \tfrac{1}{2})^n,\]

then we arrive at something like Theorem 1.2. In this paper, we address that specialization by combining the strategy and results of (P. D. Nelson 2023) with some new ideas that overcome the failure of temperedness, cuspidality and (most seriously) anisotropy.

We recall that the anisotropy of \(H\) corresponds to the compactness of the quotients \(H(\mathbb{Q}) \backslash H(\mathbb{A})\), \(H(\mathbb{Z}) \backslash H(\mathbb{R})\). There has been little work concerning the problem of bounding automorphic forms on higher-rank non-compact quotients, uniformly with respect to both

• the argument of the automorphic form as it tends into the cusp, and

• the underlying parameters of the automorphic form.

Some exceptions include the recent work of Brumley–Templier (Brumley and Templier 2020) and Blomer–Harcos–Maga (Blomer, Harcos, and Maga 2019, 2020).

We note that the present paper does not “replace” the work (P. D. Nelson 2023) treating the compact case. Indeed, the main technical results of that work (concerning the volume estimates and refined microlocal calculus) are applied here as a “black box” and combined with some new ingredients. On the other hand, the relative character asymptotics of (P. D. Nelson and Venkatesh 2021, sec. 19), which served as a crucial input to (P. D. Nelson 2023), do not enter here; the analogous role in the present treatment is played by direct analysis of local zeta integrals.

We survey the local issues addressed by this paper in §1.3.9, §1.3.10 and §1.3.11. The main global problem that we face and overcome is to give nontrivial bounds for the “local \(L^2\)-norms” of certain Eisenstein series, uniformly as we move away from the bulk of the fundamental domain. This is needed to compensate for the growth of the geometric side of a pretrace formula for \({\mathop{\mathrm{GL}}}_n\), coming from a long sum over unipotent elements of \({\mathop{\mathrm{GL}}}_n(\mathbb{Z})\). It is a familiar feature that trace-like formulas on non-compact quotients are complicated by the contributions of unipotent elements. The Arthur–Selberg trace formula gives a well-known example where the taming of the unipotent contribution requires an inclusion-exclusion scheme involving Eisenstein spectra coming from Levi subgroups. The present situation more closely resembles the Jacquet–Rallis relative trace formula for \((G, H \times H)\) (see (Hervé Jacquet and Rallis 2011; Zydor 2020; Chaudouard 2019), (Raphaël Beuzart-Plessis, Chaudouard, and Zydor 2020, sec. 3)), but differs in that, by taking for \(\Psi\) a wave packet, our integrals converge absolutely. We are thus faced not with the qualitative-analytic problem of making sense of divergent integrals, but rather the quantitative-analytic problem of obtaining satisfactory estimates for convergent ones. It was nevertheless unclear to us whether an inclusion-exclusion scheme would be necessary, or whether direct estimation would be possible. We eventually succeeded via the latter route, in the manner indicated below in §1.3.7–§1.3.8.

In the remainder of this introductory section, we give a high-level overview, intended for experts, of the new features of this paper relative to the compact quotient case treated in (P. D. Nelson 2023). In particular, we assume some familiarity with the notion of “coadjoint microlocalization” described in (P. D. Nelson and Venkatesh 2021, sec. 1) and applied in (P. D. Nelson 2023). Some readers may wish to first study (P. D. Nelson 2023, sec. 2) for an informal overview of the compact case, and (P. D. Nelson and Venkatesh 2021, sec. 1) for an overview of how microlocal analysis and the orbit method may be applied to the analysis of automorphic forms. On the other hand, we have attempted to keep the body of this paper self-contained and to give complete details. We emphasize that the following discussion is informal. In particular, notation such as \(\approx\) or \(\lessapprox\) is not defined precisely.

Setup

The main ideas may be explained in the full level case (i.e., \(C(\pi_{\mathop{\mathrm{fin}}}) = 1\)), where all the action takes place on the quotients \[:= \Gamma \backslash G := {\mathop{\mathrm{GL}}}_{n+1}(\mathbb{Z}) \backslash {\mathop{\mathrm{GL}}}_{n+1}(\mathbb{R}), \quad [H] := \Gamma_H \backslash H := {\mathop{\mathrm{GL}}}_{n}(\mathbb{Z}) \backslash {\mathop{\mathrm{GL}}}_{n}(\mathbb{R}).\] We denote by \(\mathfrak{g}, \mathfrak{h}\) the Lie algebras and \(\mathfrak{g}^\wedge, \mathfrak{h}^\wedge\) their imaginary duals. We write \(Z \leqslant G\) and \(Z_H \leqslant H\) for the centers.

We regard \(\pi\) as a space of cusp forms on \([G]\), transforming under \(Z\) via some unitary character. Similarly, \(\sigma\) is a space of Eisenstein series on \([H]\). The method of Jacquet–Piatetski-Shapiro–Shalika leads to an integral representation \[\label{eq:int-_h-varphi}\tag{1.10} \int _{[H]} \varphi \Psi \approx L(\pi,\tfrac{1}{2})^{n} Z(\varphi,\Psi),\] where \(Z(\varphi,\Psi)\) denotes the local integral obtained by integrating the archimedean Whittaker functions of \(\pi\) and \(\Psi\). To bound \(L(\pi,\tfrac{1}{2})\) using \((1.10)\) (following (J. Bernstein and Reznikov 2010; Venkatesh 2010; Philippe Michel and Venkatesh 2010)), we must choose vectors \(\varphi\) and \(\Psi\) for which we may prove that the local integral \(Z(\varphi, \Psi)\) is not too small, while the global integral \(\int_{[H]} \varphi \Psi\) is not too large. Strictly speaking, we replace \(\Psi\) with a “wave packet” or “pseudo Eisenstein series” that decays rapidly on \([H]\), which has the effect of replacing \(L(\pi,\tfrac{1}{2})^n\) by an average of \(\prod_{j=1}^n L(\pi,\tfrac{1}{2} + s_j)\) over small \(s = (s_1,\dotsc,s_n) \in \mathbb{C}^n\), but we will often ignore this point in this overview and write simply \(L(\pi,\tfrac{1}{2})^n\) in place of the more complicated expression that actually arises.

We note the recent work of Tsuzuki (Tsuzuki 2021), which applies an integral representation similar to \((1.10)\) to the nonvanishing problem.

§1.3.3. Analytic test vectors

We choose vectors following (P. D. Nelson and Venkatesh 2021) (although with completely different implementation details, for reasons discussed below):

Let \(\mathcal{O}_\pi \subseteq \mathfrak{g}^\wedge\) and \(\mathcal{O}_\sigma \subseteq \mathfrak{h}^\wedge\) denote the coadjoint orbits. (In the absence of any temperedness assumption on \(\pi\), take these to be the loci of the infinitesimal characters – see §7.1.4.) We may choose \(\tau \in \mathfrak{g}^\wedge\) and \(\theta \in \mathfrak{h}^\wedge\) so that \[T \tau \in \mathcal{O}_\pi, \quad T \theta \in \mathcal{O}_{\sigma}, \quad \tau|_{\mathfrak{h}} = \theta.\] Indeed, we may be quite explicit: since \(\sigma\) consists of Eisenstein series with trivial parameters, \(\mathcal{O}_\sigma\) is the nilcone, so we may take for \(\theta\) any nilpotent Jordan block. We then solve for \(\tau\) by an exercise in rational canonical form (§7.3.3). For example, when \(n = 3\), we might take \[\theta = \begin{pmatrix} 0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix}, \quad \tau = \begin{pmatrix} 0 & 0 & 0 & \ast \\ 1 & 0 & 0 & \ast \\ 0 & 1 & 0 & \ast \\ 0 & 0 & 1 & \ast \end{pmatrix}.\] The normalization is such that \(\tau\) remains bounded as \(T \rightarrow \infty\).

We take for the cusp form \(\varphi\) a unit vector microlocalized at \(T \tau\).

We take for the pseudo Eisenstein series \(\Psi\) a unit vector microlocalized at \(T \theta\). More precisely, \(\Psi\) is described by a function \[f : U_H \backslash H \rightarrow \mathbb{C},\] where \(U_H\) denotes a maximal unipotent subgroup. For example, if we pretend that \(\Gamma_H \backslash H = {\mathop{\mathrm{SL}}}_2(\mathbb{Z}) \backslash {\mathop{\mathrm{SL}}}_2(\mathbb{R})\), then we may identify \(U_H \backslash H\) with the punctured plane \(\mathbb{R}^2 - \{0\}\) and take \(\Psi(g) = \sum _{v \in \mathbb{Z}^2 - \{0\}} f(v g)\). The space \(U_H \backslash H\) admits a right action by \(H\) and a left action by the toral part \(A_H\) of the normalizer of \(U_H\). In practice, we take \(U_H\) lower-triangular and identify \(A_H\) with the diagonal subgroup, thus, e.g., for \(n=3\), \[U_H = \begin{pmatrix} 1 & 0 & 0 \\ \ast & 1 & 0 \\ \ast & \ast & 1 \end{pmatrix}, \quad A_H = \begin{pmatrix} \ast& 0 & 0 \\ 0 & \ast & 0 \\ 0 & 0 & \ast \end{pmatrix}.\] We take \(f\) to be:

• \(L^2\)-normalized,

• microlocalized under \(H\) at \(T \theta\),

• uniformly smooth under \(A_H\), and

• concentrated near \(T^{-\rho^\vee}\), where \(\rho^\vee\) denotes the half-sum of positive coroots for \(U_H\).

In the \({\mathop{\mathrm{SL}}}_2\) example, the resulting function on \(\mathbb{R}^2 - \{0\}\) has the rough shape \[f(x,y) \approx \begin{cases} T^{-1/2} e^{i T y/x} & \text{ if } x \asymp T^{1/2}, \, y \ll 1, \\ 0 & \text{ otherwise,} \end{cases}\] or equivalently, \[f(r \cos \alpha, r \sin \alpha) \approx \begin{cases} T^{-1/2} e^{i T \alpha} & \text{ if } r \asymp T^{1/2}, \, \alpha \ll T^{-1/2}, \\ 0 & \text{ otherwise.} \end{cases}\] We assume moreover that \(f\) is chosen so that \(\Psi\) is “nondegenerate” in certain senses (e.g., orthogonal to the residual spectrum); in the \({\mathop{\mathrm{SL}}}_2\) example, this amounts to requiring that the radial Mellin transform of \(f\) vanish at certain arguments.

Ignoring wave packet issues, we obtain with this choice \[\label{eq:zvarphi-psi-approx}\tag{1.11} Z(\varphi, \Psi) \approx T^{-n^2/4},\] leading to an integral representation roughly of the shape \[L(\pi,\tfrac{1}{2})^{n} \approx T^{n^2/4} \int _{[H]} \varphi \cdot \Psi.\]

§1.3.4. Relative trace formula

We construct, as in (P. D. Nelson 2023, sec. 1.5.2), a “convolution kernel” \(\omega \in C_c^\infty(G)\) with the following properties.

1. The associated integral operator \(\pi(\omega)\) satisfies \[\label{eq:piom-varphi-appr}\tag{1.12} \pi(\omega) \varphi \approx \varphi.\]

2. \(\omega\) concentrates on \[\left\{ g \in G : g = 1 + O(T^{-\varepsilon}), \quad {\mathop{\mathrm{Ad}}}^*(g) \tau = \tau + O(T^{-1/2-\varepsilon}) \right\}.\]

3. \(\omega\) has \(L^\infty\)-norm \(\lessapprox T^{n(n+1)/2}\).

Informally, the spectral support of \(\omega\) is “\(\pi + \operatorname{O}(1)\),” and \(\pi(\omega)\) is essentially the orthogonal projection onto \(\varphi\). The function \(\omega\) has the same shape as its adjoint \(\omega^*\) and self-convolution \(\omega \ast \omega^{\ast}\), so we denote these simply by \(\omega\) in this informal overview.

As in (P. D. Nelson 2023, sec. 1.5.3), the main point is to show that \[\label{eq:t-n2-int}\tag{1.13} T^{-n/2} \int _{h_1, h_2 \in [H]} \sum _{\gamma \in \Gamma } \bar{\Psi}(h_1) \Psi(h_2) \omega(h_1^{-1} \gamma h_2) \, d h_1 \, d h_2 = (\text{main term}) + \operatorname{O}(T^{-\delta}).\] Indeed, the pretrace formula for \([G]\) (or Cauchy–Schwarz) applied to \(\omega\) shows that the LHS of \((1.13)\) majorizes an average of \(|L(\pi,\tfrac{1}{2})|^{2 n}\) over a family of size \(T^{n(n+1)/2}\). In view of our uniform parameter growth assumption, the family size is roughly the fourth root of the conductor, so the amplification method will deliver a subconvex bound if we can give a sufficiently robust proof of \((1.13)\). The sum over \(\gamma \in H Z\) contributes an apparent main term, so we reduce to showing that the \(\gamma \notin H Z\) contribute \(\operatorname{O}(T^{-\delta})\). (There would be additional contributions to the main term had we not passed to wave packets, cf. (Ramakrishnan and Rogawski 2005).)

Bilinear forms estimates

We record an informal statement of the “bilinear forms” estimate given by (P. D. Nelson 2023, Thm 15.1), the proof of which relies ultimately upon the volume estimates (P. D. Nelson 2023, secs. 16–17) whose relevance was first observed by S. Marshall. Fix a compact neighborhood \(\mathcal{H}\) of the identity element in \(H\). Let \(\Psi_1\) and \(\Psi_2\) be measurable functions, supported on \(\mathcal{H}\), with \(\Psi_2\) satisfying the approximate equivariance condition \[|\Psi_2(h z)| \approx |\Psi_2(h)| \text{ for } (h,z) \in \mathcal{H} \times Z_H \text{ with } z = \operatorname{O}(1).\] Let \(g \in G\) be “not too close to \(H Z\).” Then \[\label{eq:t-n2-int-1}\tag{1.14} T^{-n/2} \int _{h_1, h_2 \in \mathcal{H} } \left\lvert \Psi_1(h_1) \Psi_2(h_2) \omega(h_1^{-1} g h_2) \right\rvert \, d h_1 \, d h_2 \lessapprox T^{-1/2 + \varepsilon} \|\Psi_1\|_{L^2(\mathcal{H})} \|\Psi_2\|_{L^2(\mathcal{H})}.\] The factor \(T^{-1/2}\) improves upon what one might regard as the “trivial estimate” obtained by omitting that factor. We refer to (P. D. Nelson 2023, secs. 1.5.3–1.5.7, §2.9–2.11) for an extended discussion of such estimates.

In the compact quotient case, the estimate \((1.14)\) readily yields \((1.13)\): take \(\mathcal{H}\) large enough to contain a fundamental domain for \([H]\) and observe that

• the number of \(\gamma\) contributing to \((1.13)\) is essentially \(\operatorname{O}(1)\), and

• by the discreteness of \(\Gamma\), any element of \(\Gamma - H Z\) is “not too close to \(H Z\)” in the required sense.

In the non-compact case, the same argument gives an adequate estimate for the contribution from \(x, y\) lying in any fixed compact subset of \([H]\). (Recall that \(\Psi\) is a wave packet, so we have good control over its \(L^2\)-norm.) Unfortunately, as \(x\) and/or \(y\) “escape to \(\infty\),” the sum over \(\gamma\) in \((1.13)\) features a large number of unipotent elements, so unlike in the compact case, we cannot conclude immediately.

Siegel domains

Every element of \([H]\) may be represented in the form \(h = t x\) with \((t,x) \in \mathcal{T} \times \mathcal{H}\), where \(\mathcal{T}\) denotes the set of diagonal matrices \(t\) with entries \(t_1 \geqslant\dotsb \geqslant t_n > 0\) and, as above, \(\mathcal{H}\) is a large enough fixed compact set. The Haar measure \(d h\) on \([H]\) is comparable to \[\frac{d t}{\delta_H(t)} \, d x, \quad \delta_H(t) = t_1^{n-1} t_2^{n-3} \dotsb t_n^{1-n}, \quad d t = \frac{d t_1}{t_1} \dotsb \frac{d t_n}{t_n}.\] We majorize the sum over \(\gamma \notin H Z\) in \((1.13)\) by \[T^{-n/2} \int _{t, u \in \mathcal{T} } \sum _{\gamma \in \Gamma - H Z} \int _{x,y \in \mathcal{H} } \left\lvert \Psi(t x) \Psi(u y) \omega(x^{-1} t^{-1} \gamma u y) \right\rvert \, d x \, d y \, \frac{d t \, d u}{\delta_H(t) \delta_H(u)}.\] We can apply the bilinear forms estimate \((1.14)\) with \(g = t^{-1} \gamma u\). (Many of these elements will be closer to \(H Z\) than in the compact case, where discreteness forces them all to be far away, but the dominant contribution still comes from those that are not too close.) Our task then reduces to showing that \[T^{-1/2} \int _{t, u \in \mathcal{T} } \nu(t,u) \|\Psi \|_{L^2(t \mathcal{H})} \|\Psi \|_{L^2(u \mathcal{H})} \frac{d t \, d u}{\delta_H(t) \delta_H(u)} \ll T^{-\delta},\] where \(\nu(t,u)\) denotes the number of \(\gamma \in \Gamma - H Z\) for which \(t^{-1} \gamma u = \operatorname{O}(1)\). By elementary counting arguments, we see that \(\nu(t,u)\) vanishes unless \(t \approx u\), in which case \(\nu(t,u) \approx \delta_H(t) t ^\dagger\), where \[\label{eq:t-dagger-:=}\tag{1.15} t ^\dagger := \prod_{j=1}^n \max(t_j, t_j^{-1}).\] After Cauchy–Schwarz, the main problem is then to show that \[\label{eq:t-14-int}\tag{1.16} T^{-1/2} \int _{t \in \mathcal{T} } \|\Psi \|_{L^2(t \mathcal{H})}^2 t ^\dagger \frac{d t}{\delta_H(t)} \ll T^{-\delta}.\] By comparison, the \(L^2\)-normalization of \(\Psi\) gives \[\int _{t \in \mathcal{T} } \|\Psi \|_{L^2(t \mathcal{H})}^2 \frac{d t}{\delta_H(t)} \ll 1,\] so the “enemy” is the additional factor \(t ^\dagger\).

The wave packet \(\Psi\) concentrates near \({\mathop{\mathrm{SL}}}_n(\mathbb{R})\) (since \(\det(T^{-\rho^\vee}) = 1\)), so one can focus on the contribution to \((1.16)\) from when \(\det(t) \approx 1\). Since \(\Psi\) is of rapid decay, the integrals \((1.16)\) decay rapidly in a qualitative sense, so “very large” values of \(t\) are no problem; the “intermediate” values of size larger than \(1\) must be addressed. We refer to (Blomer, Harcos, and Maga 2020, sec. 1.2) for some hint as to the precise meaning of “intermediate.” In any event, the total volume of the set of \(t\) that we must consider is logarithmic in \(T\), so the basic problem is to show that \[\label{eq:t-in-mathcalt}\tag{1.17} t \in \mathcal{T}, \det(t) = 1 \implies T^{-1/2} \|\Psi \|_{L^2(t \mathcal{H})}^2 t ^\dagger \ll \delta_H(t) T^{-\delta}.\]

§1.3.7. Theta functions and positivity

The estimate \((1.17)\) is straightforward when \(H = {\mathop{\mathrm{GL}}}_2(\mathbb{R})\), which is the case relevant for the subconvexity problem on \({\mathop{\mathrm{GL}}}_3\): every approach that we have tried works without difficulty. Already when \(H = {\mathop{\mathrm{GL}}}_3(\mathbb{R})\) (relevant for the subconvexity problem on \({\mathop{\mathrm{GL}}}_4\), hitherto unaddressed), an approach based on direct estimation of the Fourier expansion (e.g., as in (Blomer, Harcos, and Maga 2019)) seems hopeless: optimal bounds for all terms in that expansion add up to a worse than required bound. Several other natural avenues of attack (e.g., using the Rankin–Selberg integrals that represent \(L(\sigma \times \tilde{\sigma}, 1+ \varepsilon)\)) also seem inadequate. We will establish \((1.17)\) instead using “pseudo mirabolic Eisenstein series” or “Siegel theta functions,” which provide a more flexible source of nonnegative functions, taking large values in the cusp, than do their “spectral” counterparts relevant for the integral representation of \(L(\sigma \times \tilde{\sigma}, 1 + \varepsilon)\).

Let \(\phi \in C_c^\infty(\mathbb{R}^n)\) be a nonnegative function supported on a dyadic annulus of radii \(\approx t_n\). Form the series \[E(g) := \sideset{}{^*}\sum _{v \in \mathbb{Z}^n} \phi(v g),\] where \(\sum^*\) denotes a sum over primitive vectors. By considering the contribution to that series from the \(n\)th standard basis vector, we see that \(E(g) \gg 1\) for \(g \in t \mathcal{H}\). It follows that \[\|\Psi \|^2_{L^2(t \mathcal{H})} \ll \delta_H(t) \int _{[H]} |\Psi|^2 E\] The integral \(\int _{[H]} |\Psi|^2 E\) is a “pseudo Rankin–Selberg integral” (i.e., a spectral expansion of \(\Psi\) and \(E\) yields \({\mathop{\mathrm{GL}}}_n \times {\mathop{\mathrm{GL}}}_n\) Rankin–Selberg integrals with Eisenstein factors). It unfolds nicely enough. With some work, we can estimate the associated “pseudo local Rankin–Selberg integrals.” We arrive in this way at the estimate \[\label{eq:psi-_l2t-mathcalh2}\tag{1.18} \|\Psi \|_{L^2(t \mathcal{H})}^2 \ll \delta_H(t) t_n^n T^{\varepsilon}.\] We could have chosen \(\phi\) differently, i.e., concentrated on a different dyadic annulus, but the estimate \((1.18)\) turns out to be optimal among such choices.

The estimate \((1.18)\) does not imply \((1.17)\). A typical “problem case” is when \[\label{eq:t-=-diagt_1}\tag{1.19} t = \mathop{\mathrm{diag}}(t_1,\dotsc,t_n), \quad t_1 = r^{n-1}, \quad t_2 = \dotsb = t_n = r^{-1}\] where \(r\) is a small positive power of \(T\). In that case, \(t_n^n = r^{-n}\) and \(t ^\dagger \approx r^{2 (n-1)}\), so to achieve \((1.17)\), we would need \(T^{-1/2} r^{-n + 2 (n - 1)} \ll T^{-\delta}\), or equivalently, \(r^{n - 2} \ll T^{1/2 - \delta}\). The proof of (Blomer, Harcos, and Maga 2020, Thm 3) suggests that the rapid decay of \(\Psi\) kicks in only when \(r\) is a bit larger than \(T^{1/n}\). Consider then, for instance, the contribution from when \(r = T^{1/n}\). The desired estimate is that \(T^{1 - 2/n} \ll T^{1/2 - \delta}\), which holds only for \(n \leqslant 3\).

§1.3.8. Duality

We can do better by exploiting an important symmetry of the problem, coming from the outer automorphism of \({\mathop{\mathrm{GL}}}_n\) defined by the inverse transpose map \(g \mapsto g^{-% {\mathpalette\@transpose{}}% }\). (We came to appreciate the power of this symmetry while perusing (Blomer, Harcos, and Maga 2020, sec. 3.3).) We can argue either in terms of a dual mirabolic Eisenstein series \[\tilde{E}(g) = \sideset{}{^*}\sum _{v \in \mathbb{Z}^n} \phi(v g^{-% {\mathpalette\@transpose{}}% }),\] or a dual Eisenstein series \[\tilde{\Psi}(g) := \Psi(g^{-% {\mathpalette\@transpose{}}% }).\] Either way, we obtain the following bound, complementary to \((1.18)\): \[\label{eq:psi-_l2t-mathcalh2-2}\tag{1.20} \|\Psi \|_{L^2(t \mathcal{H})}^2 \ll \delta_H(t) t_1^{-n} T^{\varepsilon}.\] It is easy to see that this estimate is highly effective in “problem cases” such as \((1.19)\). In fact, the two estimates \((1.18)\) and \((1.20)\) combine to cover all cases, due to elementary inequalities such as the following (see Lemma 5.30): \[\det(1) = 1 \implies t ^\dagger \leqslant\max(t_1^n, t_n^{-n}).\] We thus obtain \[\label{eq:int-_t-in-Psi-t-dagger}\tag{1.21} \int _{t \in \mathcal{T} } \|\Psi \|_{L^2(t \mathcal{H})}^2 t ^\dagger \frac{d t}{\delta_H(t)} \ll T^{\varepsilon},\] and the proof is, to first approximation, complete.

§1.3.9. Intertwining operators and distributions related to Jacquet integrals

For the cuspidal variant of the problem considered here, the proof of \((1.21)\) is quite simple: see §18. For the Eisenstein case of primary interest, there are some further difficulties, discussed at length in Part . One of these difficulties is to understand the asymptotics of the various “partial Whittaker transforms” of \(\Psi\), including the constant term. Recalling that \(\Psi\) is attached to a function \(f \in C^\infty(U_H \backslash H)\), an important intermediate step is to understand the behavior of \(f\) under the Fourier transforms, or normalized intertwining operators, attached to all elements of the Weyl group of \(H\). We manage to avoid this step altogether by carefully constructing \(f\) so as to be exactly invariant under all normalized intertwining operators. To do this, we take for \(f\) the image, under a carefully-constructed convolution operator on \(A_H \times H\), of a distribution on \(U_H \backslash H\) closely related to the theory of Jacquet integrals. Since (by definition) normalized intertwining operators preserve Jacquet integrals, they likewise preserve \(f\) provided that we take our convolution operator on \(A_H \times H\) to be invariant under the action of the Weyl group on the first factor.

§1.3.10. Asymptotics of local zeta integrals

We have not yet commented on the key local estimates \((1.12)\) and \((1.11)\). The analogues of these already appeared in the compact case (P. D. Nelson 2023). In that case, we argued as follows. We started with the test function \(\omega\). We showed that \(\pi(\omega)\) behaves like a self-adjoint rank one idempotent, and so approximately corresponds to a scaling class of vectors \(\varphi\). The vectors \(\varphi\) themselves did not appear explicitly. In the setting of unitary groups, the local zeta integrals \(Z(\varphi, \Psi)\) are not necessarily defined. In their place, we have certain matrix coefficients integrals \(\mathcal{Q}(\varphi,\Psi)\), defined at least in the tempered case, which may be understood as avatars of \(|Z(\varphi,\Psi)|^2\). We estimated the matrix coefficient integrals using the “reduction to small elements” technique of (P. D. Nelson and Venkatesh 2021, sec. 19) and then ultimately via appeal to the Kirillov character formula for \(\pi\) and \(\sigma\). More specifially, we proved such estimates on average for \(\varphi\) in an implicit family defined by \(\omega\).

The approach of the present paper is completely different. The primary motivation for the difference is that, since we are working with wave packets, we require not just estimates for individual zeta integrals, but rather a family of integrals \(Z(\varphi, \Psi, s)\) indexed by small elements \(s \in \mathbb{C}^n\). Off the “unitary axis” \((i \mathbb{R})^n\), the connection between such zeta integrals and matrix coefficient integrals is more tenuous, and does not seem to us to lend readily to the problem of estimation. A secondary motivation for the departure from (P. D. Nelson 2023) is that we do not wish to impose temperedness assumptions on \(\pi\). The Kirillov character formulas for non-tempered representations (see, e.g., (Harris and Oshima 2020)) are more unwieldly than in the tempered case, and in any event, the estimates of (P. D. Nelson and Venkatesh 2021, sec. 19) are available only in the tempered case.

We will instead study the zeta integrals \(Z(\varphi,\Psi,s)\) directly. In doing so, it is very convenient to assume that the Whittaker function of \(\varphi\) has compactly-supported image in the Kirillov model. This assumption trivializes the problem of analytic continuation of such integrals, and also reduces the range of arguments for which we need to estimate the Whittaker functions associated to \(\Psi\). For this reason, we will view the vector \(\varphi \in \pi\) as the primary object, rather than the test function \(\omega\).

§1.3.11. Asymptotics of the Kirillov model

While the adjustment of perspective noted in §1.3.10 reduces the complexity of analyzing the zeta integrals, it introduces a new difficulty: proving that \(\omega\) has the required approximation property \((1.12)\). (Recall that this property was essentially automatic in the approach pursued in the compact case, by virtue of the implicit definition of \(\varphi\) in terms of \(\omega\).) The mirabolic subgroup \(P\) of \(G\) acts in a very simple way on the Kirillov model. We construct \(\varphi\) so that it is microlocalized with respect to \(P\) at the restriction of \(T \tau\) to \(\mathop{\mathrm{Lie}}(P)\). We need to know that \(\varphi\) is actually microlocalized with respect to \(G\) at \(T \tau\). At a heuristic level, this is clear, since \(T \tau\) is the only element of the coadjoint orbit \(\mathcal{O}_\pi\) lying above its restriction to \(\mathop{\mathrm{Lie}}(P)\). It is a bit subtle to implement this heuristic rigorously. The scenario that we need to rule out is that the “frequency” of \(\varphi\) with respect to the \(G\)-action is significantly larger than \(T\). To do this, we first apply a quantitative form of Jacquet’s arguments (Hervé Jacquet 2010, sec. 3) concerning the archimedean Kirillov model, which allows us to show that \(\varphi\) has “frequency \(T^{\operatorname{O}(1)}\).” This serves as the first step in a bootstrapping argument, involving the microlocal calculi of (P. D. Nelson and Venkatesh 2021) and (P. D. Nelson 2023), by which we eventually conclude that \(\varphi\) has frequency \(\operatorname{O}(T)\) and is microlocalized at \(T \tau\), as required. We record a more detailed overview of this part of the paper in §12. We note in particular that the Kirillov character formula makes no appearance in this paper, in contrast to its repeated usage in the previous works (P. D. Nelson and Venkatesh 2021; P. D. Nelson 2023).

Outlook

We briefly comment on the scope of the method.

It would be natural to give a common generalization of Theorems 1.2 and 1.6, valid for all unitary (or orthogonal) GGP pairs and all families of pairs of automorphic representations \((\pi,\sigma)\) with uniform parameter growth. As an imperfect analogy, one might think of such a generalization as the “convex hull” of those theorems. For example, the case of Rankin–Selberg \(L\)-functions attached to cusp forms on \({\mathop{\mathrm{GL}}}_{n+1} \times {\mathop{\mathrm{GL}}}_n\) can be approached either by dropping the anisotropy and temperedness assumptions from Theorem 1.6, or by adapting Theorem 1.2 to the case that \(\sigma\) is cuspidal, which should in most respects be simpler than the Eisenstein case treated here. However, such a generalization would not be altogether formal, as the local estimates developed here for the Eisenstein case use that they are tempered and belong locally to the principal series. We hope that our work might nevertheless serve as a useful template for such a generalization.

Over number fields with complex places, there are a couple lemmas in the literature concerning \({\mathop{\mathrm{GL}}}_n(\mathbb{R})\) that would need to be generalized to \({\mathop{\mathrm{GL}}}_n(\mathbb{C})\) (e.g., (Hervé Jacquet 2010, Lemma 3) and (Jana and Nelson 2019, Lemma 5.2)) but we do not foresee significant difficulties there. The depth aspect is often similar to the archimedean aspect, but with less technical compilation due to the availability of the exact projectors in the Hecke algebra; one might hope that it could be treated similarly.

The “horizontal” level aspect (e.g., taking \(\pi\) of prime level), the quadratic twist case of which may have applications to quadratic forms in view of the appearance of self-dual standard \(L\)-functions in formulas for Whittaker coefficients on higher rank metaplectic groups (E. Lapid and Mao 2015, 2017b, 2017a), would require new ideas concerning the construction of analytic test vectors, and, we imagine, new estimates for exponential sums over finite fields, substituting for the role played by Proposition 11.4, (P. D. Nelson 2023, Thm 4.2 (iii) and Thm 15.1) and (P. D. Nelson 2023, secs. 16–17) for the “vertical” (archimedean or depth) aspects. The works of Bykovskii (Bykovskiı̆ 1996), Blomer–Harcos (Blomer and Harcos 2008, 2014) and Fouvry–Kowalski–Michel (Fouvry, Kowalski, and Michel 2015) (resp. Sharma (Sharma 2019) and Lin–Michel–Sawin (Lin, Michel, and Sawin 2019)) concerning the case of \({\mathop{\mathrm{GL}}}_2 \times {\mathop{\mathrm{GL}}}_1\) (resp. \({\mathop{\mathrm{GL}}}_3 \times {\mathop{\mathrm{GL}}}_2\)) give some hint as to what might be required. As a first step towards generalizing the results of those works to higher rank, it might be interesting to revisit them from the perspective of integral representations (cf. (Nunes 2020; Zacharias 2021; Schumacher 2020)).

The outstanding open problem is to address the case in which \(\pi\) is not of uniform parameter growth, often known as the conductor dropping case. We have already noted the works (Duke, Friedlander, and Iwaniec 2002; Blomer, Harcos, and Michel 2007; P. Michel 2004; Philippe Michel and Venkatesh 2010; Blomer and Khan 2019) addressing that case (or its related level aspect analogue) when \(n=2\). An extension of Theorem 1.2 for \(n=3\) or \(n=6\) to the conductor dropping case in which one or two of the archimedean \(L\)-function parameters remains bounded would have well-known applications to quantum chaos, as described, for instance, in (Iwaniec and Sarnak 2000; Watson 2002; Holowinsky and Soundararajan 2010; P. D. Nelson, Pitale, and Saha 2014; Sarnak 2011; P. D. Nelson 2022). Such an extension should not be expected to follow via analysis of the specific moments considered in this paper. Indeed, to obtain such an extension, we would need to further shrink the archimedean families that we consider. Doing so is not analytically feasible, even when \(n=2\): the families are already of “size \(\operatorname{O}(1)\)” with respect to the parameters.

Organization

The paper is intended to be read linearly, but other approaches may be preferred. For this reason, we briefly indicate the organization. In Part , we begin by formulating two auxiliary results:

• a local result, Theorem 3.1, asserting the existence of elements \(\varphi\), \(f\) and \(\omega\) satisfying the desiderata indicated in the sketch of §1.3, with \(\varphi\) constructed in terms of its archimedean Whittaker function \(W\), and

• a global result, Theorem 4.1, asserting the required \(L^2\)-local growth bounds \((1.21)\) for Eisenstein series.

The proof of the former is given in Part , the latter in Part . In the remainder of Part , we deduce our main result, Theorem 1.2, from these auxiliary results, via the expected global ingredients (amplification method, counting arguments, \(\dotsc\)). Part supplies an independent ingredient (discussed in §1.3.11) required in the construction of test vectors. The dependency graph of the parts of the paper is thus as follows:

• Part : Division of the argument

  • Part : Construction of test vectors

    • Part : Asymptotics of the Kirillov model

  • Part : Growth bounds for Eisenstein series

Acknowledgments

We would like to thank V. Blomer, R. Beuzart-Plessis, P. Garrett, G. Harcos, P. Sarnak, E. Lapid, Ph. Michel, Y. Sakellaridis, A. Venkatesh and Liyang Yang for helpful discussions and feedback. Much of this paper was written while the author was an Assistant Professor at ETH Zurich, during the Hausdorff Trimester Program Harmonic Analysis and Analytic Number Theory hosted by the Hausdorf Research Institute for Mathematics. This paper was completed at the Institute for Advanced Study, where the author was supported by the National Science Foundation under Grant No. DMS-1926686. We are grateful to each of these institutions for their excellent working conditions.

§2. Preliminaries

§2.1. Asymptotic notation and terminology

§2.1.1. Overview

This paper, like its predecessor (P. D. Nelson 2023), involves the iterated application of estimates involving functions whose support, size and oscillation depend upon many parameters. To manage this iteration sensibly, we again adopt the formalism of nonstandard analysis. Specifically, we adopt the axiomatization due to E. Nelson (E. Nelson 1977) (“internal set theory”, or IST for short), but with the terminological substitution of “fixed” for “standard.” This substitution has the primary effect of making this paper read (we hope) like a typical analytic number theory paper, so as to require little adjustment of habit for the reader who is familiar with such papers. A secondary effect is to have a convenient verb form: we write “fix \(x\)” synonymously with “let \(x\) be fixed.” A tertiary effect is to avoid conflict with “standard \(L\)-functions.” We briefly recall that axiomatization, referring to loc. cit. for details and superior exposition.

In ZFC, everything is a set. ZFC consists of the binary predicate “\(\in\)” (interpreted: “is an element of”) and some axioms for working with it. IST adjoins to ZFC an additional unary predicate, which we label “fixed,” and three axioms, recalled below, for working with that. A statement of IST is internal if it is really a statement of ZFC, i.e., if it does not use the predicate “fixed,” even implicitly; otherwise, it is external. The appendix of (E. Nelson 1977) establishes the conservation theorem: any internal statement provable in IST is also provable in ZFC. More precisely, any proof in IST may be mechanically converted to one in ZFC. In some applications of nonstandard analysis, such conversion is not practical, but in this paper, it is: the proof of our main results could be rewritten in ZFC in a reasonable amount of time by a human.

We define customary asymptotic notation in terms of “fixed,” as in (P. D. Nelson 2023, sec. 3.1). If \(|A| \leqslant C |B|\) for some fixed \(C \geqslant 0\), then we write \(A \ll B\) or \(B \gg A\) or \(A = \operatorname{O}(B)\); otherwise, we write \(B \lll A\) or \(A \ggg B\) or \(B = o(A)\). We write \(A \asymp B\) as shorthand for \(A \ll B \ll A\). We introduce some further notation below, and will introduce other definitions involving “fixed” throughout the paper. The reader who is familiar with the customary use of such notation can likely skip or skim the remainder of §2.1 on a first reading, referring back in case anything seems unclear or imprecise.

Informally, “fixed” quantities behave like “absolute constants.” In particular, ordinary mathematical objects (\(\mathbb{Z}\), \(\mathbb{R}\), \(\sqrt{2}\), the set of cuspidal automorphic representations of \({\mathop{\mathrm{GL}}}_{100}(\mathbb{Q})\), \(\dotsc\)) are fixed. On the other hand, any infinite set (and any non-fixed finite set) contains non-fixed elements. To make this heuristic more precise, one could introduce an asymptotic parameter \(T\) (tending off to \(\infty\) in some directed set), take every object in one’s mathematical universe to depend upon \(T\) by default, and think of “fixed” as meaning “independent of \(T\).” This is essentially what happens in the ultrafilter approach to nonstandard analysis, which in turn forms the basis for the proof of the conservation theorem of IST. The definitions and results of this paper could be reformulated in such “parameter-dependent” language, much like in (P. D. Nelson and Venkatesh 2021), but doing so seemed to us to ultimately reduce the clarity and precision.

As detailed in (E. Nelson 1977, sec. 2), any external statement may be mechanically converted to an equivalent internal statement (subject to a minor technical caveat that applies in ordinary mathematics, and in particular, this paper). We illustrate this algorithm below in the discussion of §2.1.3 and in the proof of Lemma 2.3.

The principal results upon which we base the proof of Theorem 1.2 are formulated in the language of IST. We record their equivalent internal formulations in Appendix A. Those reformulations are clunkier, and we do not work with them in the body of the text; they are provided only for the sake of illustration.

§2.1.2. Axioms of IST

We now recall the axioms, referring again to (E. Nelson 1977) for details. In what follows, take “quantity” to be a synonym for “set,” given that we are working in ZFC.

• Transfer. Let \(A(x)\) be an internal statement involving a quantity \(x\) and possibly other fixed quantities, but no non-fixed quantities. Then \(A(x)\) is true for all \(x\) if and only if it is true for all fixed \(x\).

• Idealization. Let \(A(x,y)\) be an internal statement involving quantities \(x\), \(y\) and possibly other quantities. Then the following statements are equivalent.

  • For each fixed finite set \(x'\), there exists \(y\) so that, for each \(x \in x'\), the statement \(A(x,y)\) is true.

  • There exists \(y\) so that for each fixed \(x\), the statement \(A(x,y)\) is true.

• Standardization. Let \(A(z)\) be a statement involving the quantity \(z\). Let \(x\) be a fixed set. Then there is a fixed subset \(y \subseteq x\) so that for each fixed \(z \in x\), we have \(z \in y\) if and only if \(A(z)\) holds.

The idealization axiom simplifies when

• the variable \(x\) of the statement \(A(x,y)\) is restricted to some directed set \((X, \geqslant)\), and

• the statement \(A(x,y)\) is filtered in \(x\): if \(x_1 \geqslant x_2\), then \(A(x_1,y) \implies A(x_2,y)\).

In that case, it says that one can commute quantifiers: “for each fixed \(x\), there exists \(y\) so that \(A(x,y)\)” is equivalent to “there exists \(y\) so that for each fixed \(x\), we have \(A(x,y)\).”

A primary consequence of the standardization axiom is the following (see (E. Nelson 1977, Thm 1.3)). Suppose given fixed sets \(X\) and \(Y\) and a statement \(A(x,y)\), concerning elements \(x \in X\) and \(y \in Y\), such that for each fixed \(x \in X\), there is a fixed \(y \in Y\) for which \(A(x,y)\) holds. Then there is a fixed function \(\tilde{y} : X \rightarrow Y\) such that for all fixed \(x \in X\), we have \(A(x, \tilde{y}(x))\)

Each of the above axioms has a dual form obtained by swapping “for all” with “there exists.”

§2.1.3. Main result: reformulation

Recall that the main result to be proved is part \((b)\) of Theorem 1.2. We explain here the equivalence of that statement with the following one.

Recall the universal quantifier \(\forall\) (“for all”) and the existential quantifier \(\exists\) (“there exists”). Let us write \(\forall^*\) and \(\exists^*\) for the modified quantifiers obtained by restricting quantification to fixed elements (thus, e.g., \(\forall^* x\) means “for all \(x\), if \(x\) is fixed, then \(\dotsb\).”) The condition “\(t_j \asymp T\) for all \(j\)” expands to \[\forall j \, \exists^* C_0, c_0 \, : j \in \{1, \dotsc, n\} \implies C_0 > c_0 > 0 \text{ and } c_0 T \leqslant|t_j| \leqslant C_0 T.\] The conclusion is filtered with respect to \((C_0,c_0)\), so by idealization, the above is equivalent to \[\exists^* C_0, c_0 \, \forall j \, : j \in \{1, \dotsc, n\} \implies C_0 > c_0 > 0 \text{ and } c_0 T \leqslant|t_j| \leqslant C_0 T.\] Theorem 2.1 is thus equivalent to \[\label{eq:forall-t-reducing-nonstandard-reformulation}\tag{2.1} \forall T \, \forall^* n \, \forall \pi \, \forall^* \delta , C_0, c_0 \, \exists^* C_1 : A,\] where \(A\) denotes the internal statement “if \(T \geqslant 1\), \(n\) is a natural number, \(\pi\) is a unitary cuspidal automorphic representation of \({\mathop{\mathrm{GL}}}_n\) over \(\mathbb{Q}\), \(\delta < \delta_n ^\sharp\), \(C_0 > c_0 > 0\), \(c_0 T \leqslant|t_j| \leqslant C_0 T\) for all \(j\), and \(\delta < \delta_n^\sharp\), then \(|L(\pi,\tfrac{1}{2})| \leqslant C_1 C(\pi_{\mathop{\mathrm{fin}}})^{1/2} T^{(1-\delta) n/4}\).” By Boolean logic, \((2.1)\) is equivalent to \[\forall^* n, \delta , C_0, c_0 \, \forall T , \pi \, \exists^* C_1 \, : A.\] The internal statement \(A\) is filtered with respect to \(C_1\), so by idealization, we can commute its quantification with that for \((T,\pi)\). This yields the equivalent statement \[\label{eq:forall-n-delta}\tag{2.2} \forall^* n, \delta , C_0, c_0 \, \exists^* C_1 \, \forall T , \pi \, : A.\] By repeated application of transfer to the internal statement “\(\forall T, \pi \, : A\),” we obtain the further reformulation \[\forall n, \delta , C_0, c_0 \, \exists C_1 \, \forall T , \pi \, : A.\] The latter unwinds to the statement of part \((b)\) of Theorem 1.2.

In this way, we obtain the required equivalence with Theorem 2.1. By the conservation theorem, our task reduces to verifying the latter.

§2.1.4. Classes

As in (P. D. Nelson 2023, sec. 3.1.4), by a class, we mean a pair \((X,P)\), where \(X\) is a set and \(P\) is a boolean formula of IST that accepts an element \(x \in X\) as its argument. We note that if \(P\) is internal, then an axiom of ZFC tells us that there exists a subset of \(X\), denoted \(\{x \in X : P(x)\}\), that consists of precisely those elements for which \(P\) is true. On the other hand, ZFC says nothing about external formulas. For example, there is no subset of \(\mathbb{R}\) consisting of precisely those elements \(x\) for which \(x = \operatorname{O}(1)\). On the other hand, it makes perfect sense to form the pair \((\mathbb{R},P)\), where \(P(x)\) denotes the predicate “\(x = \operatorname{O}(1)\).” We refer to such a pair \((X,P)\) as “the class of all \(x \in X\) satisfying \(P(x)\).” For instance, we may speak of the class of all real numbers of the form \(\operatorname{O}(1)\). In the language of some axiomatizations (see, e.g., (E. Nelson 1977, sec. 3), (Kawai 1983) or (Hrbacek 1979, sec. 3)), “classes” correspond to “external subsets of interal sets,” but since we will be working with classes in a purely syntactic way, it is not necessary to adopt such additional formalism.

Classes do not appear explicitly in Part . They play a significant organizing role in Parts and , and a minor one in Part .

We may regard sets as classes (take for \(P\) the predicate “true”) and work with classes in much the same way we work with sets, except that we will never apply anything like the “power set” axiom to classes, e.g., by speaking of “the class of all subclasses of \((\dotsb)\).” An element \(x\) belongs to the class \((X,P)\) if \(x \in X\) and if \(P(x)\) is true. One class \((X,P)\) is contained in another class \((Y,Q)\) (and called, then, a “subclass”) if for every \(x \in X\) satisfying \(P(x)\), we have \(x \in Y\) and \(Q(x)\). For example, for \(T \geqslant 1\), the class of all real numbers of the form \(\operatorname{O}(T)\) is contained in the class of all real numbers of the form \(\operatorname{O}(T^2)\), say. This basic idea is routinely applied in analytic number theory papers, but since we will eventually be applying it in somewhat complicated ways, we have attempted to be precise.

Two classes are equal if they contain the same elements, or equivalently, if each is contained in the other. We may define intersections or unions of classes, class maps \(f : (X,P) \rightarrow (Y,Q)\) between classes (i.e., subclasses of the product class satisfying the map axioms), and so on.

We record a simple example of the sorts of classes we will work with. This example will not play a direct role in the paper, and is perhaps better treated without such formalism, but illustrates the basic idea.

Overspill

For our purposes, the overspill principle is a theorem of IST (E. Nelson 1977, Thm 1.1) which asserts that any subset of the natural numbers that contains every fixed natural number must contain some natural number \(n\) that is not fixed, hence for which \(n \ggg 1\). One verifies in the same way that any subset of the positive reals that contains every fixed positive real must contain some positive real of the form \(o(1)\). These conclusions would fail if we relaxed “subset” to “subclass” in the sense described above (consider, e.g., the class of natural numbers of the form \(\operatorname{O}(1)\)), so it is important to pay attention to what is actually a set rather than merely a class. We will use this principle primarily for “\(\varepsilon\) clean-up.” A sample application is given by the lemma below.

The notation \(A \ll T^{o(1)}\) (together with variants, such as \(A \ll B T^{o(1)}\)) will occur frequently. The direct meaning is that there is a fixed \(C \geqslant 0\) and an \(\varepsilon\lll 1\) such that \(|A| \leqslant C T^{\varepsilon}\). We will often make use of the following equivalent characterizations:

§2.1.6. Further notation and conventions

We conclude with some miscellaneity.

Let \(F\) be a fixed local field, with normalized absolute value \(|.| : F \rightarrow \mathbb{R}_{\geqslant 0}\). Let \(V\) be a fixed vector space (finite-dimensional) over \(F\). By a norm on \(V\), we mean a continuous function \(|.| : V \rightarrow \mathbb{R}_{>0}\) such that \(|v| = 0\) only if \(v = 0\), and \(|t v| = |t|\, |v|\) for all \((t,v) \in F \times V\). It is well-known that for any two fixed norms \(|.|_1\) and \(|.|_2\) on \(V\), we have \(|x|_1 \asymp |x|_2\) for all \(x \in V\). Given \(x,y \in F\) and a scalar \(t\) (either in \(F\) or complex), we write \(x = y + \operatorname{O}(t)\) to denote that \(|x-y| \ll |t|\) for some (equivalently, any) fixed norm \(|.|\) on \(V\). We analogously define the notations \(x = y + o(t)\), \(x \asymp y\), \(x \ll t\) and \(x \lll t\).

We often write \(T\) and \(\mathop{\mathrm{h}}\) for positive parameters with \(T \geqslant 1\) and \(\mathop{\mathrm{h}}\leqslant 1\); typically, \(T = 1/\mathop{\mathrm{h}}\) and \(T \ggg 1\), or equivalently, \(\mathop{\mathrm{h}}\lll 1\). We use notation such as \(A \ll B T^{-\infty}\) or \(A = \operatorname{O}(B T^{-\infty})\) as shorthand for the assertion that for each fixed \(m \geqslant 0\), we have \(A \ll B T^{-m}\); more explicitly, for each fixed \(m \geqslant 0\) there is a fixed \(C \geqslant 0\) so that \(|A| \leqslant C |B| T^{-m}\). We analogously define the notation \(A \ll B {\mathop{\mathrm{h}}}^\infty\). Similarly, given a subclass \(\mathcal{C}\) of a vector space, we write \(T^{-\infty} \mathcal{C}\) for the intersection over all fixed \(m \geqslant 0\) of the classes \(T^{-m} \mathcal{C}\). This notation will be used only in situations where that intersection is descending.

General notation

For a statement \(S\), we define \(1_S\) to be \(1\) if \(S\) is true and \(0\) otherwise. For a set \(X\), we write \(1_X\) for its characteristic function, thus \(1_X(x)\) is \(1\) or \(0\) according as \(x \in X\) or \(x \notin X\); equivalently, \(1_X(x) := 1_{x \in X}\).

We interchangeably write \(\# X\) and \(|X|\) for the cardinality of a set \(X\).

§2.3. General preliminaries

Fields

Let \(F\) be a local or global number field (i.e., a local or global field of characteristic zero, the only case considered here).

In the global case, we write \(\mathbb{A}\) for the adele ring of \(F\) and \(\mathbb{A}_f\) for the subring of finite adeles. We use the letter \(\mathfrak{p}\) to denote places of \(F\), possibly archimedean, and denote by \(F_\mathfrak{p}\) the associated completion. We set \(F_\infty := F \otimes_{\mathbb{Q}} \mathbb{R} = \prod_{\mathfrak{p} | \infty} F_\mathfrak{p}\), so \(\mathbb{A} = F_\infty \times \mathbb{A}_f\). When working over \(F = \mathbb{Q}\), we write \(p\) for a finite prime.

In the local case, we write \(|.| : F \rightarrow \mathbb{R}_{\geqslant 0}\) for the normalized absolute value; in the global case, \(|.| : \mathbb{A}^\times / F^\times \rightarrow \mathbb{R}^\times_+\) denotes the idelic absolute value.

In the local non-archimedean case, we denote by \(\mathfrak{o}\) the ring of integers, by \(\mathfrak{p}\) the maximal ideal, by \(\varpi \in \mathfrak{p}\) a generator, and \(q := \# \mathfrak{o}/\mathfrak{p}\).

Reductive groups

Let \(G\) be a split connected reductive group over \(F\). The examples relevant for this paper are when \(G\) is a finite product of general linear groups (e.g., arising as Levi factors of a given general linear group), but it will be convenient at times to work more generally – for instance, some points can be explained most clearly by passing temporarily to the simpler group \(G = {\mathop{\mathrm{SL}}}_2\).

In the local case, we identify \(G\) with its set of \(F\)-points, and abbreviate \(G := G(F)\). In the global case, we denote by \([G] := G(F) \backslash G(\mathbb{A})\) the associated adelic quotient.

In the local case, we write \(\mathop{\mathrm{Lie}}(G)\) for the Lie algebra. In the local archimedean case, we write \(\mathfrak{U}(G)\) for the universal enveloping algebra of the complexification of \(\mathop{\mathrm{Lie}}(G)\), and \(\mathfrak{Z}(G)\) for the center of \(\mathfrak{U}(G)\).

We occasionally take for \(G\) a split connected reductive group over \(\mathbb{Z}\) or over the ring of integers \(\mathfrak{o}\) in a non-archimedean local field \(F\); in such cases, by base restriction, we may understand \(G\) as being defined also over \(\mathbb{Q}\) or \(F\), respectively.

We always assume that \(G\) comes equipped with the following additional data:

• A closed linear embedding \(G \hookrightarrow {\mathop{\mathrm{SL}}}_r\) for some \(r\).

• A split maximal torus \(A\). We assume given a basis for its group of rational characters, hence an identification of \(A\) with a product of copies of \({\mathop{\mathrm{GL}}}_1(F)\).

• A Borel subgroup \(B\) of \(G\), containing \(A\), with unipotent radical denoted \(N\), so that \(B = A N\).

• Another Borel subgroup \(Q = A U\) containing \(A\), which may or may not coincide with \(B\). (We will use \(Q\) to define induced representations, and \(N\) to define Whittaker functions – it will be computationally convenient to keep the choices of \(Q\) and \(B\) decoupled.)

• In the local (resp. global) case, a maximal compact subgroup \(K\) of \(G\) (resp. \(K = \prod_\mathfrak{p} K_\mathfrak{p}\) of \(G(\mathbb{A})\)) in good position relative to \(B\) (resp. \(B(\mathbb{A})\)), so that \(A \cap K\) (resp. \(A(\mathbb{A}) \cap K\)) is a maximal compact subgroup of \(A\) (resp. \(A(\mathbb{A})\)) and the Iwasawa decomposition \(G = B K\) (resp. \(G(\mathbb{A}) = B(\mathbb{A}) K\)) holds (see (Mœglin and Waldspurger 1995, 113:I.1.4), (Getz and Hahn 2021, Thm A.1.1)).

• In the local archimedean case, bases \(\mathcal{B}(A)\) and \(\mathcal{B}(G)\) for \(\mathop{\mathrm{Lie}}(A)\) and \(\mathop{\mathrm{Lie}}(G)\), respectively.

When \(G\) is fixed, we may and shall assume that all of the above data is fixed, too. We denote by \(Z_G\) the center of \(G\) and by \(W_G\) the Weyl group for \((G,A)\). When \(G\) is clear from context, we abbreviate \(Z := Z_G\) and \(W := W_G\).

When working over \(\mathbb{R}\), \(\mathbb{Q}\) or \(\mathbb{Z}\), we write \(A(\mathbb{R})^0\) for the topologically connected component of the identity element of \(A(\mathbb{R})\).

§2.3.3. General linear groups

By a general linear group \(G\), we mean \(G = {\mathop{\mathrm{GL}}}_n\) for some \(n \in \mathbb{Z}_{\geqslant 0}\). We take the closed embedding \(G \hookrightarrow {\mathop{\mathrm{SL}}}_{2 n}\) given by \(g \mapsto \mathop{\mathrm{diag}}(g, g^{-% {\mathpalette\@transpose{}}% })\). We assume that \(A\) is the diagonal subgroup and \(B\) is the upper-triangular Borel subgroup. We will take \(U\) to be either \(N\) or its opposite, according to convenience. It will often be convenient to use the letter “\(n\)” for other purposes (e.g., to denote elements of the group \(N\)) or to be flexible in the indexing of our general linear groups (e.g., writing \(G = {\mathop{\mathrm{GL}}}_{n+1}\) or \(G = {\mathop{\mathrm{GL}}}_{n-1}\), according to convenience). To that end, we will make use of the formulas \[\mathop{\mathrm{rank}}(G) = n, \quad \dim(G) = n^2, \quad \dim(N) = n(n+1)/2,\] and so on (with ranks and dimensions computed over \(F\), unless otherwise indicated).

By a general linear pair \((G,H)\) over \(F\), we mean a pair of \(F\)-groups of the form \(({\mathop{\mathrm{GL}}}_{n+1},{\mathop{\mathrm{GL}}}_n)\) for some natural number \(n \in \mathbb{Z}_{\geqslant 0}\). We regard \(H\) as being included in \(G\) as the upper left \(n \times n\) block. We denote by \(A_H, B_H, N_H\) the subgroups of \(H\) defined by intersecting the corresponding subgroups of \(G\) with \(H\). We denote by \(Z_H\) the center of \(H\). We will take for \(Q_H\) a Borel subgroup of \(H\) containing \(A_H\), chosen according to convenience, and denote by \(U_H\) its unipotent radical. We define \(\mathfrak{a}_H^*\) and \(\mathfrak{a}_{H,\mathbb{C}}^*\) in terms of \(A_H\) as in \((2.5)\).

Norms

In the local case, we define a norm \(\|.\|_G\) on \(G\), denoted simply \(\|.\|\) when \(G\) is clear from context, by pulling back the operator norm on the special linear group under the given embedding \(G \hookrightarrow {\mathop{\mathrm{SL}}}_r\). In the global case, we define the norm \(\|.\|_{G(\mathbb{A})}\) on \(G(\mathbb{A})\) to be the product \(\|g\|_{G(\mathbb{A})} := \prod_{\mathfrak{p}} \|g_\mathfrak{p} \|_{G(F_\mathfrak{p})}\) of the corresponding local norms defined using the induced embeddings \(G(F_\mathfrak{p}) \hookrightarrow {\mathop{\mathrm{SL}}}_r(F_\mathfrak{p})\).

§2.3.5. Haar measures

Since this paper is ultimately concerned with estimates, the choice of Haar measure is unimportant, provided that we are consistent. We record a standard choice sufficient for our purposes.

In local and global cases, we equip \(K\) with the Haar probability measure.

In the local case, we assume given Haar measures on \(U\), \(N\), \(A\), \(Z\). In the non-archimedean case, we assume that maximal compact subgroups of each of these groups have volume one. In the archimedean case, we assume only that the given Haar measures are fixed when \(G\) is fixed. We then equip \(G\) with the Haar measure defined using the Iwasawa decomposition: \[\label{eq:int-_g-in-3}\tag{2.3} \int _{g \in G} f(g) \, d g = \int _{u \in U} \int _{a \in A} \int _{k \in K} f(u a k) \, d k \, \frac{d a}{\delta_U(a)} \, d u.\] Equivalently, the multiplication map \(A \times U \times K \rightarrow G\) is measure-preserving. We obtain a quotient measure on \(U \backslash G\), given by \[\label{eq:int-_g-in-U-bakslash-G}\tag{2.4} \int _{g \in U \backslash G} f(g) \, d g = \int _{a \in A} \int _{k \in K} f(a k) \, d k \, \frac{d a}{\delta_U(a)}.\] These last identities remain valid with \(U\) replaced by \(N\).

Sometimes, it will be convenient instead to suppose that the multiplication map \(A \times U \times N \rightarrow G\) is measure-preserving. We will indicate when this is the case.

In the global case, we choose factorizable Haar measures on \(U(\mathbb{A})\), \(N(\mathbb{A})\), \(A(\mathbb{A})\), \(Z(\mathbb{A})\) and \(K\) so that \([U]\) and \(K\) have volume one. We equip \(G(\mathbb{A})\) with the Haar measure defined by analogy to \((2.3)\). We fix compatible factorizations of the Haar measures on each of these groups, so that for each \(\mathfrak{p}\), the analogue of \((2.3)\) for \(G(F_\mathfrak{p})\) is valid and the Haar measures on the groups \(U(F_\mathfrak{p})\), \(N(F_\mathfrak{p})\), \(A(F_\mathfrak{p})\), \(Z(F_\mathfrak{p})\) and \(K_\mathfrak{p}\) satisfy the desiderata of the previous paragraph. We equip the discrete groups consisting of rational points with the counting measure, adelic quotients \([G]\), \([A]\), \([Z]\), etc., with the quotient measure.

Characters

Given a locally compact abelian group \(V\), we write \(V^\wedge\) for its Pontryagin dual. This notation applies in particular when \(F\) is local and \(V\) is a finite-dimensional vector space over \(F\), in which case \(V^\wedge\) may be regarded as a vector space over \(F\) of the same dimension. It applies also in the local (resp. global) case to \(A\) (resp. \([A]\)).

We denote \(X(A)\) the group of rational characters \(\chi : A \rightarrow {\mathop{\mathrm{GL}}}_1\). We set \[\label{eq:frak-a-star-X-A}\tag{2.5} \mathfrak{a}^* := X(A) \otimes_{\mathbb{Z}} \mathbb{R} \hookrightarrow \mathfrak{a}_{\mathbb{C}}^* := X(A) \otimes_{\mathbb{Z}} \mathbb{C}.\] We denote by \(\mathfrak{a} \hookrightarrow \mathfrak{a}_{\mathbb{C}}\) the corresponding dual spaces. For \(s \in \mathfrak{a}_{\mathbb{C}}^*\) and \(t \in \mathfrak{a}_{\mathbb{C}}\), we denote by \(t(s) = \langle s, t \rangle = \langle t,s \rangle \in \mathbb{C}\) the natural pairing.

In the local (resp. global) case, we denote by \(\mathfrak{X}(A)\) (resp. \(\mathfrak{X}([A])\)) the group of continuous complex-valued characters \(\chi\) of \(A\) (resp. \([A]\)). It is naturally a complex manifold, locally isomorphic to \(\mathfrak{a}^*_{\mathbb{C}}\). The subgroup of unitary characters is \(A^\wedge \leqslant\mathfrak{X}(A)\) (resp. \([A]^\wedge \leqslant\mathfrak{X}([A])\)).

Each \(s \in \mathfrak{a}_{\mathbb{C}}^*\) naturally defines an element \(|.|^s : a \mapsto |a|^s\) of \(\mathfrak{X}(A)\) (resp. \(\mathfrak{X}([A])\)). When \(F = \mathbb{R}\) and \(a \in A(\mathbb{R})^0\), we abbreviate \(a^s := |a|^s\). We recall that \(\chi \in \mathfrak{X}(A)\) (resp. \(\mathfrak{X}([A])\)) is unramified if it is invariant under the maximal compact subgroup of \(A\) (resp. \([A]\)); in that case, \(\chi\) is of the form \(|.|^{s}\) for some \(s \in \mathfrak{a}_{\mathbb{C}}^*\) (in the local non-archimedean case, non-uniquely). We denote by \(\mathfrak{X}^e(A)\) (resp. \(\mathfrak{X}^e([A])\)) the subgroup consisting of unramified characters. (The superscript “\(e\)” is motivated by the archimedean case, where it signifies an “evenness” condition.)

For each element \(\chi\) of \(\mathfrak{X}(A)\) (resp. \(\mathfrak{X}([A])\)), there is a unique element \(\Re(\chi) \in \mathfrak{a}^*\), called the real part of \(\chi\), for which \(|\chi| = |.|^{\sigma}\). Thus \(A^\wedge\) (resp. \([A]^\wedge\)) consists of all \(\chi\) with \(\Re(\chi) = 0\).

We equip \(A^\wedge\) (resp. \([A]^\wedge\)) with the Haar measure \(d \chi\) dual to the Haar measure \(d a\) on \(A\) (resp. \([A]\)). For each \(\sigma \in \mathfrak{a}^*\), the “vertical” subset consisting of \(\chi \in \mathfrak{X}(A)\) (resp. \(\mathfrak{X}([A])\)) with \(\Re(\chi) = \sigma\) is a coset of \(A^\wedge\) (resp. \([A]^\wedge\)); we equip it with the Haar measure \(d \chi\) obtained by transporting that on the latter group. For a function \(f\) on \(\mathfrak{X}^e(A)\) (resp. \(\mathfrak{X}^e([A])\)), we denote by \[\label{eq:int-_sigma-fs}\tag{2.6} \int _{(\sigma)} f(s) \, d \mu_A(s)\] the integral of \(f(\chi) \, d \chi\) over \(\chi\) in \(\mathfrak{X}^e(A)\) (resp. \(\mathfrak{X}^e([A])\)) of real part \(\sigma\).

We denote by \(R\) the set of roots for \((G,A)\), by \(R_U^+\) (resp. \(R_N^+\)) the subset of positive roots relative to \(U\) (resp. \(N\)), and \(\Delta_U\) (resp. \(\Delta_N\)) the associated set of simple roots. For \(\alpha \in R\), we denote by \(\alpha^\vee \in \mathfrak{a}^*\) the corresponding coroot.

We say that \(\sigma \in \mathfrak{a}^*\) is dominant if it is dominant in the sense defined by \(U\), i.e., if for each \(\alpha \in R_U^+\), we have \(\alpha^\vee(s) \geqslant 0\). We say that \(\sigma\) is strictly dominant if the stronger condition \(\alpha^\vee(s) > 0\) holds for all such \(\alpha\). Given \(\sigma_1, \sigma_2 \in \mathfrak{a}^*\), we write \[\sigma_1 \prec \sigma_2 \quad \text{ or } \quad \sigma_2 \succ \sigma_1\] to denote that the difference \(\sigma_2 - \sigma_1\) is strictly dominant.

(When we wish to speak of dominance relative to \(N\), we write more verbosely “\(N\)-dominant,” but this will rarely happen except when \(N = U\).)

We denote by \(\delta_U, \delta_N \in \mathfrak{X}(A)\) (resp. \(\mathfrak{X}([A])\)) the modulus characters attached to \(Q = A U\) and \(B = A N\), respectively. We write \(\rho_U, \rho_N \in \mathfrak{a}^*\) for the elements characterized by the identities \(\delta_U = |.|^{2 \rho_U}\), \(\delta_N = |.|^{2 \rho_N}\). Then \(\rho_U = (1/2) \sum_{\alpha \in R_U^+} \alpha\), and similarly for \(\rho_N\). We set \(\rho_U^\vee := (1/2) \sum_{\alpha \in R_U^+} \alpha^\vee\), and define \(\rho_N^\vee\) analogously.

Quotients and actions

Let \(P\) be a parabolic subgroup of \(G\) that contains \(A\). We may write \(P = M_P U_P\), where \(U_P\) is the unipotent radical of \(P\) and \(M_P\) is the unique Levi factor that contains \(A\). In the local case, we may form the quotient \(U_P \backslash G\). In the global case, we define \[_P := P(F) U_P(\mathbb{A}) \backslash G(\mathbb{A}).\] We will be almost exclusively concerned with the special cases \(P = G\) and \(P = Q\). We note that, in the global case, \([G]_G = [G]\) and \([G]_Q = Q(F) U(\mathbb{A}) \backslash G(\mathbb{A}) = A(F) U(\mathbb{A}) \backslash G(\mathbb{A})\).

In the local (resp. global) case, given a complex-valued function \(f\) on \(U_P \backslash G\) (resp. \([G]_P\)) and \(g \in G\), we denote by \(R(g) f\) the function on the same space obtained via right translation: \[R(g) f(x) := f(x g).\] The spaces \(U \backslash G\) (resp. \([G]_Q\)) admit natural left action by \(A\) (resp. \([A]\)), and we denote by \(L\) the associated normalized action on function spaces, namely, for \(a \in A\) (resp. \([A]\)), \[\label{eq:normalized-left-translation}\tag{2.7} L(a) f(x) := \delta_U^{-1/2}(a) f(a x).\] In the local archimedean case, we denote also simply by \(R\) (resp. \(L\)) the induced actions of \(\mathop{\mathrm{Lie}}(G)\) and \(\mathfrak{U}(G)\) (resp. \(\mathop{\mathrm{Lie}}(A)\) and \(\mathfrak{U}(A)\)) on the corresponding space of smooth functions. In the global setting, we use similar notation for the differential actions induced by \(G(F_\infty)\) and \(A(F_\infty)\).

§2.6. Schwartz spaces

We refer to Casselman (Casselman 1989) and the thesis of Moore (Moore 2018) for background, although we note that in non-archimedean aspects, we do not adopt the same definitions as the latter reference.

Continue to let \(P = M_P U_P\) be a parabolic containing \(A\).

Recall that in the local (resp. global) case, we have defined a norm \(\|.\|\) on \(G\) (resp. \(G(\mathbb{A})\)). In the local case, we define a norm on \(U_P \backslash G\) by \[\|g\|_{U_P \backslash G} := \inf_{u \in U_P} \|u g\|.\] In the global case, we define a norm on \([G]_P\), hence on \([G]_Q\) and \([G]\), by \[\label{eq:g_g-:=-inf_gamma}\tag{2.8} \|g\|_{[G]_P} := \inf_{h \in P(F) U_P(\mathbb{A})} \|h g\|,\]

Suppose for the moment that \(F\) is fixed (or that \(F\) is non-archimedean local and “everything is unramified”). In the local case, for \(g = u a k\) with \((u,a,k) \in U \times A \times K\), we then have \(\|g\|_{U \backslash G} \asymp \|a\|\). In the global case, we have under analogous hypotheses \[\|g\| _{[G]_Q} \asymp \|a\|_{[A]} := \inf_{h \in A(F)} \|h a\|.\] (The proofs of these estimates may be reduced, via the given linear embedding of \(G\), to the case \(G = {\mathop{\mathrm{GL}}}_n\), where they may be verified directly.)

Suppose for the moment that \(F\) is local. We define the Schwartz space \(\mathcal{S}(U_P \backslash G)\) as follows. In the non-archimedean case, it consists of locally constant compactly-supported functions \(f : U_P \backslash G \rightarrow \mathbb{C}\). In the archimedean case, it consists of smooth functions satisfying, for all \(x \in \mathfrak{U}(G)\) and \(m \in \mathbb{Z}_{\geqslant 0}\), \[\sup_{g \in U_P \backslash G} \|g\|_{U_P \backslash G}^m | R(x) f(a) | < \infty.\] In particular, by specializing to \(P = G\) and \(P = Q\), we obtain definitions of \(\mathcal{S}(G)\) and \(\mathcal{S}(U \backslash G)\).

In the global case, we define the Schwartz space \(\mathcal{S}([G]_P)\) to be the union, over all open subgroups \(J\) of \(G(\mathbb{A}_f)\), of the space of smooth functions \(f : [G]_P \rightarrow \mathbb{C}\) that are right-invariant under \(J\) and satisfy, for each \(m \geqslant 0\) and \(x \in \mathfrak{U}(G(F_\infty))\), \[\sup_{g \in [G]_P} \|g\|_{[G]_P}^m |R(x) f(g)| < \infty.\] In particular, we obtain definitions of \(\mathcal{S}([G])\) and \(\mathcal{S}([G]_Q)\).

In the local (resp. global) case, the space \(\mathcal{S}(U \backslash G)\) (resp. \(\mathcal{S}([G]_Q)\)) admits a natural left action by \(A\) (resp. \([A]\)). We denote by \(\mathcal{S}^e(U \backslash G)\) (resp. \(\mathcal{S}^e([G]_Q)\)) the subspaces consisting of elements that are left-unramified, i.e., left-invariant under the maximal compact subgroup of \(A\) (resp. \([A]\)).

All of the above definitions apply in particular to \(G = A\). In particular, in the local case, we may define the Schwartz space \(\mathcal{S}(A)\) and its subspace \(\mathcal{S}^e(A)\) consisting of elements invariant under the maximal compact subgroup.

In the global case, we define the space \(\mathcal{T}([G]_P)\) of functions of uniform moderate growth to be the union, over \(d \in \mathbb{Z}_{\geqslant 0}\) and open subgroups \(J\) of \(G(\mathbb{A}_f)\), of the space of smooth functions \(f : [G]_P \rightarrow \mathbb{C}\) that are right-invariant by \(J\) and satisfy, for each \(x \in \mathfrak{U}(G(F_\infty))\), \[\sup_{g \in [G]_P} \|g\|^{-d}_{[G]_P} | R(x) \varphi(g)| < \infty.\]

Each of the function spaces just defined is naturally a Fréchet space in the local archimedean case and an inductive limit of Fréchet spaces otherwise.

§2.7. Principal series representations

In the local (resp. global) case, for \(\chi \in \mathfrak{X}(A)\) (resp. \(\mathfrak{X}([A])\)), we denote by \[\mathcal{I}(\chi) = {\mathop{\mathrm{Ind}}}_Q^G(\chi) \subseteq C^\infty(U \backslash G) \quad \text{ (resp. $\mathcal{I}(\chi) = {\mathop{\mathrm{Ind}}}_{Q(\mathbb{A})}^{G(\mathbb{A})}(\chi) \subseteq C^\infty([G]_Q)$) }\] the corresponding normalized principal series representation, consisting of smooth complex-valued functions \(f\) on \(G\) (resp. \(G(\mathbb{A})\)) satisfying \(f(u a g) = \delta_U^{1/2}(a) \chi(a) f(g)\) for all \((u,a,g)\) in \(U \times A \times G\) (resp. \(U(\mathbb{A}) \times A(\mathbb{A}) \times G(\mathbb{A})\)). Here “smooth” carries the following convention:

• In the archimedean case, it means that all derivatives (left-invariant, say) exist.

• In the non-archimedean (resp. global) case, it means that there is an open subgroup \(J\) of \(G\) (resp. of \(G(\mathbb{A}_f)\)) under which \(f\) is right-invariant.

In the global case, for a prime \(\mathfrak{p}\) of \(F\), we denote by \(\chi_\mathfrak{p}\) the local component of \(\chi\) at \(\mathfrak{p}\) and by \(I_\mathfrak{p}(\chi_\mathfrak{p})\) (or simply \(\mathcal{I}_\mathfrak{p}(\chi)\)) the corresponding principal series representation of \(G(F_\mathfrak{p})\); then \(\mathcal{I}(\chi)\) identifies naturally with the restricted tensor product of the \(\mathcal{I}_\mathfrak{p}(\chi_\mathfrak{p})\), restricted with respect to the spherical elements taking the value \(1\) at the identity.

An invariant pairing between \(\mathcal{I}(\chi)\) and \(\mathcal{I}(\chi^{-1})\) is given by \((f_1,f_2) \mapsto \int_{K} f_1 f_2\). In the local case, up to a constant factor depending upon measure normalizations, this coincides with the pairing given by \(\int_N f_1 f_2\).

We denote by \(\|.\|\) the norm on \(\mathcal{I}(\chi)\) given by \(\|f\|^2 := \int _{K} |f|^2\), which is invariant when \(\Re(\chi) = 0\).

When \(\chi\) is unramified, i.e., \(\chi = |.|^s\) for some \(s \in \mathfrak{a}_{\mathbb{C}}^*\), we often write simply \[\mathcal{I}(s) := \mathcal{I}(|.|^s)\] in both local and global cases. (This notational abbreviation has the potential to introduce confusion – should \(\mathcal{I}(1)\) denote the induction of the trivial character \(1 = |.|^0\), or that of \(|.|^1\)? – so we will be careful when using it.)

§2.8. Mellin analysis and Paley–Wiener theory

Warm-up

We briefly recall the case of the real line. We say that a holomorphic function \(\tilde{f} : \mathbb{C} \rightarrow \mathbb{C}\) has rapid vertical decay if for each finite interval \([a,b] \subset \mathbb{R}\) and \(m \geqslant 0\), the quantity \[\label{eq:sup_s-in-mathbbc}\tag{2.9} \sup_{s \in \mathbb{C} : \Re(s) \in [a,b]} (1 + |s|)^m |\tilde{f}|(s)\] is finite. The Schwartz space \(\mathcal{S}(\mathbb{R}^\times_+)\) is defined to consist of all smooth \(f : \mathbb{R}^\times_+ \rightarrow \mathbb{C}\) with the property that for each \(m, k \geqslant 0\), we have \[\label{eq:sup_y-in-mathbbrt}\tag{2.10} \sup_{y \in \mathbb{R}^\times_+} (y + y^{-1})^m \left\lvert \left( y \frac{d}{d y} \right)^k f(y) \right\rvert < \infty.\] For each \(f \in \mathcal{S}(\mathbb{R}^\times_+)\), the Mellin transform \(\tilde{f}(s) := \int _{y \in \mathbb{R}^\times_+} f(y) y^{-s} \, d^\times y\), where \(d^\times y := \frac{d y}{y}\), converges absolutely, and defines a holomorphic function \(\tilde{f}\) of rapid vertical decay; conversely, every such function \(\tilde{f}\) arises in this way. Moreover, the topology on \(\mathcal{S}(\mathbb{R}^\times_+)\), as described by the seminorms \((2.10)\), is equivalent to the topology described by the seminorms \((2.9)\). Finally, we have \[\int _{y \in \mathbb{R}^\times_+} |f(y)|^2 \, d^\times y = \int _{\Re(s) = 0} \left\lvert \tilde{f}(s) \right\rvert^2 \, \frac{d s}{2 \pi i }.\]

The above assertions summarize parts of classical Paley–Wiener theory, expressed in terms of the Mellin transform. The proofs consist of elementary complex analysis. A similar theory relates decay properties of functions \(f : \mathbb{Z} \rightarrow \mathbb{C}\) to the regularity of the associated Laurent series \(\tilde{f}(z) = \sum_{n \in \mathbb{Z}} f(n) z^n\).

We temporarily drop the general notation of §2. In §2.8.2 and §2.8.3, we consider groups \(A\) acting on spaces \(X\) with the following properties:

• \(A\) is isomorphic \(\mathbb{R}^{n_1} \times \mathbb{Z}^{n_2} \times \Gamma\) for some \(n_1, n_2 \geqslant 0\) and some profinite group \(\Gamma\).

• There is a compact group \(K\) and a surjective \(A\)-equivariant homomorphism \(A \times K \rightarrow X\) with compact kernel.

In each case, we describe a form of Paley–Wiener theory for the action of \(A\) on certain function spaces on \(X\). The proofs remain elementary. Analogous results appear throughout (Mœglin and Waldspurger 1995, 113:II.1).

§2.8.2. Tori

We now return to the general thread of §2. When \(F\) is archimedean, we use the given linear embedding of \(G\) to equip the real Lie group \(\mathop{\mathrm{Lie}}(A)\), hence its real dual \(\mathop{\mathrm{Lie}}(A)^*\), with a \(W\)-invariant norm \(\|.\|\). We denote by \(d \chi \in \mathop{\mathrm{Lie}}(A)^*\) the differential at the identity of a character \(\chi \in \mathfrak{X}(A)\), and define the analytic conductor \(C(\chi) := 3 + \|d \chi\|\). (It is easy to see that this definition is comparable to the “standard” one, obtained by identifying \(A\) with a product of copies of \({\mathop{\mathrm{GL}}}_1(F)\), hence \(\chi\) with a tuple of characters \(\chi_j\) of \(F^\times\), and multiplying together the analogous conductors of the \(\chi_j\) as defined in §(Philippe Michel and Venkatesh 2010, sec. 3.1.8).)

We consider here the local case, which is all that we require. We say that a complex-valued holomorphic function \(\tilde{f}\) on \(\mathfrak{X}(A)\) has rapid vertical decay if

1. in the non-archimedean case, there is an open subgroup \(J\) of \(A\) so that \(\tilde{f}(\chi) = 0\) unless \(\chi|_J\) is trivial, and

2. in the archimedean case, for each compact subset \(\mathcal{D} \subseteq \mathfrak{a}^*\) and \(m \geqslant 0\), the supremum \[\label{eq:sup_chi-in-mathfr}\tag{2.11} \sup_{ \substack{ \chi \in \mathfrak{X}([A]) : \\ \Re(\chi) \in \mathcal{D} } } C(\chi)^m |\tilde{f}(\chi)|\] is finite.

For \(f \in \mathcal{S}(A)\), the Mellin transform \(\tilde{f}(\chi) := \int _{a \in A} f(\chi) \chi^{-1}(a) \, d a\) converges absolutely, and defines a holomorphic function \(\tilde{f} : \mathfrak{X}(A) \rightarrow \mathbb{C}\) of rapid decay; conversely, every such function \(\tilde{f}\) arises in this way. In the archimedean case, the topology on \(\mathcal{S}(A)\) is the same as that defined by the seminorms \((2.11)\).

We have \(f \in \mathcal{S}^e(A)\) if and only if \(\tilde{f}\) is supported on \(\mathfrak{X}^e(A)\). In that case, \(\tilde{f}\) is described entirely by the numbers \[\tilde{f}(s) := \tilde{f}(|.|^s).\] (We will be careful in using such notation for the same reasons as indicated at the end of §2.7.)

§2.8.3. Affine quotients

In the local (resp. global) case, let \(E\) be an open subset of \(\mathfrak{X}(A)\) (resp. \(\mathfrak{X}([A])\)). By a holomorphic family \(\{f[\chi]\}_{\chi \in E}\) of vectors \(f[\chi] \in \mathcal{I}(\chi)\), we mean a family of such vectors that arises as the pointwise specialization \(f[\chi](g) := f(\chi,g)\) of some complex-valued function \(f\) on \(E \times U \backslash G\) (resp. \(E \times [G]_Q\)) that is holomorphic in the first variable and smooth in the second variable, where “smooth” follows the same convention as in §2.7, with the subgroup \(J\) taken independent of the first argument. This definition applies in particular when \(E\) is the full character group \(\mathfrak{X}(A)\) (resp. \(\mathfrak{X}([A])\)), in which case we write simply “holomorphic family \(\{f[\chi]\}\).”

We say that a holomorphic family \(\{f[\chi]\}\) has rapid vertical decay if

• in the non-archimedean local case, there is an open subgroup \(J\) of \(A\) so that \(f[\chi] = 0\) unless \(\chi|_J\) is trivial,

• in the archimedean local case, for each compact subset \(\mathcal{D} \subseteq \mathfrak{a}^*\), \(m \geqslant 0\), and \(x \in \mathfrak{U}(G)\), the supremum \[\label{eq:sup_chi-in-mathfr-1}\tag{2.12} \sup_{ \substack{ \chi \in \mathfrak{X}(A) : \\ \Re(\chi) \in \mathcal{D} } } C(\chi_\infty)^m \sup_{k \in K} |R(x) f[\chi](k)|\] is finite, and

• in the global case, there is an open subgroup \(J\) of \(A(\mathbb{A}_f)\) so that \(f[\chi] = 0\) unless \(\chi|_J\) is trivial, and for each compact subset \(\mathcal{D} \subseteq \mathfrak{a}^*\), \(m \geqslant 0\) and \(x \in \mathfrak{U}(G(F_\infty))\), the supremum \[\label{eq:sup_-substack-chi}\tag{2.13} \sup_{ \substack{ \chi \in \mathfrak{X}([A]) : \\ \Re(\chi) \in \mathcal{D} } } C(\chi)^m \sup_{k \in K} |R(x) f[\chi](k)|\] is finite.

We note that in the archimedean local and global cases, replacing \(\sup_{k \in K} |R(x) f[\chi](k)|\) with \(\|R(x) f[\chi] \|\) yields the same definition, thanks to the Sobolev lemma for \(K\). We note also that the vanishing conditions imposed on \(f[\chi]\) in the non-archimedean local and global cases can be seen to follow from our definition of “holomorphic family”; we have stated it, redundantly, for emphasis.

In the local (resp. global) case, given \(f \in \mathcal{S}(U \backslash G)\) (resp. \(\mathcal{S}([G]_Q)\)), we define the Mellin components \(f[\chi] \in \mathcal{I}(\chi)\) by the absolutely convergent integrals \[f[\chi](g) := \int \delta_U^{1/2}(a) \chi(a) f(a^{-1} g) \, d a\] taken over \(a \in A\) (resp. \([A]\)). These integrals define a holomorphic family of vectors, of rapid vertical decay; conversely, every such family arises in this way. In the archimedean local (resp. global) case, the topology on \(\mathcal{S}(U \backslash G)\) (resp. \(\mathcal{S}([G]_Q)\)) is described by the family of seminorms \((2.12)\) (resp. \((2.13)\)).

We have for each \(\sigma \in \mathfrak{a}^*\) the Mellin decomposition \[\label{eq:f-=-int}\tag{2.14} f = \int _{(\sigma)} f[\chi] \, d \chi.\] When \(\sigma = 0\), this extends to a decomposition of \(L^2(U \backslash G)\) (resp. \(L^2([G]_Q)\)) as the direct integral of the \(\mathcal{I}(\chi)\) with \(\Re(\chi) = 0\) (cf. (Gelfand, Graev, and Pyatetskii-Shapiro 2016, sec. 3.4.9, §3.5.2)). In particular, for \(f_1, f_2 \in \mathcal{S}(U \backslash G)\) (resp. \(\mathcal{S}([G]_Q)\)), we have the Parseval formula \[\label{eq:leftlangle-f_1-f_2}\tag{2.15} \left\langle f_1, f_2 \right\rangle = \int _{(0)} \left\langle f_1[\chi], f_2[\chi] \right\rangle \, d \chi.\]

If \(f \in \mathcal{S}^e(U \backslash G)\) (resp. \(\mathcal{S}^e([G]_Q)\)), then \(f[\chi] = 0\) unless \(\chi\) is unramified, and so \(f\) is again described by the vectors \[f[s] := f[|.|^s] \in \mathcal{I}(s) := \mathcal{I} (|.|^s).\]

§2.9. Sobolev norms

Here we restrict to the local archimedean case. The topology on \(\mathcal{S}^e(U \backslash G)\) is described by the following family of seminorms, indexed by compact subsets \(\mathcal{D} \subseteq \mathfrak{a}^*\) and \(\ell \in \mathbb{Z}_{\geqslant 0}\): \[\label{eq:nu_mathcald-ellf-:=}\tag{2.16} \nu_{\mathcal{D},\ell}(f) := \sup _{ \substack{ s \in \mathfrak{a}_{\mathbb{C}}^* : \\ \Re(s) \in \mathcal{D} } } \sum_{m=0}^{\ell} \sum_{x_1,\dotsc,x_m \in \mathcal{B}(G)} (1 + |s|)^{\ell} \|R(x_1 \dotsb x_m) f[s] \|.\] Here \(\|.\|\) denotes the norm on \(\mathcal{I}(s)\), induced as in §2.7 from \(L^2(K)\). For each \(T \geqslant 1\), we defined a rescaled variant of such seminorms: \[\nu_{\mathcal{D},\ell,T}(f) := \sup _{ \substack{ s \in \mathfrak{a}_{\mathbb{C}}^* : \\ \Re(s) \in \mathcal{D} } } \sum_{m=0}^{\ell} \sum_{x_1,\dotsc,x_m \in \mathcal{B}(G)} (1 + |s|)^{\ell} \frac{\|R(x_1 \dotsb x_m) f[s] \|}{T ^{\left\langle \rho_U^\vee , \Re(s) \right\rangle + m}}.\] The informal idea here is that \(\nu_{\mathcal{D},\ell,T}(f) = \operatorname{O}(1)\) provided that \(f\) concentrates near \(T^{-\rho_U^\vee}\), is uniformly smooth under left translation by \(A\), has \(L^2\)-norm \(\operatorname{O}(1)\), and has “frequency” \(\operatorname{O}(T)\) under right translation by \(G\) (see, e.g., Lemmas 20.2 and 21.26 for details).

Let \(\pi\) be a unitary representation of \(G\). More precisely, we write \(\pi\) for the space of smooth vectors in the underlying Hilbert space representation. For each \(d \in \mathbb{Z}_{\geqslant 0}\), we define a Sobolev norm \(\mathcal{S}_d\) on \(\pi\) by the formula: for \(v \in \pi\), \[\label{eq:mathc-:=-sum_x_1}\tag{2.17} \mathcal{S}_d(v)^2 := \sum_{x_1,\dotsb,x_d \in \mathcal{B}(G) \cup \{1\}} \| \pi(x_1 \dotsb x_d) v\|^2,\] where \(\|.\|\) denotes the given norm on \(\pi\).

For each \((d_1,d_2) \in \mathbb{Z}_{\geqslant 0} \times \mathbb{Z}_{\geqslant 0}\), we define a Sobolev norm \(\mathcal{S}_{d_1,d_2}\) on \(\mathcal{S}(U \backslash G)\) by the formula: for \(f \in \mathcal{S}(U \backslash G)\), \[\mathcal{S}_{d_1,d_2}(f)^2 := \sum _{ \substack{ x_1, \dotsb, x_{d_1} \in \mathcal{B}(A) \cup \{1\} \\ y_1, \dotsc, y _{d _2 } \in \mathcal{B}(G) \cup \{1\} } } \| L(x_1 \dotsb x_{d_1}) R(y_1 \dotsb y_{d_2}) f \|^2,\] where \(\|.\|\) denotes the norm on \(L^2(U \backslash G)\). For example, \(\mathcal{S}_{d,0}(f) = \mathcal{S}_{d}(f)\) as defined via \((2.17)\), taking for \(\pi\) the right regular representation \(R\) of \(G\) on \(\mathcal{S}(U \backslash G)\). By the Parseval relation on \(A\), we have for \(f \in \mathcal{S}^e(U \backslash G)\) the relation \[\label{eq:sobolev-parseval-A-U-G}\tag{2.18} \mathcal{S}_{d_1,d_2}(f)^2 = \sum _{ y_1, \dotsc, y _{d _2 } \in \mathcal{B}(G) \cup \{1\} } \int _{ \substack{ s \in \mathfrak{a}_{\mathbb{C}}^* : \\ \Re(s) = 0 } } E_{d_1}(s) \| R(y_1 \dotsb y_{d_2}) f[s] \|^2 \, d \mu_A(s),\] where \[\label{eq:e_ds-:=-sum}\tag{2.19} E_d(s) := \sum _{ x_1, \dotsc, x _{d } \in \mathcal{B}(A) \cup \{1\} } |\langle s, x_1 \dotsb x_{d} \rangle|^2.\]

We note that for all \(a \in A\) and \((d_1,d_2,f)\) as above, \[\label{eq:mathcals_d_1-d_2la-f}\tag{2.20} \mathcal{S}_{d_1,d_2}(L(a) f) = \mathcal{S}_{d_1,d_2}(f),\] as follows from the commutativity of \(A\) and the unitarity of the operator \(L(a)\) on \(L^2(U \backslash G)\).

Given a parameter \(T \geqslant 1\), we define the rescaled Sobolev norm \(\mathcal{S}_{d_1,d_2,T}\) on \(\mathcal{S}(U \backslash G)\) by \[\label{eq:mathcals_d_1-d_2-tf}\tag{2.21} \mathcal{S}_{d_1,d_2,T}(f) := \max_{0 \leqslant k \leqslant d_2} T^{-k} \mathcal{S}_{d_1,k}(f).\]

§2.10. Zeta functions

In the local case, we denote by \(\zeta_F\) the meromorphic function on \(\mathbb{C}\) given by \[\zeta_F(s) := \begin{cases} \pi^{-s/2} \Gamma(s/2) & \text{ if } F = \mathbb{R}, \\ 2 (2 \pi ) ^{- s} \Gamma (s) & \text{ if } F = \mathbb{C}, \\ (1 - q^{-s})^{-1} & \text{ if } F: \text{non-archimedean}. \end{cases}\] We write \(\zeta_F(U,\cdot)\) for the meromorphic function on \(\mathfrak{a}_{\mathbb{C}}^*\) given by \[\zeta_F(U,s) := \prod_{\alpha \in R_U^+} \zeta_F(1 + \alpha^\vee(s)).\] We note that \(\zeta_F(U,s)\) depends only upon the character \(\chi = |.|^s\) of \(A\). Thus the shorthand \[\zeta_F(U,\chi) := \zeta_F(U,s)\] defines a meromorphic function \(\zeta_F(U,\cdot)\) on \(\mathfrak{X}^e(A)\).

In the global case, we abbreviate \(\zeta_\mathfrak{p} := \zeta_{F_\mathfrak{p}}\). Given a finite set of places \(S\) containing the archimedean ones, we write \(\zeta_F^{(S)}(U,s)\) or \(\zeta_F^{(S)}(U,\chi)\) for the product over \(\mathfrak{p} \notin S\) of the corresponding local factors, with the product defined either via meromorphic continuation from the domain where \(\Re(s)\) is strictly dominant or by the analogous product of partial Dedekind zeta functions. When \(F = \mathbb{Q}\) and \(S = \{\infty\}\), we write simply \(\zeta(U,s)\) or \(\zeta(U,\chi)\).

§2.11. Additive characters

We denote by \(\psi_{\mathbb{Q}}\) the “standard” nontrivial unitary character of \(\mathbb{A}/\mathbb{Q}\), characterized by \(\psi_{\mathbb{Q}}|_{\mathbb{R}} = e^{2 \pi i (\cdot)}\). For each completion \(F\) of \(\mathbb{Q}\), we write \(\psi_F\) for the restriction of \(\psi_{\mathbb{Q}}\) to \(F\). Each local field \(F\) of characteristic zero is a finite extension of some completion \(F_0\) of \(\mathbb{Q}\); we define \(\psi_F\) to be the pullback of \(\psi_{F_0}\) via the trace map \(F \rightarrow F_0\).

We denote by \(\mathop{\mathrm{U}}(1) := \{z \in \mathbb{C}^\times : |z| = 1\}\) the unitary group in one variable.

In the local (resp. global) case, let \(\psi : N \rightarrow \mathop{\mathrm{U}}(1)\) (resp. \(\psi : N(\mathbb{A}) \rightarrow \mathop{\mathrm{U}}(1)\)) be a nondegenerate unitary character, trivial on \(N(F)\) in the global case. For example, if \(G = {\mathop{\mathrm{GL}}}_r\), then \[\label{eq:psi-n-vs-psi-F-whittaker}\tag{2.22} \psi(n) = \psi_F\left(\sum_{i=1}^{r-1} c_i n_{i,i+1}\right)\] for some \(c_1,\dotsc,c_{r-1} \in F^\times\).

Suppose \(G = {\mathop{\mathrm{GL}}}_r\). In the local (resp. global) case, the standard nondegenerate unitary character \(\psi\) of \(N\) (resp. of \(N(\mathbb{A})\), trivial on \(N(F)\)) is then defined to be \[\label{eq:psin-=-psi_f}\tag{2.23} \psi(n) = \psi_{F} \left( \sum_{i=1}^{r-1} n_{i,i+1} \right).\]

Suppose now that \(F\) is local and that \(G\) arises from some split reductive group over \(\mathfrak{o}\), which we continue to denote by \(G\). We may speak then of \(\psi\) being unramified. For example, if \(G = {\mathop{\mathrm{GL}}}_r\) and \(\psi_F\) itself is unramified (i.e., \(F = \mathbb{Q}_p\) or an unramified extension), then this says that each \(|c_i| = 1\) in the expression \((2.22)\).

For a reductive group \(G\) over \(\mathbb{Z}\) and a nondegenerate unitary character \(\psi\) of \(N(\mathbb{A})\), trivial on \(N(\mathbb{Q})\), we say that \(\psi\) is unramified if its restriction to each \(N(\mathbb{Q}_p)\) is unramified. For example, when \(G = {\mathop{\mathrm{GL}}}_n\), the standard character and its inverse are unramified, as is any \(\psi\) for which each \(c_i = \pm 1\) in \((2.22)\).

§2.12. Whittaker functions

Let \(\psi\) be as in §2.11.

In the local case, we denote by \(C^\infty(N \backslash G, \psi)\) the space of smooth functions \(W : G \rightarrow \mathbb{C}\) satisfying \(W(u g) = \psi(u) W(g)\) for all \((u,g) \in N \times G\). (We are using the letter \(W\) to denote both the Weyl group and Whittaker functions, but it should always be clear by context what we mean.) In the global case, we analogously define \(C^\infty(N(\mathbb{A}) \backslash G(\mathbb{A}), \psi)\).

We denote by \(w_{\bar{U},N} \in W\) the unique Weyl group element with \(w_{\bar{U},N} N = \bar{U} w_{\bar{U},N}\), where \(\bar{U}\) denotes the maximal unipotent subgroup opposite to \(U\). We choose a representative for \(w _{\overline{U}, N}\) in \(G(F)\). The definitions that follow depend upon this choice. When \(G\) is a general linear group (the case ultimately relevant for this paper), we always take the standard choice, given by a permutation matrix; more generally, if \(U\) is opposite to \(N\), then we take for \(w_{\overline{U}, N}\) the identity element of \(G(F)\). In the local case, \(w _{\overline{U},N}\) then lies in the standard maximal compact subgroup.

In the local case, for \(\chi \in \mathfrak{X}(A)\) and \(f \in \mathcal{I}(\chi)\), we denote by \[W[f,\psi] \in C^\infty(N \backslash G, \psi)\] the corresponding Jacquet integral, defined for \(\Re(\chi) \succ 0\) by the absolutely convergent integral \[W[f, \psi](g) := \int _{n \in N} \psi^{-1}(n) f(w_{\bar{U},N} n g) \, d n\] and in general by analytic continuation along a holomorphic family of vectors (see (Hervé Jacquet and Shalika 1983, sec. 3), (Hervé Jacquet 2009, sec. 7.2), (Wallach 1992, vol. 132, sec. 15), (Shahidi 1980)).

In the global case, for a continuous function \(\varphi\) on \([G]\), we define \[W[\varphi,\psi] \in C^\infty(N(\mathbb{A}) \backslash G(\mathbb{A}), \psi)\] by the analogous global integral: \[W[\varphi, \psi](g) := \int _{n \in [N]} \psi^{-1}(n) \varphi(w_{\bar{U},N} n g) \, d n.\]

Assume now that \(F\) is non-archimedean local, that \(G\) arises from some split reductive group over \(\mathfrak{o}\), which we continue to denote by \(G\), that \(K = G(\mathfrak{o})\), and that \(w_{\bar{U},N} \in K\). Assume that \(\psi\) is unramified, \(\chi\) is unramified, and \(f^0[\chi]\) is the unramified vector with \(f^0[\chi](1) = 1\). Recall our convention that the Haar measure on \(N\) assigns volume one to a maximal compact subgroup. The Casselman–Shalika formula (Casselman and Shalika 1980, Thm 5.4) then reads \[\label{eq:wf0chi-psi1-casselman-shalika}\tag{2.24} W[f^0[\chi],\psi](1) = \frac{1}{\zeta_F(U,\chi)},\] where the RHS is as defined in §2.10.

§2.13. Intertwining operators

§2.13.1. Basic definition

Let \(w \in W\).

In the local case, for \(\chi \in \mathfrak{X}(A)\), we denote by \[M_w[\chi] : \mathcal{I}(\chi) \rightarrow \mathcal{I}(w \chi),\] or simply \(M_w\) when \(\chi\) is clear from context, the standard local intertwining operator, defined initially for \(\Re(\chi) \succ 0\) by \[M_w f[\chi](g) = \int _{u \in w U w^{-1} \cap U \backslash U} f[\chi](w^{-1} u g) \, d u\] and in general via meromorphic continuation (see (Wallach 1992, 132:10.1.14), (Shahidi 1981, sec. 2.2)). We note that the definition depends upon a lift of \(w\) to \(G\).

In the global case, for a function \(f\) on \([G]_Q\), we denote by \(M_w f : [G]_Q \rightarrow \mathbb{C}\) the image of \(f\) under the standard intertwining operator \(M_w\), defined by the usual integral, if it converges absolutely: \[M_w f(g) = \int _{(w U w^{-1} \cap U \backslash U)(\mathbb{A})} f(w^{-1} u g) \, d u.\] Here we lift \(w\) to an element of \(G(F)\). We recall the basic properties of such integrals (see (Arthur 1979; J. Bernstein and Lapid 2019; E. M. Lapid 2008)). They converge if \(f \in \mathcal{S}([G]_Q)\), in which case they induce continuous maps \(\mathcal{S}([G]_Q) \rightarrow \mathcal{T}([G]_Q)\). They also converge if \(f \in \mathcal{I}(\chi)\) for some \(\chi\) with \(\Re(\chi) \succ \rho_U\). For a holomorphic family \(\{f[\chi]\}_{\chi \in \mathfrak{X}([A])}\), the assignment \(\chi \mapsto M_w f[\chi]\) admits a meromorphic continuation to all \(\chi\), inducing a meromorphic family of maps \[M_w := M_w[\chi] : \mathcal{I}(\chi) \rightarrow \mathcal{I}(w \chi)\] This family of operators is holomorphic in a neighborhood of the subgroup \([A]^\wedge\) of unitary characters, and is unitary on that axis: for \(\chi \in [A]^\wedge\) and \(f_1, f_2 \in \mathcal{I}(\chi)\), we have \[\label{eq:langle-f_1-f_2}\tag{2.25} \langle f_1, f_2 \rangle = \langle M_w f_1, M_w f_2 \rangle.\] For \(w_1, w_2 \in W\) and \(\chi \in \mathfrak{X}(A)\), we have the composition formula \[\label{eq:m_w_1w_2-chi-circ}\tag{2.26} M_{w_1}[w_2 \chi] \circ M_{w_2}[\chi] = M_{w_1 w_2}[\chi] : \mathcal{I}(\chi) \rightarrow \mathcal{I}(w_1 w_2 \chi).\]

§2.13.2. Fourier transforms

In the local case, given a nondegenerate unitary character \(\psi\) of \(N\), we denote by \[\mathcal{F}_{w,\psi}[\chi] : \mathcal{I}(\chi) \rightarrow \mathcal{I}(w \chi),\] or simply \(\mathcal{F}_{w}[\chi]\), \(\mathcal{F}_{w,\psi}\) or \(\mathcal{F}_w\) when \(\psi\) and/or \(\chi\) are understood by context, the normalized intertwining operator as in (Shahidi 1981, p336), i.e., the scalar multiple of \(M_w[\chi]\) for which \[\label{eq:wmathcalf_w-fchi-psi}\tag{2.27} W[\mathcal{F}_w f, \psi] = W[f,\psi] \quad \text{ for all } f \in \mathcal{I}(\chi).\] We note that, while \(M_w\) depends upon a choice of representative for \(w\), the normalized operators \(\mathcal{F}_{w}\) are independent of that choice, although they do depend upon the representatives \(w_{\bar{U},N}\) used to define our Whittaker functionals.

By (Shahidi 1981, Prop 3.1.4), we have \[\label{eq:mathc-1-mathc}\tag{2.28} \mathcal{F}_{w^{-1}} \mathcal{F}_w f = f \quad \text{ for all } f \in \mathcal{I}(\chi)\] and \[\label{eq:f_1chi-f_2chi-1}\tag{2.29} (f_1, f_2) = (\mathcal{F}_w f_1, \mathcal{F}_w f_2) \quad \text{ for all } (f_1,f_2,w) \in \mathcal{I}(\chi) \times \mathcal{I}(\chi^{-1}) \times W.\] In particular, \(\mathcal{F}_w[\chi]\) is unitary when \(\chi\) is unitary, and so the family \(\{\mathcal{F}_w[\chi] : \chi \in A^\wedge \}\) induces a unitary operator \(\mathcal{F}_w\) on \(L^2(U \backslash G)\) (cf. (Gelfand, Graev, and Pyatetskii-Shapiro 2016, sec. 3.5), (Braverman and Kazhdan 1999, 2002; Shahidi 2018)). It is also known that \[\mathcal{F}_{w_1} \circ \mathcal{F}_{w_2} = \mathcal{F}_{w_1 w_2},\] see, e.g., (Arthur 1989, Thm 2.1, \((R_2)\)).

For \(w \in W\), we denote by \[\label{eq:mathc-backsl-gw}\tag{2.30} \mathcal{S}(U \backslash G)^w\] the space of all \(f \in \mathcal{S}(U \backslash G)\) satisfying the equivalent conditions of Lemma 2.6. We define \(\mathcal{S}^e(U \backslash G)^w\) analogously, and then define \(\mathcal{S}(U \backslash G)^W\) and \(\mathcal{S}^e(U \backslash G)^W\) by taking the intersection over all \(w \in W\).

§2.13.3. Euler factorization

Suppose for the moment that \(F\) is global. Intertwining operators may be factored, in a certain sense, over the places of \(F\). Recall that \(\mathcal{I}(\chi)\) identifies with the restricted tensor product of the local principal series representations \(\mathcal{I}_\mathfrak{p}(\chi_\mathfrak{p})\). The restriction is with respect to the normalized spherical elements \(f_\mathfrak{p}^0[\chi_\mathfrak{p}]\) defined for almost all \(\mathfrak{p}\), with the normalization being \(f_\mathfrak{p}^0[\chi_\mathfrak{p}](1) = 1\). For a pure tensor \(f = \otimes_\mathfrak{p} f_\mathfrak{p}\) in this restricted tensor product, we have \[M_w f = \otimes_\mathfrak{p} ^* M_w f_\mathfrak{p},\] where \(M_w\) is the local intertwining operator, defined for \(\Re(\chi_\mathfrak{p}) \succ 0\) by integrating over \(U(F_\mathfrak{p})\), and the meaning of \(\otimes_\mathfrak{p}^*\) is that we regularize the tensor product using the ratio of \(L\)-functions describing the images \(M_w f_\mathfrak{p}^0\) of the unramified vectors, as given in (Robert P. Langlands 1971, 1:(4)).

Let \(\psi\) be a nondegenerate unitary character of \(N(\mathbb{A})\), trivial on \(N(F)\). Write \(\psi_\mathfrak{p} : N(F_\mathfrak{p}) \rightarrow \mathop{\mathrm{U}}(1)\) for its local component at a place \(\mathfrak{p}\). As explained in (Shahidi 1981, sec. 4), the functional equation for Eisenstein series (see, e.g., \((2.43)\) below) implies that \[\label{eq:m_w-f-=}\tag{2.32} M_w f = \otimes_{\mathfrak{p}}^* \mathcal{F}_{w} f_\mathfrak{p},\] where \[\mathcal{F}_{w} f_\mathfrak{p} := \mathcal{F}_{w,\psi_\mathfrak{p}} f_\mathfrak{p}\] and \(\otimes^*_{\mathfrak{p}}\) is defined as above, but using a different ratio of \(L\)-values. More precisely, suppose that \(S\) is a finite set of places taken large enough that \(f_\mathfrak{p} = f_\mathfrak{p}^0[\chi_\mathfrak{p}]\) and \(\psi_\mathfrak{p}\) is unramified for all \(\mathfrak{p} \notin S\). Then \[\begin{align} \mathcal{F}_{w} \zeta_{F_\mathfrak{p}}(U, \chi_\mathfrak{p}) f_\mathfrak{p}^0[\chi_\mathfrak{p}] = \zeta_{F_\mathfrak{p}}(U,w \chi_\mathfrak{p})f_\mathfrak{p}^0[w \chi_\mathfrak{p}], \end{align}\] for \(\mathfrak{p} \notin S\). The precise meaning of \((2.32)\) is then that \[\label{eq:m_w-f-=-1}\tag{2.33} M_w f = \frac{ \zeta_F^{(S)}(U,w \chi) }{ \zeta_F^{(S)}(U,\chi) } (\otimes_{\mathfrak{p} \in S} \mathcal{F}_w f_\mathfrak{p} ) \otimes ( \otimes_{\mathfrak{p} \notin S} f_\mathfrak{p}^0).\]

A more convenient formulation of \((2.33)\) may be given as follows. Introduce the modified spherical elements \[f_\mathfrak{p}^\sharp[\chi_\mathfrak{p}] := \zeta_{F_\mathfrak{p}}(U,\chi_\mathfrak{p}) f_\mathfrak{p}^0[\chi_\mathfrak{p}].\] For \(f \in \mathcal{I}(\chi)\), write \[f = \otimes^\sharp f_\mathfrak{p}\] to signify that for some large enough finite set of places \(S\), we have \(f_\mathfrak{p} = f_\mathfrak{p} ^\sharp [\chi_\mathfrak{p}]\) for all \(\mathfrak{p} \notin S\), and \[f = \zeta_F^{(S)}(U,\chi) (\otimes_{\mathfrak{p} \in S} f_\mathfrak{p} ) \otimes (\otimes_{\mathfrak{p} \notin S} f_\mathfrak{p}^0 ).\] Then for \(\mathfrak{p} \notin S\), we have \[\label{eq:normalized-F-w-preserves-basic-vector}\tag{2.34} \mathcal{F}_w f_\mathfrak{p}^\sharp[\chi_\mathfrak{p}] = f_\mathfrak{p} ^\sharp [w \chi_\mathfrak{p}],\] and so we may rewrite \((2.33)\) as follows: \[\label{eq:m_w-f-=-2}\tag{2.35} M_w f = \otimes ^\sharp \mathcal{F}_w f_\mathfrak{p}.\]

Representation theory of general linear groups

§2.14.1. Kirillov model

Suppose now that \(F\) is local and that \(G\) is a nontrivial general linear group.

We recall that an irreducible (smooth admissible) representation \(\pi\) of \(G\) is generic if it admits an equivariant embedding into the space \(C^\infty(N \backslash G, \psi)\) for some (equivalently, any) nondegenerate unitary character \(\psi\) of \(N\); such an embedding is then uniquely determined up to a scalar (see (Shalika 1974), (Piatetski-Shapiro 1979, sec. 1)), so its image, denoted \(\mathcal{W}(\pi,\psi)\), is well-defined.

We extend \(G\) to a general linear pair \((G,H)\), with \(H\) possibly trivial. The restriction \(\psi|_{N_H}\) is a nondegenerate unitary character of \(N_H\); for notational simplicity, we denote that restriction simply by \(\psi\). The restriction map from \(\mathcal{W}(\pi,\psi)\) to the space \(C^\infty(N \backslash H, \psi)\) is injective (see (J. N. Bernstein 1984, sec. 6.5), (Baruch 2003a, sec. 10.2)), with image called the Kirillov model of \(\pi\). If \(\pi\) is unitarizable, then an invariant norm on \(\mathcal{W}(\pi,\psi)\) may be given by (see (J. N. Bernstein 1984, sec. 6.4), (Baruch 2003a, sec. 10.2)) \[\|W\|^2 := \int _{N_H \backslash H} |W|^2.\] We discuss the finer details of the archimedean Kirillov model in §8.1.

§2.14.2. Essential vector

We retain the setting of §2.14.1, but assume that \(F\) is non-archimedean and \(\psi\) is unramified. Let \(\pi\) be a generic irreducible representation of \(G\), with Whittaker model \(\mathcal{W}(\pi,\psi)\).

For a nonzero integral \(\mathfrak{o}\)-ideal \(\mathfrak{q}\), we write \(K_1(\mathfrak{q}) \leqslant K_0(\mathfrak{q}) \leqslant K\) for the standard congruence subgroups, consisting of those elements with bottom row lying in \((\mathfrak{q},\dotsc,\mathfrak{q},1 + \mathfrak{q})\) and \((\mathfrak{q},\dotsc,\mathfrak{q},\mathfrak{o})\), respectively. Then \(K_1(\mathfrak{q})\) is a normal subgroup of \(K_0(\mathfrak{q})\), with quotient isomorphic to \(\mathfrak{o}^\times / (\mathfrak{o}^\times \cap (1 + \mathfrak{q}))\).

For a subgroup \(J\) of \(G\), we denote by \(\pi^J\) the subspace of \(\pi\) consisting of vectors invariant by \(J\).

We recall some results of (H. Jacquet, Piatetski-Shapiro, and Shalika 1981; Matringe 2013; Hervé Jacquet 2012).

We refer to \(\mathfrak{c}\) as the conductor ideal of \(\pi\) and to its absolute norm \(C(\pi) := \# \mathfrak{o} / \mathfrak{c}\) as the conductor of \(\pi\). We refer to any nonzero element of that space as an essential vector. We will often impose the normalization \(W(1) = 1\).

If \(\mathfrak{c} = \mathfrak{o}\), then \(K_1(\mathfrak{c}) = K_0(\mathfrak{c}) = K\). Otherwise, writing \(\pi|_Z : F^\times \rightarrow \mathbb{C}^\times\) for the central character of \(\pi\), the group \(K_0(\mathfrak{c})\) acts on \(\pi^{K_1(\mathfrak{c})}\) via the character \(g \mapsto \pi|_Z(d_g)\) of \(K_1(\mathfrak{c})\), where \(d_g\) denotes the lower-right entry of \(g\). In particular, the conductor ideal of the central character \(\pi|_Z\) divides that of the representation \(\pi\).

§2.14.3. Standard \(L\)-factors

Let \(F\) be local, and let \(G\) be a general linear group. Let \(\pi\) be a generic irreducible admissible representation of \(G\). The standard \(L\)-factor \(L(\pi,s)\) was defined first by Godement–Jacquet (Godement and Jacquet 1972) as the greatest common divisor of a family of zeta integrals attached to the matrix coefficients of \(\pi\). It was later interpreted more broadly, by Jacquet–Piatetski-Shapiro–Shalika (see (H. Jacquet, Piatetskii-Shapiro, and Shalika 1983, sec. 5.1), (Hervé Jacquet, Piatetski-Shapiro, and Shalika 1979a, Thm 4.3)) as the \({\mathop{\mathrm{GL}}}_n \times {\mathop{\mathrm{GL}}}_1\) case of Rankin–Selberg convolutions for \({\mathop{\mathrm{GL}}}_n \times {\mathop{\mathrm{GL}}}_m\). We refer to the summaries (Hervé Jacquet 1979) and (Rudnick and Sarnak 1996, sec. 2 and Appendix) for further discussion.

Such \(L\)-factors may be expressed as finite products \(L(\pi,s) = \prod_{j=1}^m \zeta_F(s + \mu_{j})\), where \(0 \leqslant m \leqslant n\) and \(\mu_j \in \mathbb{C}\). If \(F\) is archimedean, or if \(F\) is non-archimedean and \(\pi\) is unramified, then \(m = n\). We remark that if \(\pi\) is unitary, then each \(\Re(\mu_j) > -1/2\) (H. Jacquet and Shalika 1981, sec. 2.5), (Barthel and Ramakrishnan 1994, Prop 2.1), so \(L(\pi,s)\) is holomorphic in the region \(\Re(s) \geqslant 1/2-\varepsilon\) for some \(\varepsilon> 0\).

§2.14.4. Zeta integrals

Let \(F\) be local and let \((G,H)\) be a general linear pair. We summarize here some basic facts concerning local Rankin–Selberg integrals for \(G \times H\).

As above, we choose a nondegenerate unitary character \(\psi\) of \(N\), and denote also by \(\psi\) its restriction to \(N_H\).

Let \(\pi\) and \(\sigma\) be generic irreducible representations of \(G\) and \(H\), respectively. The following discussion applies also to Whittaker-type representations in the sense of (Cogdell and Piatetski-Shapiro 2017, sec. 1.5), i.e., inductions of essentially square-integrable representations. In particular, it applies to principal series representations in the sense of §2.7.

Recall (§2.3.5) that we have chosen Haar measures on \(H\) and \(N_H\) that, in the non-archimedean case, assign volume one to maximal compact subgroups.

For \((W,V,u) \in \mathcal{W}(\pi,\psi) \times \mathcal{W}(\sigma,\psi^{-1}) \times \mathbb{C}\), we define the zeta integral \[Z(W, V, u) := \int _{h \in N_H \backslash H} W \begin{pmatrix} h & \\ & 1 \end{pmatrix} V(h) |\det h|^{u-1/2} \, d h,\] provided that it converges.

We recall some results of Jacquet, Piatetski-Shapiro and Shalika.

The definition of \(Z(W,V,u)\) is particularly simple in the special case that the restriction of the Whittaker function \(W\) to the subgroup \(H\) has compact support modulo \(N_H\). In that case, no meromorphic continuation is necessary - the zeta integral is entire in \(u\). This trivial case of Theorem 2.12 is in fact the only case required for the purposes of this paper.

We now specialize to the case that \(\sigma\) is a principal series representation \(\mathcal{I}(s)\) of \(H\), as defined in §2.7 for \(s \in \mathfrak{a}_{H,\mathbb{C}}^*\). Write \((G,H) = ({\mathop{\mathrm{GL}}}_{n+1}(F), {\mathop{\mathrm{GL}}}_n(F))\). Using standard diagonal coordinates, we may identify \(\mathfrak{a}_{H,\mathbb{C}}^*\) with \(\mathbb{C}^n\), hence \(s\) with an \(n\)-tuple \((s_1,\dotsc,s_n)\) of complex numbers \(s_j\).

We abbreviate \[Z(W,V) := Z(W,V,u)|_{u=1/2},\] where the RHS is interpreted as the value (possibly infinite) of the meromorphic function \(u \mapsto Z(W,V,u)\).

Given \(f \in \mathcal{I}(s)\), we set \[Z(W,f) := Z(W, W[f,\psi^{-1}]),\] where \(W[f,\psi^{-1}] \in \mathcal{W}(\mathcal{I}(s), \psi^{-1})\) denotes the image of \(f\) under the Jacquet integral, as defined in §2.12.

§2.14.5. Standard \(L\)-functions

For simplicity, we assume here that \(F = \mathbb{Q}\).

Let \(G = {\mathop{\mathrm{GL}}}_n\) be a general linear group, \(\pi\) a cuspidal automorphic representation of \({\mathop{\mathrm{GL}}}_n(\mathbb{A})\). We recall some properties of the standard \(L\)-function \(L(\pi,s)\) introduced by Godement–Jacquet (Godement and Jacquet 1972, Lemma 10.20). We refer to Jacquet (Hervé Jacquet 1979), Rudnick–Sarnak (Rudnick and Sarnak 1996, sec. 2), and Iwaniec–Kowalski (Iwaniec and Kowalski 2004, vol. 53, sec. 5) for further background and details.

For \(s \in \mathbb{C}\), we may form the twisted representation \(\pi \otimes |.|^s\), consisting of all functions \(g \mapsto \varphi(g) |\det g|^s\) with \(\varphi \in \pi\). There is a unique \(\sigma \in \mathbb{R}\) for which the twist \(\pi \otimes |.|^\sigma\) has unitary central character, and in that case, the Petersson inner product equips \(\pi\) with the structure of a unitary representation. There is thus little loss of generality in supposing \(\pi\) to be unitary.

For each place \(\mathfrak{p}\), the local component \(\pi_\mathfrak{p}\) is a generic irreducible representation of \({\mathop{\mathrm{GL}}}_n(\mathbb{Q}_\mathfrak{p})\), unitary if \(\pi\) is. By the discussion of §2.14.3, we obtain standard local \(L\)-factors \(L(\pi_\mathfrak{p},s)\). As shown by Godement–Jacquet (Godement and Jacquet 1972, Lemma 10.20), the Euler product \[L(\pi,s) := \prod_p L(\pi_p,s)\] converges for \(\Re(s)\) sufficiently large (indeed, for \(\Re(s) > 1\) if \(\pi\) is unitary, as shown by Jacquet–Shalika (H. Jacquet and Shalika 1981, I, Thm 5.3)). The completed \(L\)-function \(\Lambda(\pi,s) := L(\pi_\infty,s) L(\pi,s)\) satisfies a functional equation \(\Lambda(\pi,s) = \varepsilon(\pi,s) \Lambda(\tilde{\pi},1-s)\), where \(\tilde{\pi}\) denotes the contragredient representation (for unitary \(\pi\), the complex conjugate representation) and \(\varepsilon(\pi,s)\) has the form \(\varepsilon(\pi,s) = \varepsilon(\pi) C(\pi_{\mathop{\mathrm{fin}}})^{1/2-s}\), with \(|\varepsilon(\pi)| = 1\) for unitary \(\pi\).

The finite conductor \(C(\pi_{\mathop{\mathrm{fin}}}) \in \mathbb{Z}_{\geqslant 1}\) of \(\pi\) is defined to be the product \(\prod_p C(\pi_p)\) of the local conductors, as defined in §2.14.2; only finitely many of these local conductors are not equal to one, so the product is really a finite product. Recall from §1.2 that if \(\pi\) has archimedean \(L\)-function parameters \(t_1,\dotsc,t_n\), i.e., if \(L(\pi_\infty, s) = \prod_{j=1}^n \Gamma_{\mathbb{R}}(s + t_j)\), then its archimedean conductor is defined by \(C(\pi_\infty) = \prod_{j=1}^n (3 + |t_j|)\), and its global analytic conductor by \(C(\pi) = C(\pi_\infty) C(\pi_{\mathop{\mathrm{fin}}})\).

The completed \(L\)-function \(\Lambda(\pi,s)\) is bounded in vertical strips, so \(L(\pi,s)\) is of order one. The absolute convergence of the Euler product, the functional equation, and the Phragmen–Lindelöf convexity principle then imply that \(L(\pi,s)\) is polynomially bounded in vertical strips (see (Iwaniec and Kowalski 2004, Lemma 5.2)): for \(\Re(s) \ll 1\), \[\label{eq:lpi-tfrac12-+-2}\tag{2.36} L(\pi, \tfrac{1}{2} + s) \ll C(\pi)^{\operatorname{O}(1)} (1 + |s|)^{\operatorname{O}(1)},\] provided that \(n \geqslant 2\); when \(n = 1\) and \(L(\pi,s) = \zeta(s)\), this estimate must be modified to take into account the pole at \(s=1\).

We recall the convexity bound:

The twisted representation \(\pi \otimes |.|^s\) has archimedean \(L\)-function parameters \[t_1+s,\dotsc,t_n+s\] and the same finite conductor as \(\pi\), hence \[C(\pi \otimes |.|^s) = C(\pi_{\mathop{\mathrm{fin}}}) \prod_{j=1}^n (3 + |t_j + s|).\] Since \(C(\pi \otimes |.|^s) \ll (1 + |s|)^{n} C(\pi)\), the convexity bound \((2.37)\) contains the estimate: for \(s \in \mathbb{C}\) with \(|\Re(s)| \leqslant 1/2\), \[\label{eqn:convexity-polynomially-in-s}\tag{2.38} L(\pi, \tfrac{1}{2} +s) \ll C(\pi)^{(1 - 2 s_j)/4 + o(1)} (1 + |s|)^{\operatorname{O}(1)}.\]

§2.15. Eisenstein series

Let \(F\) be a number field.

Definition and convergence

We will be concerned primarily with “minimal” Eisenstein series coming from the Borel subgroup \(Q\). (We will eventually need to consider “mirabolic” or “Siegel-type” Eisenstein series, as well, but we treat those separately as they arise.)

For a function \(f\) on \([G]_Q\), we denote by \(\mathop{\mathrm{Eis}}(f) : [G] \rightarrow \mathbb{C}\) the usual series, provided that it converges absolutely: \[\mathop{\mathrm{Eis}}(f)(g) := \sum _{\gamma \in Q(F) \backslash G(F)} f(\gamma g).\] When we wish to explicate which groups are being considered, we write more verbosely \({\mathop{\mathrm{Eis}}}_Q^G\).

Standard convergence arguments as in (Borel 2007, sec. 12) give that for each \(\varepsilon> 0\), there exists \(m \geqslant 0\) so that \[\label{eq:sup_g-in-gmathbba}\tag{2.39} \sup_{g \in G(\mathbb{A})} \|g\|^{-m} \int_{ (\sigma) } \sum_{\gamma \in Q(F) \backslash G(F)} |\mathfrak{a}(\gamma g)|^{(1+\varepsilon) \rho_U} \, d \chi < \infty,\] where the integral is taken over \(\chi \in \mathfrak{X}([A])\) with \(\Re(\chi) = \sigma\), following the notation \((2.6)\). It follows that \(\mathop{\mathrm{Eis}}(f)\) converges absolutely if \(f \in \mathcal{I}(\chi)\) for some \(\chi \in \mathfrak{X}([A])\) with \(\Re(\chi) \succ \rho_U\), in which case \(\mathop{\mathrm{Eis}}(f) \in \mathcal{T}([G])\). The arguments of (Mœglin and Waldspurger 1995, vol. 113, secs. II.1.7, II.1.10) show that \(\mathop{\mathrm{Eis}}(f)\) converges for \(f \in \mathcal{S}([G]_Q)\), in which case \(\mathop{\mathrm{Eis}}(f) \in \mathcal{S}([G])\), and the induced map \[\mathop{\mathrm{Eis}}: \mathcal{S}([G]_Q) \rightarrow \mathcal{S}([G])\] is continuous. (By the definition of the topologies, we can check the continuity after passing to invariants for some compact open subgroup \(J\) of \(G(\mathbb{A}_f)\). We then appeal to (Casselman 1989, Prop 1.13).)

Meromorphic continuation and Mellin expansion

For any holomorphic family \(\{f[\chi]\}_{\chi \in \mathfrak{X}([A])}\), the assignment \(\chi \mapsto \mathop{\mathrm{Eis}}(f[\chi])\), defined for \(\Re(\chi) \succ \rho_U\) by the series, extends to a meromorphic vector-valued map \(\mathfrak{X}([A]) \rightarrow \mathcal{T}([G])\). We refer to (E. M. Lapid 2008) for precise statements and proofs, and also to (J. Bernstein and Krötz 2014, Thm 11.7).

Let \(f \in \mathcal{S}([G]_Q)\), with corresponding holomorphic family of Mellin components \(\{f[\chi]\}_{\chi \in \mathfrak{X}([A])}\) (necessarily of rapid vertical decay). Let \(\sigma \in \mathfrak{a}^*\) with \(\sigma \succ \rho_U\). By \((2.39)\), there exists \(m \geqslant 0\) so that \[\sup_{g \in G(\mathbb{A})} \|g\|^{-m} \int_{ (\sigma) } \sum_{\gamma \in Q(F) \backslash G(F)} |f[\chi](\gamma g)| \, d \chi < \infty,\] and similarly for archimedean derivatives. By interchanging integration with summation, it follows that \[\label{eq:eisf-=-int}\tag{2.40} \mathop{\mathrm{Eis}}(f) = \int _{(\sigma) } \mathop{\mathrm{Eis}}(f[\chi]) \, d \chi,\] where the integral converges in \(\mathcal{T}([G])\). In particular, for any \(\varphi \in \mathcal{S}([G])\), we have \[\label{eq:int-_g-eisf}\tag{2.41} \int _{[G]} \mathop{\mathrm{Eis}}(f) \varphi = \int _{(\sigma)} \left( \int _{[G]} \mathop{\mathrm{Eis}}(f[\chi]) \varphi \right) \, d \chi.\]

§2.15.3. Whittaker functions

Let \(\psi : N(\mathbb{A}) \rightarrow \mathop{\mathrm{U}}(1)\) be a nondegenerate character, trivial on \(N(F)\). Provided that \(\mathop{\mathrm{Eis}}(f)\) converges absolutely and locally uniformly, we have the standard unfolding calculation (see, e.g., (Shahidi 2010, Prop 7.1.3)) \[\label{eq:weisf-psig-=}\tag{2.42} W[\mathop{\mathrm{Eis}}(f),\psi](g) = W[f,\psi](g) := \int _{u \in N(\mathbb{A})} f(w_G^{-1} u g) \psi^{-1}(u) \, d u.\]

Functional equation

For \(f \in \mathcal{I}(\chi)\), we have the functional equation \[\label{eq:eisenstein-series-functional-equation}\tag{2.43} \mathop{\mathrm{Eis}}(M_w f[\chi]) = \mathop{\mathrm{Eis}}(f[\chi])\] whenever both sides are defined.

§2.15.5. Constant terms

Let \(P\) be a parabolic subgroup of \(G\) that contains \(A\). We factor \(P = M U_P\), with \(M\) the unique Levi factor containing \(A\).

The constant term with respect to \(P\) of a continuous function \(\varphi\) on \([G]\) is the function \(\varphi_P : [G]_P \rightarrow \mathbb{C}\) defined by \(\varphi_P(g) := \int _{[U_P]} \varphi(u g) \, d u\).

For \(f \in \mathcal{S}([G]_Q)\), or for \(f \in \mathcal{I}(\chi)\) with \(\Re(\chi) \succ \rho_U\), we have by standard unfolding calculations (Mœglin and Waldspurger 1995, 113:p101 and §II.1.7) that \[\label{eq:int_g-eisf-varphi}\tag{2.44} \int_{[G]} \mathop{\mathrm{Eis}}(f) \varphi = \int_{[G]_Q} f \varphi_Q \quad \text{ for } \varphi \in \mathcal{S}([G])\] and \[\label{eq:eisf_b-=-sum}\tag{2.45} \mathop{\mathrm{Eis}}(f)_Q = \sum _{w \in W} M_w f.\] Such identities extend meromorphically to \(f \in \mathcal{I}(\chi)\) for almost all \(\chi\).

For general \(P\) as above, the intersection \(Q_{M} := Q \cap M\) is a Borel subgroup of \(M\), and for \(f \in \mathcal{S}([G]_Q)\), we have \[\label{eq:eis_bgf_p-=-sum}\tag{2.46} {\mathop{\mathrm{Eis}}}_Q^G(f)_P = \sum _{w \in W(A,M)} {\mathop{\mathrm{Eis}}}_{Q_{M}}^{M} (M_w f),\] where \(W(A,M)\) denotes the set of all \(w \in W\) such that \(w^{-1}(\alpha) > 0\) for all positive roots \(\alpha\) for \((M,A)\), and where each Eisenstein series appearing on the RHS of \((2.46)\) converges absolutely. This again follows from an unfolding calculation as in loc. cit, the main point being, as noted in loc cit., that the set \(W(A,M)\) defines a system of representatives for \(Q(F) \backslash G(F) / P(F)\).

For a holomorphic family \(\{f[\chi]\}_{\chi \in \mathfrak{X}([A])}\), the Eisenstein series \(\mathop{\mathrm{Eis}}(f[\chi])\) is holomorphic in \(\chi\) if and only its constant term \(\mathop{\mathrm{Eis}}(f[\chi])_Q\) with respect to the Borel \(Q\) has the same property (see (Mœglin and Waldspurger 1995, vol. 113, sec. I.4.10 and Remark p167)). By \((2.45)\), the latter condition holds provided that \(M_w f[\chi]\) is holomorphic in \(\chi\) for each \(w \in W\).

Inner product formulas

We require the following consequence of the spectral theory of Eisenstein series.

§2.16. Completed Eisenstein series

For simplicity, we now take \(F = \mathbb{Q}\) and assume that \(G\) arises from a split reductive group over \(\mathbb{Z}\), which we continue to denote by \(G\). (The results of this section represent the special case \(F = \mathbb{Q}\) and \(S = \{\infty\}\), adequate for our immediate applications, of more general results concerning any number field \(F\) and large enough finite set of places \(S\).) We assume that \(K_p = G(\mathbb{Z}_p)\) for each finite prime \(p\).

§2.16.1. Notation and basic definition

Recall from §2.7 that, for \(s \in \mathfrak{a}_{\mathbb{C}}^*\), we denote by \(\mathcal{I}(s) = {\mathop{\mathrm{Ind}}}_{Q(\mathbb{A})}^{G(\mathbb{A})} (|.|^s)\) the corresponding principal series representation of \(G(\mathbb{A})\), which is the restricted tensor product, over places \(\mathfrak{p}\) of \(\mathbb{Q}\), of the local principal series representations \(\mathcal{I}_\mathfrak{p}(s) = {\mathop{\mathrm{Ind}}}_{Q(\mathbb{Q}_\mathfrak{p})}^{G(\mathbb{Q}_\mathfrak{p})}(|.|_{\mathfrak{p}}^s)\). For each finite prime \(p\) and all \(s \in \mathfrak{a}_{\mathbb{C}}^*\), we temporarily write \[\label{eq:holomorphic-families-f-p-of-s}\tag{2.47} f_p[s] \in \mathcal{I}_p(s) \subseteq C^\infty(U(\mathbb{Q}_p) \backslash G(\mathbb{Q}_p))\] for the spherical vector normalized by requiring that \(f_p[s](1) = 1\).

As in §2.10, for \(s \in \mathfrak{a}_{\mathbb{C}}^*\), we define \(\zeta(U,s)\), initially for \(\Re(s) \succ 0\) by the convergent Euler product \[\label{eq:zeta-U-s-global-over-Z}\tag{2.48} \zeta(U,s) := \prod_p \zeta_p(U,s) = \prod_{\alpha \in R_U^+} \zeta(1 + \alpha^\vee(s)),\] then in general via meromorphic continuation. We also define the \(W\)-invariant polynomial \[\mathcal{P}_G(s) := \prod_{\alpha \in R} \alpha^\vee(s) (1 - \alpha^\vee(s)) = \prod_{\alpha \in R_U^+} \alpha^\vee(s)^2 (\alpha^\vee(s)^2 - 1).\] The product \(\mathcal{P}_G(s) \zeta(U,s)\) is then entire and of polynomial growth in vertical strips, hence acts as a multiplier on, e.g., \(\mathcal{S}^e(U(\mathbb{R}) \backslash G(\mathbb{R}))\).

Let \(f_\infty \in \mathcal{S}^e(U(\mathbb{R}) \backslash G(\mathbb{R}))\), with associated Mellin components \(f_\infty[s] \in \mathcal{I}_\infty(s)\).

For \(s \in \mathfrak{a}_{\mathbb{C}}^*\), define \(\Phi[f_\infty][s] \in \mathcal{I}(s)\) by the formula \[\label{eq:phis-:=-qn}\tag{2.49} \Phi[f_\infty][s] = \mathcal{P}_G(s) \zeta(U,s) \otimes_{\mathfrak{p}} f_\mathfrak{p}[s].\] This defines a holomorphic family of vectors of rapid vertical decay. Per the discussion of §2.8.3, this family arises by taking the Mellin components of some element \(\Phi[f_\infty]\) of the Schwartz space \(\mathcal{S}^e([G]_Q)\). We obtain a \(G(\mathbb{R})\)-equivariant linear map \[\Phi : \mathcal{S}^e(U(\mathbb{R}) \backslash G(\mathbb{R})) \rightarrow \mathcal{S}^e([G]_Q)\] \[f_\infty \mapsto \Phi[f_\infty].\] This map may be used to define the “completed” pseudo Eisenstein series \[\begin{align} \mathop{\mathrm{Eis}}[\Phi[f_\infty]] &= \int _{(\sigma)} \mathop{\mathrm{Eis}}[\Phi[f_\infty][s]] \, d \mu_{[A]}(s) \quad \quad (\sigma \succ \rho_U) \\ &= \int _{(\sigma)} \mathcal{P}_G(s) \zeta(U,s) \mathop{\mathrm{Eis}}[\otimes_{\mathfrak{p}} f_\mathfrak{p}[s]] \, d \mu_{[A]}(s). \end{align}\] In what follows, we drop the subscript \(\infty\), writing simply \(f\) for an element of \(\mathcal{S}^e(U(\mathbb{R}) \backslash G(\mathbb{R}))\) and \(\Phi[f]\) for the associated element of \(\mathcal{S}^e([G]_Q)\).

We note that for \(a \in A(\mathbb{R}) \hookrightarrow A(\mathbb{A})\), we have \[\Phi[L(a) f] = L(a) \Phi[f].\] Indeed, both sides have Mellin components \[|a|^s \mathcal{P}_G(s) \zeta(U,s) f[s] \otimes (\otimes_p f_p[s]).\]

Equivariance under Fourier transforms

In what follows, we let \(\psi\) be a nondegenerate unitary character of \(N(\mathbb{A})\), trivial on \(N(\mathbb{Q})\), that is unramified in the sense of §2.11. This condition determines \(\psi\) “up to signs.” Recall the notation \(\mathcal{S}^e(U(\mathbb{R}) \backslash G(\mathbb{R}))^w\) from §2.13.2. We define this notation with respect to the restriction \(\psi_{\infty}\) of \(\psi\) to \(N(\mathbb{R})\), thus \(\mathcal{S}^e(U(\mathbb{R}) \backslash G(\mathbb{R}))^w\) is the subspace of \(\mathcal{S}^e(U(\mathbb{R}) \backslash G(\mathbb{R}))\) consisting of \(\mathcal{F}_{w,\psi_\infty}\)-invariant elements.

§2.16.3. Matrix coefficient bounds

Let \(\mathcal{H} := \mathcal{H}_G\) denote the spherical Hecke algebra for \(G(\mathbb{Z})\), i.e., the restricted product over all primes \(p\) of the spherical Hecke algebras for \((G(\mathbb{Q}_p), G(\mathbb{Z}_p))\).

By definition, \(\mathcal{H}\) consists of the smooth compactly-supported complex measures on \(G(\mathbb{A}_f) = \prod ' G(\mathbb{Q}_p)\) that are bi-invariant under \(\prod_p G(\mathbb{Z}_p)\). Using the Haar measure on \(G(\mathbb{A}_f)\) that assigns volume one to \(G(\mathbb{Z}_p)\), we may identify elements of \(\mathcal{H}\) with functions on \(G(\mathbb{A}_f)\).

We may speak of an element \(t \in \mathcal{H}\) being nonnegative; this says that it defines a positive measure, or equivalently, that the corresponding function is nonnegative.

By the Satake isomorphism, characters (i.e., algebra homomorphisms) \(\lambda : \mathcal{H} \rightarrow \mathbb{C}\) are in bijection with collections \(A = (A_p)_p\), indexed by the primes \(p\), of semisimple conjugacy classes \(A_p\) in the dual group \({}^L G(\mathbb{C})\). (In the case \(G = {\mathop{\mathrm{GL}}}_n\) ultimately of interest to us, we have \({}^L G = {\mathop{\mathrm{GL}}}_n\), so such classes may be identified further with \(n\)-tuples of nonzero complex numbers.)

For \(s \in \mathfrak{a}_{\mathbb{C}}^*\), let \(\lambda_s\) denote the character of \(\mathcal{H}\) attached to the principal series representation \(\mathcal{I}(s)\).

The representation \(\mathcal{I}(s)\) is tempered when \(\Re(s) = 0\). It follows from the results of (Cowling, Haagerup, and Howe 1988) (or more elementarily) that if \(t \in \mathcal{H}\) is nonnegative and \(\Re(s) = 0\), then \[\label{eq:lambd-leq-lambd}\tag{2.56} |\lambda_s(t)| \leqslant\lambda_0(t).\]

Crude growth bounds

§2.17. Siegel domains

We take here for \(G\) a general linear group over \(\mathbb{Z}\), with \(K_p = G(\mathbb{Z}_p)\).

For \(c > 0\), we write \(A(\mathbb{R})^0_{\geqslant c}\) for the set of all \(a \in A(\mathbb{R})\) satisfying \(|a|^{\alpha} \geqslant c\) for all \(\alpha \in \Delta_N\), i.e., \(a_i/a_{i+1} \geqslant c\) for \(1 \leqslant i < \mathop{\mathrm{rank}}(G)\).

Each proposition follows readily from standard results on Siegel domains. We could not quickly locate a suitable reference, so we record the short proofs. First of all, it is known that \(G(\mathbb{Q}) G(\mathbb{R}) \prod_p K_p = G(\mathbb{A})\) (Borel 1963, Prop 2.2) and \(G(\mathbb{Z}) = G(\mathbb{Q}) \cap \prod_p K_p\), so we can reduce to verifying the corresponding assertions concerning \(\phi\) on \(G(\mathbb{Z}) \backslash G(\mathbb{R})\), where we integrate now over \(\Omega\) rather than \(\Omega \times \prod_p K_p\). For \(b, c > 0\), let \(\Omega_N(b)\) denote the set of all \(n \in N(\mathbb{R})\) satisfying \(|n_{i j}| \leqslant b\) for all \(i < j\), and denote by \[\mathfrak{S}(b,c) := \Omega_N(b) A(\mathbb{R})^0_{\geqslant c} K_\infty \subseteq G(\mathbb{R})\] the corresponding standard Siegel domain. A theorem of Siegel (see (Borel and Harish-Chandra 1962, Prop 4.5)) implies that

1. if \(b\) is large enough and \(c\) is small enough (e.g., \(b \geqslant 1/2\) and \(c \leqslant\sqrt{3}/2\)), then \(G(\mathbb{Z}) \mathfrak{S}(b,c) = G(\mathbb{R})\), and

2. for all \(b\) and \(c\), we have \[\label{eq:nubc}\tag{2.65} \nu(b,c) := \{\gamma \in G(\mathbb{Q}) : \gamma \mathfrak{S}(b,c) \cap \mathfrak{S}(b,c) \neq \emptyset \} < \infty.\]

The proof of Proposition 2.26 requires some further preliminaries.

§2.18. Petersson norms vs. Whittaker norms

Let \(G\) be a general linear group over \(\mathbb{Z}\). Let \(\psi\) be an unramified (§2.11) nondegenerate unitary character of \(N(\mathbb{A})\), trivial on \(N(\mathbb{Q})\). Let \(\pi\) be a unitary cuspidal automorphic representation of \(G(\mathbb{A})\). For each prime number \(p\), let \(W_p \in \mathcal{W}(\pi_p, \psi_p)\) denote the essential vector (§2.14.2), normalized by \(W_p(1) = 1\). Denote as usual by \(C(\pi_{\mathop{\mathrm{fin}}}) \in \mathbb{Z}_{\geqslant 1}\) the finite conductor of \(\pi\), i.e., the product of the local conductors \(C(\pi_p)\) defined in §2.14.2.

For each archimedean Whittaker function \(W_\infty \in \mathcal{W}(\pi_\infty, \psi_\infty)\), there is a unique automorphic form \(\varphi \in \pi\) whose Whittaker function \(W_\varphi(g) = \int_{[N]} \varphi(u g) \psi^{-1}(u) \, d u\) is the product of the \(W_\mathfrak{p}\): \[W_{\varphi}(g) = \prod_{\mathfrak{p}} W_{\mathfrak{p}}(g_\mathfrak{p}).\] With respect to our fixed Haar measures, we define the Petersson norm \(\|\varphi \|^2 := \int _{[G/Z]} |\varphi|^2\) and the Whittaker norm \(\|W_\infty \|^2 := \int_{N_H(\mathbb{R}) \backslash H(\mathbb{R})} |W_\infty|^2\), as in §2.14.1. They may be compared via the following standard estimates, obtained by combining (Venkatesh 2006, sec. 7) and (Xiannan Li 2010, sec. 1.3).

§2.19. The distance function \(d_H\)

We specialize the definition of (P. D. Nelson 2023, sec. 4.2) to the case that \((G,H)\) is a fixed general linear pair over \(\mathbb{R}\), say \((G,H) = ({\mathop{\mathrm{GL}}}_{n+1}(\mathbb{R}), {\mathop{\mathrm{GL}}}_n(\mathbb{R}))\). Set \(\bar{G} := G/Z = {\mathop{\mathrm{PGL}}}_{n+1}(\mathbb{R})\). We may regard \(H\) as a subgroup of \(\bar{G}\). For a matrix \(x\), we write \(|x|\) for the maximum of the absolute values of its entries. In particular, for \(g \in \bar{G}\), we may regard \(\mathop{\mathrm{Ad}}(g)\) as a matrix (using the standard basis for \(\mathop{\mathrm{Lie}}(G)\)) and define \(|\mathop{\mathrm{Ad}}(g) - 1|\), which quantifies the distance from \(g\) to the identity element.

Observe that \(d_H(g) = 0\) if and only if \(g\) lies in (the image of) \(H\). We may thus regard \(d_H(g)\) as a quantification of the distance from \(g\) to \(H\).

We recall (P. D. Nelson 2023, Lemma 4.1):

§1. Division of the argument

§3. Construction of test vectors: statement

The following is the main result of Part , which relies in turn on Part . The statement is regrettably complicated, but has the advantage of being a purely local assertion concerning representations of real groups. The sketch of §1.3 may provide a useful roadmap for the relevance of many assertions in the following result to our global argument. We note that, in the language of that sketch, the Whittaker function \(W\) below describes the global cusp form \(\varphi\). The reader may also wish to compare with (P. D. Nelson 2023, Thm 4.2), the analogous result in the compact quotient case, which is simpler.

We refer to §A for an elementary reformulation of this result, without the language of nonstandard analysis.

Growth bounds for Eisenstein series: statement

We retain the setting and notation of §2.17, working with \(G := {\mathop{\mathrm{GL}}}_n\) over \(\mathbb{Z}\) for some natural number \(n\).

Let \(\Omega\) be a compact subset of \(G(\mathbb{R})\). For each \(t \in A(\mathbb{R})^0_{\geqslant 1}\), we define a seminorm \(\|.\|_{t,\Omega}\) on \(L^2([G])\) by \[\label{eq:varphi-2_r-:=}\tag{4.1} \|\varphi \|^2_{t,\Omega} := \int _{g \in \Omega \times \prod_p K_p } |\varphi(t g)|^2 \, d g.\] As \(t\) varies with \(\Omega\) fixed, we may understand these seminorms as describing the local \(L^2\)-norm of \(\varphi\) near a given point in \([G]\).

The following is the main result of Part

We refer again to §A for an elementary reformulation of this result, without the language of nonstandard analysis.

§5. Reduction of the proof

We aim now to deduce Theorem 2.1 from Theorems 3.1 and 4.1.

§5.1. Setup and preliminary reductions

Let \((G,H)\) be a fixed general linear pair over \(\mathbb{Z}\), with \(H\) nontrivial. In this section, we label indices as follows: \[(G,H) = ({\mathop{\mathrm{GL}}}_{n+1}, {\mathop{\mathrm{GL}}}_n).\] Thus \(n\) is a fixed natural number \(\geqslant 1\).

Let \(T \geqslant 1\). Let \(\pi\) be a unitary cuspidal automorphic representation of \(G\), each of whose archimedean \(L\)-function parameters are \(\asymp T\). We must show, for suitable fixed \(\delta > 0\), that \[\label{eq:lpi-tfrac12-ll}\tag{5.1} L(\pi,\tfrac{1}{2}) \ll C(\pi_{\mathop{\mathrm{fin}}})^{1/2} T^{(1 - \delta) (n+1)/4}.\]

Consider first the case that \(T \ll 1\). Then \(C(\pi_\infty) \ll 1\), so \(C(\pi) \asymp C(\pi_{\mathop{\mathrm{fin}}})\). In that case, the convexity bound (Lemma 2.14) implies that \(L(\pi,\tfrac{1}{2}) \ll C(\pi_{\mathop{\mathrm{fin}}})^{1/4 + o(1)}\), while the RHS of \((5.1)\) is \(\asymp C(\pi_{\mathop{\mathrm{fin}}})^{1/2}\). The required estimate \((5.1)\) thus holds in that case. We thereby reduce to the case that \[\label{eq:t-ggg-1}\tag{5.2} T \ggg 1.\]

We may similarly reduce to the case that \[\label{eq:n_pi-=-to1}\tag{5.3} C(\pi_{\mathop{\mathrm{fin}}}) \leqslant T^{\operatorname{O}(1)}.\] Otherwise, the RHS of \((5.1)\) is readily seen to be \(\gg C(\pi)^{1/3}\), say, and the required bound follows again from convexity.

We appeal throughout this section to Theorem 3.1.

§5.2. Construction of automorphic forms

We let \(\psi\) denote the standard nondegenerate unitary character of \(N(\mathbb{A})\), trivial on \(N(\mathbb{Q})\) (§2.11). We denote also simply by \(\psi\) its restriction to \(N(\mathbb{Q}_\mathfrak{p})\) for each place \(\mathfrak{p}\) of \(\mathbb{Q}\), and also to \(N_H(\mathbb{A}), N_H(\mathbb{Q}_\mathfrak{p})\).

Let \(W_{\infty} \in \mathcal{W}(\pi_\infty, \psi)\) denote the smooth vector furnished by Theorem 3.1. Let \(\varphi \in \pi\) denote the automorphic form attached to \(W_{\infty}\), as in §2.18, by requiring that its local components at finite primes be normalized essential vectors \(W_{\pi,p} \in \mathcal{W}(\pi_p, \psi)\). By the estimates of that section (cf. Lemma 2.30) and our assumption \((5.3)\), we then have the Petersson norm estimate \[\label{eq:varphi-2-asymp}\tag{5.4} \|\varphi \| \leqslant T^{o(1)}.\]

Since \(\varphi\) is cuspidal, its restriction to \([H]\) decays rapidly, so the integral \(\int_{[H]} \varphi \Psi\) converges absolutely for any automorphic function \(\Psi\) on \([H]\) of moderate growth. We may unfold that integral using the Whittaker expansion (cf. (Shalika 1974, Thm 5.9)) \[\varphi(g) = \sum _{\gamma \in N_H(\mathbb{Q}) \backslash H(\mathbb{Q})} W_\varphi(\gamma g).\] Working formally for the moment, we obtain \[\label{eqn:unfolding-rs-1}\tag{5.5} \int_{[H]} \varphi \Psi = \int_{N_H(\mathbb{Q}) \backslash H(\mathbb{A})} W_\varphi \Psi = \int_{N_H(\mathbb{A}) \backslash H(\mathbb{A})} W_\varphi \tilde{W}_{\Psi},\] where the dual Whittaker function \(\tilde{W}_{\Psi}\) is defined by \[\tilde{W}_{\Psi}(g) = \int_{[N_H]} \Psi(u g) \psi(u ) \, d u.\] Such calculations \((5.5)\) are the starting point for the study of \({\mathop{\mathrm{GL}}}_{n+1} \times {\mathop{\mathrm{GL}}}_n\) zeta integrals due to Jacquet, Piatetski-Shapiro and Shalika.

For each \(s \in \mathfrak{a}_{H,\mathbb{C}}^*\), we denote by \(\mathcal{I}(s) = {\mathop{\mathrm{Ind}}}_{Q_H(\mathbb{A})}^{H(\mathbb{A})} (|.|^s)\) the corresponding principal series representation of \(H(\mathbb{A})\), as defined in §2.7; we recall that it is the restricted tensor product of local principal series representations \(\mathcal{I}_\mathfrak{p}(s) = {\mathop{\mathrm{Ind}}}_{Q_H(\mathbb{Q}_\mathfrak{p})}^{H(\mathbb{Q}_\mathfrak{p})}(|.|_{\mathfrak{p}}^s)\).

Let \(f, f_{\ast} \in \mathcal{S}^e(U_H(\mathbb{R}) \backslash H(\mathbb{R}))^{W_H}\) denote the vectors furnished by Theorem 3.1, with associated Mellin components \(f[s], f_{\ast}[s] \in \mathcal{I}_\infty(s)\). Recall the elements \[\label{eq:phi-:=-phif}\tag{5.6} \Phi[f] \in \mathcal{S}^e([H]_{Q_H})\] constructed in §2.16, with Mellin components \[\Phi[f][s] \in \mathcal{I}(s).\] This notation and the following discussion applies also with \(f\) replaced by \(f_{\ast}\).

Suppose for the moment that \(\Re(s) \succ \rho_U\), so that the series definition of, e.g., \(\mathop{\mathrm{Eis}}[\Phi[f][s]] \in \mathcal{T}([H])\) applies. By the unfolding identity \((2.42)\), the definition (§2.16.1) of \(\Phi[f][s]\) and the Casselman–Shalika formula \((2.24)\), we see that the global Whittaker function of \(\mathop{\mathrm{Eis}}(\Phi[f][s])\) is given by \[\label{eq:weisphis-psi-1g}\tag{5.7} W[\mathop{\mathrm{Eis}}[\Phi[f][s]], \psi^{-1}](g) = \mathcal{P}_H(s) W[f[s], \psi^{-1}](g_\infty) \prod_p W_{s,p}(g_p)\] where \(W_{s,p} \in \mathcal{W}(\mathcal{I}_p(s), \psi^{-1})\) denotes the spherical element of the local Whittaker model satisfying the normalization \({W}_{s,p}(1) = 1\).

Let \(Y\) be an element of the identity component \(A_H(\mathbb{R})^0\) of \(A_H(\mathbb{R})\). We identify \(Y\) with an element of \(A_H(\mathbb{A})\) via the standard inclusion. We define the element \[L(Y) \Phi[f] \in \mathcal{S}([H]_{B_H}),\] as in \((2.7)\), by normalized left translation: \[\label{eq:ly-phig-=}\tag{5.8} L(Y) \Phi[f](g) = \delta_{U_H}^{-1/2}(Y) \Phi[f](Y g),\] or equivalently, via the Mellin integral \[L(Y) \Phi[f] = \int _{(\sigma)} Y^{s} \Phi[f][s] \, d \mu_{[A_H]}(s),\] convergent for all \(\sigma \in \mathfrak{a}_H^*\). (Here \(Y^s\) abbreviates \(|Y|^s\), noting that the entries of \(Y\) are positive.) We define the associated pseudo Eisenstein series \[\Psi[Y] := \mathop{\mathrm{Eis}}[L(Y) \Phi[f]] \in \mathcal{S}([H]),\] which in turn admits the Mellin integral representation \[\begin{align} \Psi[Y] &= \int _{(\sigma)} Y^{s} \mathop{\mathrm{Eis}}[\Phi[s]] \, d \mu_{[A_H]}(s) \\ &= \int _{(\sigma)} Y^{s} \mathcal{P}_H(s) \zeta(U_H,s) \mathop{\mathrm{Eis}}[f[s] \otimes (\otimes_{p} f_p[s])] \, d \mu_{[A_H]}(s). \end{align}\] We analogously define \[\Psi_{\ast}[Y] := \mathop{\mathrm{Eis}}[L(Y) \Phi[f_{\ast}]],\] which admits an analogous integral representation.

§5.3. Reduction to period bounds

Since \(\varphi|_{[H]}\) and \(\Psi[Y]\) are (both) of rapid decay, the integral of their product \[I(Y) := \int_{[H]} \varphi \Psi_{\ast}[Y]\] converges absolutely. The following lemma reduces our task of bounding \(L(\pi,\tfrac{1}{2})\) to one of bounding the periods \(I(Y)\) for \(Y\) “fairly close” to the identity element \(1\).

By Lemma 5.1, our task reduces to estimating the period integrals \[\int _{[H]} \varphi \Psi_{\ast}[Y]\] for \(Y \in A_H(\mathbb{R})^0\) with \(T^{-\kappa} \leqslant Y_j \leqslant T^\kappa\), where \(\kappa \in (0,1/2)\) is a fixed parameter. Since we focus in what follows on individual values of \(Y\), we abbreviate \[\Psi_{\ast} := \Psi_{\ast}[Y], \quad \Psi := \Psi[Y].\]

§5.4. Matrix coefficients of \(\Psi\)

Let \(\mathcal{H}_H\) denote the spherical Hecke algebra for \(H(\mathbb{Z})\), as in §2.16.3.

§5.5. Introduction of an approximate idempotent

Let \(\omega_{0,\infty} \in C_c^\infty(G(\mathbb{R}))\) be the element produced by Theorem 3.1 (denoted there by “\(\omega_0\)”). Recall that it is supported in each fixed neighborhood of the identity element. In particular, it is supported in some fixed compact subset \(E_{G(\mathbb{R})} \subseteq G(\mathbb{R})\).

Let \(p\) be a prime number. For each \(q \in \{1, p, p^2, \dotsc\}\), we may define the congruence subgroup \(K_0(q)_p \leqslant G(\mathbb{Z}_p)\) consisting of matrices with bottom row congruent to \((0,\dotsc,0,\ast)\) modulo \(q\). Let \[d : K_0(q)_p \rightarrow \mathbb{Z}_p^\times / (\mathbb{Z}_p^\times \cap (1 + q \mathbb{Z}_p))\] denote the homomorphism assigning to a matrix the class of its lower-right entry. Let \(C(\pi_p) \in \{1, p, p^2, \dotsc \}\) denote the conductor of the component \(\pi_p\). Let \(\pi_p|_{Z} : \mathbb{Q}_p^\times \rightarrow \mathop{\mathrm{U}}(1)\) denote the central character of \(\pi_p\). Let \(\omega_{0,p} \in C_c^\infty(G(\mathbb{Q}_p))\) denote the function supported on \(K_0(C(\pi_p))_p\) and given there by the following \(L^2\)-normalized character: \[g \mapsto \mathop{\mathrm{vol}}(K_0(C(\pi_p))_p)^{-1/2} \pi_p|_{Z}(d(g))^{-1}.\] Note that this character is well-defined due to the fact the conductor of \(\pi_p|_{Z}\) divides that of \(\pi_p\) (§2.14.2). The significance of this normalization is that the self-convolution of \(\omega_{0,p}\) is bounded in magnitude by one (see Lemma 5.10 below).

Let \(\omega_0 = \otimes_{\mathfrak{p}} \omega_{0,\mathfrak{p}} \in C_c^\infty(G(\mathbb{A}))\) denote the tensor product of the functions defined above.

We have \(C(\pi_{\mathop{\mathrm{fin}}}) = \prod_p C(\pi_p)\). Set \(K_0(\pi) := \prod_p K_0(C(\pi_p))_p\). Then \[\label{eq:volk_0pi-=-cpiinfty}\tag{5.17} \mathop{\mathrm{vol}}(K_0(\pi)) = C(\pi_{\mathop{\mathrm{fin}}})^{-n+o(1)}.\]

We thereby reduce to estimating the integrals \[\int_{[H]} \pi(\omega_0) \varphi \cdot \Psi_{\ast}.\]

§5.6. Introduction of an amplifier

Let \(S\) be the finite set of places of \(\mathbb{Q}\) consisting of the infinite place \(\infty\) together with any finite primes at which \(\pi\) is ramified. Let \(\mathcal{H}_G^S\) denote the restricted product over \(p \notin S\) of the spherical Hecke algebras for \((G(\mathbb{Q}_p), G(\mathbb{Z}_p))\).

The algebra \(\mathcal{H}_G^S\) is generated by the elements \(T_p(a)\), defined for primes \(p \notin S\) and \((n+1)\)-tuples of integers \(a = (a_1,\dotsc,a_{n+1}) \in \mathbb{Z}^{n+1}\), given by the characteristic functions of the double cosets \[G(\mathbb{Z}_p) \mathop{\mathrm{diag}}(p^{a_1}, \dotsc, p^{a_{n+1}}) G(\mathbb{Z}_p) \subseteq G(\mathbb{Q}_p)\] (see for instance (P. D. Nelson 2023, sec. 5.2.1)). For \(j \geqslant 0\), we introduce the shorthand \(T_p[j] := T_p ((j,0,\dotsc,0))\) as well as its normalized variant \[t_{p,j} := p^{ \frac{n j}{2} } T_p[j].\]

Let \(L > 1\) with \(\log L / \log T\) fixed and positive (to be determined later, in §5.11). Let \(\mathbb{P}\) denote the set of all primes \(p \notin S\) that lie in the interval \([L,2 L]\). By the divisor bound and the prime number theorem, we then have \[T^{-o(1)} L \leqslant|\mathbb{P}| \leqslant L.\]

Let \(\lambda_\pi : \mathcal{H}_G^S \rightarrow \mathbb{C}\) denote the algebra homomorphism describing the eigenvalues for the action of \(\mathcal{H}_G^S\) on the \(\prod_{p \notin S} G(\mathbb{Z}_p)\)-invariant subspace of \(\pi\). By (P. D. Nelson 2023, (5.6)) (which relies in turn on (Blomer and Maga 2016, Cor 4.3)), we have \(\sum_{j=1}^{n+1} \lvert \lambda_{\pi}(t_{p,j}) \rvert \gg 1\) for each \(p \notin S\). By the pigeonhole principle, we may thus find \(j_0 \in \{1, \dotsc, n+1\}\) so that \[\sum_{p \in \mathbb{P}} \lvert \lambda_{\pi}(t_{p,j_0}) \rvert \gg |\mathbb{P}|.\]

Write \(\omega_{0,S} := \otimes_{p \in S} \omega_{0,p}\). Then \(\omega_0 = \omega_{0,S} \otimes (\otimes_{p \notin S} \omega_{0,p})\), with \(\omega_{0,p}\) unramified for each \(p \notin S\). For \(p \in \mathbb{P}\), set \(x_p := \overline{\mathop{\mathrm{sgn}}(\lambda_\pi(t_{p,j_0}))}\), where \(\mathop{\mathrm{sgn}}(0) := 0\) and \(\mathop{\mathrm{sgn}}(z) := z/|z|\) for \(z \neq 0\). Then \(|x_p| \leqslant 1\), and \[\pi \left( \omega_{0,S} \otimes \sum _{p \in \mathbb{P} } x_p t_{p,j_0} \right) \varphi = \left( \sum _{p \in \mathbb{P} } |\lambda_\pi(t_{p,j_0}) | \right) \pi(\omega_0) \varphi.\]

By combining the above identities and estimates with \((5.17)\) and \((5.18)\), we deduce the following implication: \[\begin{align} &\mathcal{R} := \left\lvert \int _{[H]} \pi \left( \omega_{0,S} \otimes \sum _{p \in \mathbb{P} } x_p t_{p,j_0} \right) \varphi \cdot \Psi_{\ast} \right\rvert^2 \ll T^{n/2 - 2 \delta } L^2 \label{eq:leftlvert-int-_h}\tag{5.20} \\ &\quad \implies \int _{[H]} \varphi \Psi_{\ast} \ll C(\pi_{\mathop{\mathrm{fin}}})^{n/2} T^{n/4 - \delta + o(1)}. \nonumber \end{align}\] Our task thereby reduces to suitably estimating the quantity \(\mathcal{R}\).

§5.7. Pretrace inequality and multiplication of Hecke operators

Following (P. D. Nelson 2023, sec. 6.3) line-by-line, we apply the pretrace inequality given in (P. D. Nelson 2023, Lemma 5.5) (which explicitly avoids requiring compactness for the quotients \([G]\) and \([H]\)) and multiply the Hecke operators arising in the geometric expansion. We summarize the estimate obtained in this way in the lemma stated below.

We henceforth focus on individual values of \(\mathfrak{j}, p_1, p_2\) appearing on the RHS of \((5.25)\), abbreviate \[\omega := \omega[p_1,p_2,\mathfrak{j}],\] and aim to estimate \(\mathcal{R}(\omega)\). As a first step, we decompose \[\label{eq:mathc-=-mathc}\tag{5.26} \mathcal{R}(\omega) = \mathcal{M} + \mathcal{E},\] where \(\mathcal{M}\) denotes the contribution from \(\gamma\) lying in \(H(\mathbb{Q}) \hookrightarrow \bar{G}(\mathbb{Q})\) and \(\mathcal{E}\) denotes the remaining contribution.

We will estimate \(\mathcal{M}\) in §5.8 and \(\mathcal{E}\) in §5.10. We then combine the estimates so obtained in §5.11.

We conclude this subsection with some basic estimates for the factors \(\omega_\mathfrak{p} ^\sharp\), to be applied below.

§5.8. Main term estimates

§5.9. Matrix counting

We pause to record some estimates to be applied in the following subsection. We note that the quantitative details of this subsection are relevant only for explicating the precise numerical savings \(\delta_n^\sharp\) obtained in our results. The reader who is primarily interested in understanding the proof of a qualitative subconvex bound is thus encouraged to skim this subsection.

Recall that \((G,H) = ({\mathop{\mathrm{GL}}}_{n+1},{\mathop{\mathrm{GL}}}_n)\) over \(\mathbb{Z}\), with \(n \geqslant 1\) fixed. For \(t \in A_H(\mathbb{R})^0\), we set \[t ^\dagger := \prod_{i=1}^n \max(t_i, t_i^{-1}).\] Recall that \(T \ggg 1\) and \(\bar{G} = G/Z\).

For the following lemmas, we retain the setting of Definition 5.13, we fix a compact symmetric (i.e., inversion-invariant) neighborhood \(E \subseteq \bar{G}(\mathbb{R})\) of the identity element, we fix a compact symmetric subset \(\Omega \subseteq H(\mathbb{R})\), and we abbreviate \[\Sigma := \Sigma(t,u,\ell, \ell', X, \Omega E \Omega).\]

We conclude §5.9 by recording the combinatorial argument required to complete the proof of Lemma 5.15. The following discussion will not otherwise be applied in this paper.

§5.10. Error term estimates

By definition, \[\label{eq:mathcale-:=E--int}\tag{5.47} \mathcal{E} := \int _{x, y \in [H]} \overline{\Psi_{\ast}(x)} \Psi_{\ast}(y) \sum _{\gamma \in \bar{G}(\mathbb{Q}) - H(\mathbb{Q})} \omega^\sharp(x^{-1} \gamma y) \, d x \, d y.\] We recall the notation:

• \(\Psi_{\ast}\) is shorthand for \(\Psi_{\ast}[Y]\), the pseudo Eisenstein series attached in §5.2 to some \(Y \in A_H(\mathbb{R})^0\) satisfying \(T^{-\kappa} \leqslant Y_j \leqslant T^{\kappa}\) for some fixed \(\kappa \in (0,1/2)\).

• \(\omega^\sharp \in C_c^\infty(\bar{G}(\mathbb{A}))\), \(\bar{G} = G/Z\), is the integral over \(Z(\mathbb{A})\), weighted by the central character of \(\pi\), of the test function \(\omega \in C_c(G(\mathbb{A}))\) attached to some triple \((p_1,p_2,\mathfrak{j})\) (see §5.7), which is in turn an “amplified” self-convolution of the test function \(\omega_{0,\infty}\) produced by Theorem 3.1.

The main result of §5.10 is as follows.

The proof will be broken up into several steps, and occupies the remainder of §5.10. The crucial ingredients are

• the “bilinear forms” or “transversality” estimate supplied by part \((vii)\) of Theorem 3.1,

• the “growth bounds for Eisenstein series” estimate supplied by Theorem 4.1, and

• the simple estimates for matrix counts given above.

Siegel domains, triangle inequality

We begin by bounding the integrand in absolute value and applying the Siegel domain majorization afforded by Proposition 2.25 to both variables. Recall that \(A_H(\mathbb{R})^0_{\geqslant 1}\) denotes the set of \(N\)-dominant elements of the connected component of the diagonal subgroup \(A_H(\mathbb{R})\). Each factor in the integrand of the integral \((5.47)\) defining \(\mathcal{E}\) is \(\prod_{p} H(\mathbb{Z}_p)\)-invariant in both variables. We deduce that there is a fixed compact subset \(\Omega\) of \(H(\mathbb{R})\) so that \[\label{eq:mathcale-ll-int}\tag{5.48} \mathcal{E} \ll \int _{t, u \in A_H(\mathbb{R})^0_{\geqslant 1} } \sum_{\gamma \in \bar{G}(\mathbb{Q}) - H(\mathbb{Q})} I_{t, u}(\gamma) \, \frac{d t \, d u}{\delta_H(t) \delta_H(u)},\] where \[I_{t,u}(\gamma) := \int _{x \in t \Omega} \int _{y \in u \Omega} \left\lvert \Psi_{\ast}(x) \Psi_{\ast}(y) \omega^\sharp (x ^{-1} \gamma y) \right\rvert \,d x \,d y,\] where \(d x\) and \(d y\) denote our chosen Haar measure on \(H(\mathbb{R})\).

In what follows, and unless otherwise stated, \(t\) and \(u\) denote elements of \(A_H(\mathbb{R})^0_{\geqslant 1}\).

Factorization

By part \((i)\) of Theorem 3.1 and the \(H(\mathbb{R})\)-equivariance of the association \(f \mapsto \Psi\), we may write \[\label{eq:psiy-=-psi_0}\tag{5.49} \Psi_{\ast}(y) = (\Psi \ast \phi_{Z_H})(y) := \int _{z \in Z_H(\mathbb{R})} \Psi(y z ^{-1}) \phi_{Z_H}(z) \, d z,\] where

• \(\phi_{Z_H} \in C_c^\infty(Z_H(\mathbb{R}))\) is fixed,

• \(\Psi : [H] \rightarrow \mathbb{C}\), attached as in §2.16 to the element \(f \in \mathcal{S}^e(U_H(\mathbb{R}) \backslash H(\mathbb{R}))^{W_H}\) furnished by Theorem 3.1 via the recipe \[\Psi := \mathop{\mathrm{Eis}}[L(Y) \Phi[f]] = \mathop{\mathrm{Eis}}[\Phi[L(Y) f]] \in \mathcal{S}([H]),\] has the same essential properties as \(\Psi_{\ast}\) (indeed, all except the explicit factorization property \((5.49)\)); in particular, it is smooth and of rapid decay.

Application of transversality

Here we apply the “bilinear forms” estimate afforded by part \((vii)\) of Theorem 3.1, whose proof relies in turn on the volume estimates established in (P. D. Nelson 2023, secs. 15–16).

Recall that part \((vii)\) of Theorem 3.1 asserts “for each fixed compact \(\Omega_{H} \subseteq H(\mathbb{R})\), there exists a compact \(\Omega_{H}' \subseteq H(\mathbb{R})\) so that \(\dotsb\).” Let \(\Omega ' \subseteq H(\mathbb{R})\) denote the set obtained as “\(\Omega_{H}'\)” by taking for “\(\Omega_H\)” our given set \(\Omega\). By enlarging \(\Omega '\) if needed, we may and shall assume that \(\Omega '\) is symmetric (i.e., invariant by inversion). For \(t \in A_H(\mathbb{R})^0_{\geqslant 1}\), we define the seminorms \(\|.\|_t\) on \(L^2(H(\mathbb{Z}) \backslash H(\mathbb{R}))\) as in \((4.1)\), but using \(\Omega '\), i.e., \[\|\phi\|^2_t := \int _{t \Omega' } |\phi|^2 = \int _{h \in \Omega' \subseteq H(\mathbb{R})} |\phi(t h)|^2 \, d h,\] where the integral is taken with respect to the Haar measure on \(H(\mathbb{R})\).

For \(t, u \in A_H(\mathbb{R})^0_{\geqslant 1}\), set \[\Sigma_{t,u} := \left\{ \gamma \in \bar{G}(\mathbb{Q}) - H(\mathbb{Q}) : \text{$\omega^\sharp(x^{-1} \gamma y) \neq 0$ for some $(x,y) \in t \Omega \times u \Omega $} \right\}.\] For \(0 < X \leqslant 1\), let \(\Sigma_{t,u}^X \subseteq \Sigma_{t,u}\) denote the subset defined by the further condition \(d_H(t^{-1} \gamma u) \leqslant X\). For example, \(\Sigma_{t,u}^1 = \Sigma_{t,u}\). Define \[\label{eq:i-:=-int}\tag{5.50} I := \int _{t, u \in A_H(\mathbb{R})^0_{\geqslant 1}} \|\Psi\|_t \|\Psi\|_u \sum _{\gamma \in \Sigma_{t,u}} \min \left( T^{1/2}, \frac{1}{d_H(t^{-1} \gamma u)} \right) \, \frac{d t \, d u}{\delta_H(t) \delta_H(u)}.\] From \((5.48)\) and Lemma 5.22, we see that \[\mathcal{E} \ll T ^{n/2 - 1/2 + o(1)} L^{- n \mathfrak{j}} I.\] The proof of Proposition 5.21 thereby reduces to that of the estimate \[\label{eq:i-ll-tnkappa}\tag{5.51} I \ll T^{n \kappa + o(1)} L^{(3 (n+1)^2 + 2 n) \mathfrak{j}}.\]

Application of matrix counting

The sets \(\Sigma_{t,u}^X\) are contained in those considered in §5.9. More precisely, let \(E \subseteq \bar{G}(\mathbb{R})\) denote the image of the set \(\{x y^{-1} : x,y \in E_{G(\mathbb{R})}\}\), where \(E_{G(\mathbb{R})} \subseteq G(\mathbb{R})\) is any fixed symmetric compact subset that contains the support of \(\omega_0\).

By construction, the sets \(E\) and \(\Omega\) satisfy the assumptions required by §5.9. The results of that section thus apply here, giving the following.

§5.10.5. Application of crude bounds for Eisenstein series

We now introduce some a priori bounds concerning \(\|\Psi\|_t\). These will permit us to reduce the range of pairs \((t,u)\) to consider.

Clean-up

Since an additive error \(T^{-C}\), with \(C\) large enough but fixed, is acceptable for the purposes of proving \((5.51)\), we reduce to showing that the estimate \((5.51)\) holds for the modification \(I_{\mathcal{C}}\) of \(I\) obtained by restricting the integral over \(t,u\) to some set \(\mathcal{C}\) as in the conclusion of Lemma 5.27: \[I_{\mathcal{C}} := \int _{ t, u \in \mathcal{C} } \|\Psi\|_t \|\Psi\|_u \sum _{\gamma \in \Sigma_{t,u}} \min \left( T^{1/2}, \frac{1}{d_H(t^{-1} \gamma u)} \right) \, \frac{d t \, d u}{\delta_H(t) \delta_H(u)}.\] We henceforth focus on one such set \(\mathcal{C}\). By Cauchy–Schwarz, we have \(I \leqslant I_0\), where \[I_0 := \int _{t, u \in \mathcal{C} } \|\Psi\|_u^2 \sum _{\gamma \in \Sigma_{t,u}} \min \left( T^{1/2}, \frac{1}{d_H(t^{-1} \gamma u)} \right) \, \frac{d t \, d u}{\delta_H(t) \delta_H(u)}.\]

The proof of our earlier reduction \((5.51)\) of Proposition 5.21 thereby reduces to that of the estimate \[\label{eq:int-_t-in}\tag{5.70} \int _{t \in \mathcal{C} } \|\Psi\|_t^2 t ^\dagger \, \frac{d t}{\delta_H(t)} \ll T^{n \kappa + o(1)}.\]

§5.10.7. Application of growth bounds for Eisenstein series

We now apply Theorem 4.1 to deduce the key estimate \((5.70)\), following some preliminary lemmas.

In view of our assumption that \(Y_j \in [T^{-\kappa}, T^{\kappa}]\), we have \(\det(Y) \in [T^{- n \kappa}, T^{n \kappa}]\). The desired estimate \((5.70)\) is thus an immediate consequence of the following.

§5.11. Optimization

We now optimize (as in (P. D. Nelson 2023, sec. 6.6)). The results of §5.8 and §5.10 combine to give \[\mathcal{M} + \mathcal{E} \ll T^{n/2 + o(1)} ( L^{-\mathfrak{j}}+ T^{-1/2 + n \kappa} L^{(3(n+1)^2 + n) \mathfrak{j}})\] We insert this bound into the decompositions \((5.25)\) and \((5.26)\) to see that the quantity \(\mathcal{R}\) defined in \((5.20)\) satisfies, for fixed \(\varepsilon> 0\), \[\begin{align} \frac{\mathcal{R}}{L^2 T^{n/2}} &\ll T^{o(1)} \sum_{\mathfrak{j}=1}^{n+1} \left( L^{-\mathfrak{j}} + T^{-1/2 + n \kappa} L^{(3(n+1)^2 + n) \mathfrak{j}} \right) \\ &\quad + T^{o(1)} L^{-1} \sum _{\mathfrak{j} = 0}^{n+1} \left( L^{- \mathfrak{j}} + T^{-1/2 + n \kappa} L^{(3(n+1)^2 + n) \mathfrak{j}} \right). \end{align}\] By the results of §5.3 and §5.6, we have the following implication, for fixed \(\kappa \in (0, 1/2)\) and \(\delta > 0\): \[\frac{\mathcal{R}}{L^2 T^{n/2}} \ll T^{- 2 \delta + o(1)} \implies L(\pi,\tfrac{1}{2})^n \ll C(\pi_{\mathop{\mathrm{fin}}})^{n/2} T^{n(n+1)/4 - \min(\kappa/2,\delta) + o(1)}.\] To optimize, we take \(\kappa = 2 \delta\), which forces \(\delta < 1/4\), and \(L = T^{2 \delta}\). Our estimate for \(\mathcal{R}\) then reads \[\frac{\mathcal{R}}{L^2 T^{n/2}} \ll T^{- 2 \delta + o(1)} ( 1 + T^{\alpha} )\] with \[\begin{align} \alpha &:= 2 \delta -1/2 + n \kappa + (3 (n+1)^2 + n)(n+1) \delta \\ &= (2 + 2 n + (3 (n+1)^2 + n)(n+1)) \delta - 1/2. \end{align}\] We have \(\alpha \leqslant 0\) provided that \[\delta \leqslant\frac{1}{2 (2 + 2 n + (3 (n+1)^2 + n)(n+1))},\] in which case we obtain \[L(\pi,\tfrac{1}{2})^n \ll C(\pi_{\mathop{\mathrm{fin}}})^{n/2} T^{n(n+1)/4 - \delta + o(1)}.\] Equivalently, for any fixed \[\delta < \delta_{n+1} ^\sharp := \frac{2}{n(n+1) (2 + 2 n + (3 (n+1)^2 + n)(n+1))}.\] we have \[L(\pi,\tfrac{1}{2}) \ll C(\pi_{\mathop{\mathrm{fin}}})^{1/2} T^{\frac{n+1}{4}(1 - \delta)}.\] We verify readily that \(\delta_n ^\sharp\) simplifies to the fraction indicated in the introduction, i.e., that, \[n(n+1) (2 + 2 n + (3 (n+1)^2 + n)(n+1)) = 3 ( n+1)^5 - 2 (n+1)^4 - (n+1)^2.\] 0◻

This completes the deduction of Theorem 2.1 from Theorems 3.1 and 4.1. The remainder of the paper is devoted to proofs of the latter results. Theorem 3.1 is the subject of Parts and , while Theorem 4.1 is taken up in Part .

§2. Construction of test vectors

Here we develop the proof of Theorem 3.1. That result asserts the existence of, among other things, vectors \(f \in \mathcal{S}^e(U_H \backslash H)^{W_H}\) and \(W \in \mathcal{W}(\pi,\psi)\) having favorable properties. The main point of Part is to construct \(f\) and \(W\). The construction of \(W\) is given in §8, that of \(f\) in §9. Some ingredients and language common to both constructions are given in §6.

§6. Lie-theoretic preliminaries

Let \(F\) be a fixed completion of \(\mathbb{Q}\), thus \(F\) is either \(\mathbb{R}\) or \(\mathbb{Q}_p\) for some fixed prime \(p\). This section concerns Lie theory over \(F\). We remark that any algebraic group over a local field of characteristic zero may be regarded as a Lie group over some such \(F\), so there is little loss of generality in our assumptions. Our aim is to record some basic properties of “microlocalized” functions on Lie groups over \(F\). It seemed clearest to develop the main arguments first in the category of analytic manifolds, and then to specialize at the end to the setting of Lie groups.

The \(p\)-adic case of the present discussion is not applied in this paper. It could be useful for generalizating the results of this paper to the depth aspect, although simpler approaches are likely possible for that. Our main motivation for discussing the \(p\)-adic case is that many statements and proofs are simpler in that case, and we feel that including them helps motivate the ideal that we are striving for with our treatment of the real case.

The main exports of this section are Definition 6.38 and Propositions 6.37, 6.46, 6.56, 6.57 and 6.58. These give a well-defined notion of, roughly, a “smooth modulated bump concentrated on \(p + \operatorname{O}(T^{-1/2})\) and oscillating with frequency \(\ll T\)” with good functorial properties (under pushforward, the action of a Lie group, convolution, etc.). The proofs are direct applications of Taylor’s theorem, using that the quadratic and higher terms do not contribute significantly in the indicated range.

Preliminaries

§6.1.1. Pontryagin duality for vector spaces

In this section, “vector space” always means “finite-dimensional vector space over \(F\).”

Let \(V\) be a vector space. We denote by \(V^*\) the dual space, consisting of \(F\)-linear maps \(V \rightarrow F\). We denote by \(V^\wedge\) the Pontryagin dual, consisting of continuous group homomorphisms \(V \rightarrow \mathop{\mathrm{U}}(1)\), with \(\mathop{\mathrm{U}}(1) \subseteq \mathbb{C}^\times\) the circle group. Using (say) the “standard” nontrivial unitary character \(\psi_F : F \rightarrow \mathop{\mathrm{U}}(1)\) (§2.11), we may identify \(V^*\) with \(V^\wedge\): to \(\ell \in V^*\) corresponds \([V \ni v \mapsto \psi_F(\ell(v)) \in \mathop{\mathrm{U}}(1)] \in V^\wedge\). In particular, \(V^\wedge\) is naturally a vector space of the same dimension as \(V\). We will generally avoid making such identifications here.

As a matter of notation, given \(\xi \in V^\wedge\), we denote by \[\psi_\xi : V \rightarrow \mathop{\mathrm{U}}(1)\] the homomorphism canonically associated to it. In the archimedean case \(F = \mathbb{R}\), we define the bilinear pairing \[\langle , \rangle : V \otimes V^\wedge \rightarrow i \mathbb{R}\] to be the differential of the canonical pairing \(V \times V^\wedge \rightarrow \mathop{\mathrm{U}}(1)\), i.e., \[\langle x, \xi \rangle := \partial_{t=0} \psi_{t \xi}(x) = \partial_{t=0} \psi_{\xi}(t x),\] so that \(\psi_{\xi}(x) = \exp(\langle x, \xi \rangle)\). In this way, \(V^\wedge\) identifies naturally with the imaginary dual of \(V\).

§6.1.2. Local maps

Given topological spaces \(X\) and \(Y\), a local map \(f : X % \mathrel{% \mathpalette{\da@xarrow{}{}{}\mathchar"0\hexnumber@\symAMSa 4B {\,}{}}{}% }% Y\) is a pair \((\mathop{\mathrm{dom}}(f), f)\), where

• \(\mathop{\mathrm{dom}}(f)\), called the domain of \(f\), is an open subset of \(X\), and

• \(f : \mathop{\mathrm{dom}}(f) \rightarrow Y\) is a continuous map.

We say that \(f\) is defined at an element \(p\) of \(X\) if \(p \in \mathop{\mathrm{dom}}(f)\).

Given two local maps \(f : X % \mathrel{% \mathpalette{\da@xarrow{}{}{}\mathchar"0\hexnumber@\symAMSa 4B {\,}{}}{}% }% Y\) and \(g : Y % \mathrel{% \mathpalette{\da@xarrow{}{}{}\mathchar"0\hexnumber@\symAMSa 4B {\,}{}}{}% }% Z\), we may define their composition \(g \circ f : X % \mathrel{% \mathpalette{\da@xarrow{}{}{}\mathchar"0\hexnumber@\symAMSa 4B {\,}{}}{}% }% Z\), which is a local map with (possibly empty) domain \[\mathop{\mathrm{dom}}(g \circ f) := \mathop{\mathrm{dom}}(f) \cap f^{-1}(\mathop{\mathrm{dom}}(g)).\]

§6.1.3. Multi-index notation

Let \(V\) be a vector space. By choosing a basis for \(V\), we may identify it with \(F^n\) for some \(n\). By a multi-index for \(V\), we mean an \(n\)-tuple \(\alpha = (\alpha _1, \dotsc, \alpha _n ) \in \mathbb{Z} _{\geqslant 0} ^n\) of nonnegative integers. For a multi-index \(\alpha\) and \(x \in V\), we set \(x^\alpha := \prod_i x_i^{\alpha_i}\) and \(|\alpha| := \sum_i \alpha_i\). In the archimedean case (thus \(V = \mathbb{R}^n\)), we denote by \(\partial^\alpha := \prod_i \frac{\partial^{\alpha_i}}{\partial ^{\alpha_i} x_i}\) the translation-invariant differential operator on \(C^\infty(V)\) associated to a multi-index \(\alpha\).

§6.1.4. Analytic maps

Let \(V\) be a vector space. By choosing a basis, we may identify it with \(F^n\) for some \(n \in \mathbb{Z}_{\geqslant 0}\). We denote by \(|.|\) the norm on \(F^n\) given by \(|x| := \max_{i} |x_i|\).

Let \(V_1\) and \(V_2\) be vector spaces, and let \(f : V_1 % \mathrel{% \mathpalette{\da@xarrow{}{}{}\mathchar"0\hexnumber@\symAMSa 4B {\,}{}}{}% }% V_2\) be a local map. We recall that

• \(f\) is analytic at a point \(p \in \mathop{\mathrm{dom}}(f)\) if it may be expressed in some neighborhood of \(p\) by a convergent power series, and

• \(f\) is analytic if it is analytic at every point of its domain.

By choosing bases, we may identify \(V_i\) with \(F^{n_i}\) for some \(n_i \in \mathbb{Z}_{\geqslant 0}\). For analytic \(f\), the power series representation of \(f\) at \(p \in \mathop{\mathrm{dom}}(f)\) may be written \[\label{eq:phip-+-x}\tag{6.1} f(p + x) = \sum _{\alpha } c_\alpha x^\alpha,\] where \(\alpha\) runs over multi-indices for \(V_1\) and \(c_\alpha \in V_2 = F^{n_2}\). The analyticity condition implies that there exist \(C \geqslant 0\) and \(r > 0\) so that for all \(\alpha\), \[\label{eq:c_alpha-leq-c_0}\tag{6.2} |c_\alpha| \leqslant C r^{-|\alpha|}.\] We say then that \(f\) is \((C,r)\)-controlled at \(p\).

The following elementary results may be verified by effectivizing the proofs of (Serre 2006, 1500:p69, Theorem) and (Serre 2006, 1500:p70, Theorem), respectively.

§6.1.5. Manifolds

We recall that an analytic manifold \(M\) over \(F\) is a topological space equiped with a maximal atlas of coordinate charts, taking values in finite-dimensional vector spaces over \(F\), for which the transition maps are analytic (see (Serre 2006, vol. 1500, pt. II, Chapter III) for details). In what follows, “manifold” always means “analytic manifold over \(F\),” and “chart” refers to the analytic structure.

A morphism \(f : M_1 \rightarrow M_2\) of manifolds is an analytic map, i.e., a map that is described in local coordinates, with respect to every pair (equivalently, sufficiently many pairs) of charts \(c_1\) on \(M_1\) and \(c_2\) on \(M_2\), by an analytic map of vector spaces as in §6.1.4. We then define isomorphisms in the usual way as the morphisms that admit two-sided inverses.

For each \(p \in M\), we denote by \(T_p(M)\) the tangent space, \(T_p^*(M)\) the cotangent space, and \(T_p^\wedge(M)\) the Pontryagin dual of the tangent space.

§6.1.6. Pointed manifolds

By a pointed manifold \((M,p)\), we mean a pair consisting of a manifold \(M\) and a point \(p \in M\). A morphism of pointed manifolds \((M_1, p_1) \rightarrow (M_2, p_2)\) is a morphism of manifolds \(M_1 \rightarrow M_2\) that maps \(p_1\) to \(p_2\). A local morphism of pointed manifolds \(f : (M_1,p_1) % \mathrel{% \mathpalette{\da@xarrow{}{}{}\mathchar"0\hexnumber@\symAMSa 4B {\,}{}}{}% }% (M_2,p_2)\) is a local map for which

• \(p_1\) belongs to the domain of \(f\),

• \(f\) is analytic on its domain, and

• \(f(p_1) = p_2\).

It would be cleaner for some purposes to pass to germs, but we do not do so here.

A composition of local morphisms of pointed manifolds, defined as in §6.1.2, is likewise a local morphism.

Let \(f : (M_1,p_1) % \mathrel{% \mathpalette{\da@xarrow{}{}{}\mathchar"0\hexnumber@\symAMSa 4B {\,}{}}{}% }% (M_2,p_2)\) and \(g : (M_2,p_2) % \mathrel{% \mathpalette{\da@xarrow{}{}{}\mathchar"0\hexnumber@\symAMSa 4B {\,}{}}{}% }% (M_1,p_1)\) be local morphisms of pointed manifolds. We say that \(g\) is an inverse for \(f\) if \(\mathop{\mathrm{dom}}(g) = \mathop{\mathrm{image}}(f)\), \(\mathop{\mathrm{image}}(g) = \mathop{\mathrm{dom}}(f)\), and \(f \circ g\) (resp. \(g \circ f\)) coincides with the identity map on \(\mathop{\mathrm{dom}}(g)\) (resp. \(\mathop{\mathrm{dom}}(f)\)); in that case, \(g\) is uniquely determined, and we denote it by \(f^{-1}\). We say that \(f\) is a local isomorphism if it admits an inverse. We note that if \(f\) satisfies the weaker condition that there exists \(g\) for which \(f \circ g\) (resp.) \(g \circ f\) coincides with the identity map on some neighborhood of \(p_2\) (resp. \(p_1\)), then by shrinking the domain of \(f\) to a suitable neighborhood of \(p\), we obtain a local isomorphism in the indicated sense.

In this terminology, a chart \(c\) for \(M\) at \(p\) is a local isomorphism \((M,p) % \mathrel{% \mathpalette{\da@xarrow{}{}{}\mathchar"0\hexnumber@\symAMSa 4B {\,}{}}{}% }% (V,q)\) for some vector space \(V\). By composing with a translation, we may always arrange that \(q\) is the origin \(0 \in V\). The derivative of \(c'(p)\) of \(c\) at \(p\) induces a linear isomorphism between \(T_p(M)\) and \(T_{c(p)} V = V\). There is thus little loss of generality in taking \(c\) of the form \[\label{eq:c-:-m}\tag{6.3} c : (M,p) % \mathrel{% \mathpalette{\da@xarrow{}{}{}\mathchar"0\hexnumber@\symAMSa 4B {\,}{}}{}% }% (T_p(M), 0), \quad c'(p) = \text{ the identity map on } T_p(M).\] For example, if \(M\) is a Lie group and \(p\) the identity element, then the logarithm defines a chart of this form.

§6.1.7. Inverse function theorem

Let \(\rho : (M_1, p_1) % \mathrel{% \mathpalette{\da@xarrow{}{}{}\mathchar"0\hexnumber@\symAMSa 4B {\,}{}}{}% }% (M_2, p_2)\) be a morphism of pointed manifolds. We recall that \(\rho\) is submersive at \(p_1\) if the map \((d \rho)_{p_1} : T_{p_1}(M_1) \rightarrow T_{p_2}(M_2)\) is surjective. In that case, the implicit function theorem implies that there are vector spaces \(V_j\), coordinate charts \(c_j : (M_j, p_j) % \mathrel{% \mathpalette{\da@xarrow{}{}{}\mathchar"0\hexnumber@\symAMSa 4B {\,}{}}{}% }% (V_j,0)\) and a surjective linear map \(\ell : V_1 \rightarrow V_2\) so that the diagram \[\begin{CD} M_1 @>\phi>>M_2\\ @Vc_1VV @VVc_2V \\ V_1 @>>\ell>V_2\\ \end{CD}\] is defined and commutes in some neighborhood of \(p_1\).

We recall that a morphism \(\rho : M_1 \rightarrow M_2\) of manifolds is submersive if for each \(p \in M\), the induced morphism \((M_1, p) \rightarrow (M_2, \rho(p))\) is submersive in the above sense.

Smooth measures

Let \(V\) be a vector space. We may speak of smooth functions \(V \rightarrow \mathbb{C}\). In the archimedean case, this carries the usual meaning (“smooth over \(\mathbb{R}\)”). In the non-archimedean case, it means “locally constant;” we will primarily consider compactly-supported measures, for which it is equivalent to ask for invariance under some open subgroup of \(V\). By a smooth measure on \(V\), we mean a complex-valued measure obtained by multiplying a Haar measure by a smooth function.

Let \(M\) be a manifold. By a smooth measure \(\gamma\) on \(M\), we mean a complex-valued measure with the following property: for each \(p \in M\), there is a coordinate chart \(c : (M,p) % \mathrel{% \mathpalette{\da@xarrow{}{}{}\mathchar"0\hexnumber@\symAMSa 4B {\,}{}}{}% }% (V, 0)\) and a smooth measure \(\gamma_c\) on \(V\) so that \[c_*(\gamma|_{\mathop{\mathrm{dom}}(c)}) = \gamma_c |_{\mathop{\mathrm{image}}(c)}.\] We denote by \(\mathcal{M}(M)\) (resp. \(\mathcal{M}_c(M)\)) the space of smooth measures (resp. the space of smooth measures of compact support) on \(M\).

Let \(\phi : M_1 \rightarrow M_2\) be an isomorphism of manifolds. The pushforward map \(\phi_*\) on measures induces isomorphisms \(\phi_* : \mathcal{M}(M_1) \rightarrow \mathcal{M}(M_2)\), preserving the compactly-supported subsets. We write \[\phi^* := \phi_*^{-1} = (\phi^{-1})_* : \mathcal{M}(M_2) \rightarrow \mathcal{M}(M_1)\] for the inverse map. It will be convenient to describe such maps in local coordinates. In doing so, we may assume that each \(M_i\) is an open subset of some vector space \(V_i\). Choose Haar measures \(d x\) on \(V_1\) and \(d y\) on \(V_2\). Then, writing \(y = \phi(x)\), we have \[d y = |\phi '(x)| \, d x,\] where \(|\phi '(x)|\) denotes the normalized absolute value of the Jacobian of \(\phi\) with respect to \(d x\) and \(d y\); if we identify both \(V_1\) and \(V_2\) with \(F^n\) and choose \(d x\) and \(d y\) compatibly, then \(|\phi '(x)|\) is the absolute value of the determinant of the matrix of partial derivatives. Let \(\gamma \in \mathcal{M}(M_2)\). We identify \(\gamma\) (resp. \(\phi^* \gamma\)) with a smooth function on \(M_2\) (resp. \(M_1\)) by dividing by the Haar measure \(d y\) (resp. \(d x\)). Then \[\label{eq:phi-gammax-=jacobian}\tag{6.4} \phi^* \gamma(x) = |\phi '(x)| \gamma(\phi(x)).\]

In the archimedean case, given a vector field \(X\) on \(M\) and a smooth measure \(\gamma\), we may define the smooth measure \(X \gamma\) via duality. We explicate this definition in local coordinates. Suppose that \(M\) is an open subset of \(\mathbb{R}^n\). Identify \(\gamma\) with a smooth function on \(M\) by dividing by Lebesgue measure. Write \(\nabla_X \gamma\) for the ordinary derivative of the smooth function \(\gamma\) with respect to \(X\). Let \(\operatorname{div}X : M \rightarrow \mathbb{R}\) denote the divergence of \(X\) with respect to the given coordinates. Then the smooth measure \(X \gamma\), identified with a smooth function by dividing by Lebesgue measure, is given by \[X \gamma = \nabla_X \gamma + (\operatorname{div}X) \gamma.\] The verification of this identity is an application of integration by parts.

Let \(\rho : M_1 \rightarrow M_2\) be a submersive morphism of manifolds. Using the implicit function theorem, we see that the pushforward map \(\rho_*\) on measures induces a surjective linear map \[\rho_* : \mathcal{M}_c(M_1) \rightarrow \mathcal{M}_c(M_2).\]

Some nonstandard formalism

Infinitesimal neighborhoods

Let \(X\) be a fixed topological space (e.g., a fixed manifold \(M\)).

We refer to the element \(q\), as in the conclusion of the lemma, as the fixed part of \(p\). This element has the following property: a fixed open subset \(E\) of \(X\) contains \(q\) if and only if it contains \(p + o(1)\).

§6.2.2. Control

We may prepend the adjective “fixed” to any of the objects defined above: “fixed manifold,” “fixed chart,” etc. To allow more flexibility, we define “controlled” analogues of such objects.

The terminology applies in particular to pointed vector spaces \((V,p)\).

The basic axioms of a category are satisfied (although for set-theoretic reasons, we will not introduce that language explicitly):

We now extend the above discussion to manifolds. Let \(f : M_1 \rightarrow M_2\) be a morphism (i.e., analytic map) between manifolds \(M_1\) and \(M_2\). Thus, for each \(p_1 \in M_1\), each chart \(c_1 : (M_1, p_1) % \mathrel{% \mathpalette{\da@xarrow{}{}{}\mathchar"0\hexnumber@\symAMSa 4B {\,}{}}{}% }% (V_1, 0)\) and each chart \(c_2 : (M_2, f(p_2)) % \mathrel{% \mathpalette{\da@xarrow{}{}{}\mathchar"0\hexnumber@\symAMSa 4B {\,}{}}{}% }% (V_2,0)\), the composition \[f_{c_1,c_2} : (V_1,0) % \mathrel{% \mathpalette{\da@xarrow{}{c_1^{-1}}{}\mathchar"0\hexnumber@\symAMSa 4B {\,}{}}{}% }% (M_1, p_1) % \mathrel{% \mathpalette{\da@xarrow{}{f}{}\mathchar"0\hexnumber@\symAMSa 4B {\,}{}}{}% }% (M_2, p_2) % \mathrel{% \mathpalette{\da@xarrow{}{c_2}{}\mathchar"0\hexnumber@\symAMSa 4B {\,}{}}{}% }% (V_2,0)\] is analytic in the sense of §6.1.4.

§6.2.3. Distances

We henceforth write \({\mathop{\mathrm{dist}}}_M\), or \(\mathop{\mathrm{dist}}\) for short, for some function of the form \({\mathop{\mathrm{dist}}}_{c,|.|}\) as above, which is thus “essentially well-defined.” In particular, for \(C \ll 1\) or \(\varepsilon\lll 1\), the conditions “\(\mathop{\mathrm{dist}}(x,y) \lll C\)” and “\(\mathop{\mathrm{dist}}(x,y) \ll \varepsilon\)” are well-defined, i.e., independent of the choice of \((c,|.|)\). The same conditions will often be denoted respectively by \[x = y + o(C), \quad x = y + \operatorname{O}(\varepsilon).\]

Similar definitions apply to elements of \(T_p^*(M)\) and \(T_p^\wedge(M)\). For example, given \(C \geqslant 0\) and \(\theta \in T_p^\wedge(M)\), we say that \(\theta \ll C\) if for some (equivalently, any) controlled chart \(c : (M,p) % \mathrel{% \mathpalette{\da@xarrow{}{}{}\mathchar"0\hexnumber@\symAMSa 4B {\,}{}}{}% }% (V,0)\), the following condition holds. Let \(c_* \theta \in T_{c(p)}^\wedge(V)\) denote the pushforward of \(\theta\). Using the standard identification of \(T_{c(p)} V\) with \(V\), we may identify \(c_* \theta\) with an element of \(V^\wedge\). Then, with respect to some (equivalently, any) fixed norm \(|.|\) on \(V^\wedge\), we have \(|c_* \theta| \ll C\).

Controlled morphisms satisfy a uniform Lipschitz condition.

Bumps

We now extend to manifolds:

§6.3. Modulated bumps on manifolds

Basic definition

Explication in local coordinates

Insensitivity

For the remainder of this section, we retain the setting of the above definitions concerning \(M, p, \theta, T, \delta\). We focus moreover on individual values of \(T\) and \(\delta\), and accordingly abbreviate \(\mathfrak{C}(M,p,\theta,T,\delta)\) to \(\mathfrak{C}(M,p,\theta)\). We will also write, e.g., \(p_1, p_2\) or \(\theta_1, \theta_2\) for quantities satisfying the same conditions as \(p\) and \(\theta\).

Total variation norm

§6.3.5. Modulation

Pushforward

Let \(\rho : (M_1, p_1) % \mathrel{% \mathpalette{\da@xarrow{}{}{}\mathchar"0\hexnumber@\symAMSa 4B {\,}{}}{}% }% (M_2, p_2)\) be a fixed local morphism of fixed pointed manifolds. (The discussion to follow applies with “fixed” replaced by “controlled,” but is slightly more involved, and we do not require such generality.) Then, if \(\rho\) is submersive, the charts \(c_1, c_2\) and the linear map \(\ell\) arising from application of the inverse function theorem (§6.1.7) may also be taken fixed.

Let \(\theta_2 \in T_{p_2}(M_2)^\wedge\) with \(\theta_2 \ll 1\). Set \(\theta_1 := \rho^* \theta_2 \in T_{p_1}(M_1)^\wedge\). As noted earlier, we then have \(\theta_1 \ll 1\).

Negligible bumps

Given any complex scalar \(t\), we may define the rescaled class \(t \mathfrak{C}(M,p,\theta,T,\delta)\), consisting of smooth measures on \(M\) of the form \(t \gamma\) for some \(\gamma \in \mathfrak{C}(M,p,\theta,T,\delta)\). This notation applies in particular when \(t\) is a power of \(|T|\). We denote by \[|T|^{-\infty} \mathfrak{C}(M,p,\theta,T,\delta)\] the intersection over all fixed \(m \geqslant 0\) of the classes \(|T|^{-m} \mathfrak{C}(M,p,\theta,T,\delta)\) (cf. §2.1.6). Passing to local coordinates as in Example 6.41, we readily obtain the following explicit description of that class:

The following technical lemma will be needed at one point, in the proof of Proposition 17.3.

Fourier-analytic description

For a vector space \(V\) over \(F\), we denote by \(\mathcal{S}(V)\) the Schwartz space of \(V\), by \(\mathcal{M}_{\mathcal{S}}(V)\) the space of Schwartz measures on \(V\) (i.e., measures of the form \(\varphi(x) \, d x\), where \(\varphi \in \mathcal{S}(V)\) and \(d x\) is a Haar measure on \(V\)), and by \(V^\wedge\) the Pontryagin dual of \(V\). We denote by \(f \mapsto f^\wedge\) the Fourier transform \(\mathcal{M}_{\mathcal{S}}(V) \rightarrow \mathcal{S}(V^\wedge)\) given by \[f^\wedge(\xi) = \int_{x \in V} f(x) \psi_\xi(x),\] and by \(f \mapsto f^\vee\) its inverse. Given \(T \in F^\times\), we define the rescaled Fourier transform \[\mathcal{F}_T : \mathcal{M}_{\mathcal{S}}(V) \rightarrow \mathcal{S}(V^\wedge)\] by the formula \(\mathcal{F}_T(f)(\xi) := f^\wedge(T \xi)\). We denote by \(\mathcal{F}_T^{-1}\) its inverse.

Action by vector fields

§6.3.10. Action by Lie groups

Let \(G\) be a fixed analytic group over \(F\), i.e., a topological group that is also a manifold (analytic, by our conventions) for which the multiplication and inversion maps are analytic (Serre 2006, vol. 1500, sec. II.IV). We denote by \(1 \in G\) the identity element and by \(\mathop{\mathrm{Lie}}(G)\) the Lie algebra, which is a vector space over \(F\). The group logarithm defines a (fixed) chart \((G,1) % \mathrel{% \mathpalette{\da@xarrow{}{}{}\mathchar"0\hexnumber@\symAMSa 4B {\,}{}}{}% }% (\mathop{\mathrm{Lie}}(G), 0)\).

Assume given a fixed action \(G \times M \rightarrow M\) our on manifold \(M\). We denote this action simply by juxtaposition. This action induces one of \(G\) on functions on \(M\), measures on \(M\), and so on.

In particular, for each \(g \in G\) with \(g = 1 + o(1)\), we may define \(\psi_{\mu(T \theta)}(\log(g)) \in \mathop{\mathrm{U}}(1)\).

We have the following consequence of Lemma 6.53:

We now record the effect of the group action on the classes \(\mathfrak{C}(M,p,\theta)\).

§6.4. Modulated bumps on Lie groups

We may specialize Definition 6.38 to the case that \((M,p) = (G,1)\) for some fixed analytic group \(G\) with identity element \(1\). The group \(G\) acts on itself in three standard ways: by left translation, right translation, and conjugation. The discussion of §6.3.10 applies to all three actions (although in the case of conjugation, more precise estimates are possible).

We often focus on the case that \(p\) is the identity element \(1\), and accordingly abbreviate \[\label{eq:mathfrakcg-theta-t}\tag{6.24} \mathfrak{C}(G,\theta,T,\delta) := \mathfrak{C}(G,1,\theta,T,\delta)\] We further abbreviate \(\mathfrak{C}(G,\theta) := \mathfrak{C}(G, \theta, T,\delta)\) when \(T\), \(\delta\) are clear from context.

Recall that \(\mathcal{M}_c(G)\) denotes the space of compactly-supported smooth measures on \(G\), of which \(\mathfrak{C}(G,\theta)\) is a subclass. For \(\gamma \in \mathcal{M}_c(G)\), we denote by \(\gamma^*\) its adjoint, i.e., the complex conjugate of the pushforward under the inversion map. For \(\gamma_1, \gamma_2 \in \mathcal{M}_c(G)\), we denote by \(\gamma_1 \ast \gamma_2\) their convolution, i.e., the pushforward of \(\gamma_1 \times \gamma_2\) under the multiplication map \(G \times G \rightarrow G\).

§7. Parameters and stability

In this section, given a Lie group \(G\) over \(\mathbb{R}\), we often denote simply by \(\mathfrak{g} := \mathop{\mathrm{Lie}}(G)\) its Lie algebra and by \(\mathfrak{g}^\wedge := \mathop{\mathrm{Lie}}(G)^\wedge\) the Pontryagin dual of the Lie algebra. As explained in §6.1.1, we may identify \(\mathfrak{g}^\wedge\) with the imaginary dual \(i \mathfrak{g}^* = \mathop{\mathrm{Hom}}(\mathfrak{g}, i \mathbb{R})\) of \(\mathfrak{g}\). We write \(\mathfrak{g}_\mathbb{C} := \mathfrak{g}\otimes_{\mathbb{R}} \mathbb{C}\) for the complexification of the Lie algebra and \(\mathfrak{g}_\mathbb{C}^*\) for the complex dual, which identifies naturally with the complexification of \(\mathfrak{g}^\wedge\).

We recall that \(\mathfrak{U}(G)\) and \(\mathfrak{Z}(G)\) denote the universal enveloping algebra and center for \(\mathfrak{g}_{\mathbb{C}}\). We similarly define \(\mathfrak{U}(\mathfrak{h})\), \(\mathfrak{U}(\mathfrak{h}_{\mathbb{C}})\) for any Lie algebra \(\mathfrak{h}\).

§7.1. Preliminaries on real reductive groups

Let \(G\) be a connected reductive group over \(\mathbb{R}\). In particular, we may regard \(G\) as a Lie group over \(\mathbb{R}\).

§7.1.1. Regular elements

A subscripted \(\mathop{\mathrm{reg}}\), as in \(\mathfrak{g}_{\mathop{\mathrm{reg}}}\) or \(\mathfrak{g}^\wedge_{\mathop{\mathrm{reg}}}\), denotes the subset of regular elements, i.e., those with \(\mathfrak{g}\)-centralizer of minimal dimension.

Invariant-theoretic quotients

We denote by \([\mathfrak{g}_\mathbb{C}^*]\) the geometric invariant theory quotient \(\mathfrak{g}_\mathbb{C}^* /\!\!/G\), i.e., the spectrum of the ring of \(G\)-invariant polynomials \(\mathfrak{g}_\mathbb{C}^* \rightarrow \mathbb{C}\), and \([\mathfrak{g}^\wedge]\) for its real form corresponding to polynomials that are real-valued on \(\mathfrak{g}^\wedge\). These come with natural maps \[\mathfrak{g}_\mathbb{C}^* \rightarrow [\mathfrak{g}_\mathbb{C}^*], \quad \mathfrak{g}^\wedge \rightarrow [\mathfrak{g}^\wedge],\] which we denote by \(\xi \mapsto [\xi]\).

There is a natural scaling action of \(\mathbb{R}\) on \([\mathfrak{g}^\wedge]\) and of \(\mathbb{C}\) on \([\mathfrak{g}_{\mathbb{C}}^*]\), compatible with the vector space operations of \(\mathbb{R}\) on \(\mathfrak{g}^\wedge\) and of \(\mathbb{C}\) on \(\mathfrak{g}_{\mathbb{C}}^*\).

Harish–Chandra isomorphism

The Harish–Chandra isomorphism (see, e.g., (P. D. Nelson and Venkatesh 2021, sec. 9.4)) identifies the center \(\mathfrak{Z}(\mathfrak{g}_\mathbb{C})\) of the universal enveloping algebra with the ring \(\mathop{\mathrm{Sym}}(\mathfrak{g}_{\mathbb{C}})^G\) of regular functions on \([\mathfrak{g}_\mathbb{C}^*]\). We recall its construction. Choose a Cartan subalgebra \(\mathfrak{t} \leqslant\mathfrak{g}_\mathbb{C}\) and a decomposition \(\mathfrak{g}_\mathbb{C} = \mathfrak{n}_{\mathbb{C}}^- \oplus \mathfrak{t}_\mathbb{C} \oplus \mathfrak{n}_{\mathbb{C}}\) into positive and negative root spaces. We may identify \(\mathfrak{U}(\mathfrak{t}_\mathbb{C})\) with the symmetric algebra \(\mathop{\mathrm{Sym}}(\mathfrak{t}_\mathbb{C})\). Then \[\label{eqn:decomposition-for-HC}\tag{7.1} \mathfrak{U}(\mathfrak{g}_\mathbb{C}) = (\mathfrak{n}_{\mathbb{C}}^- \mathfrak{U}(\mathfrak{g}_{\mathbb{C}}) + \mathfrak{U}(\mathfrak{g}_{\mathbb{C}}) \mathfrak{n}_{\mathbb{C}}) \oplus \mathop{\mathrm{Sym}}(\mathfrak{t}_\mathbb{C}).\] Let \(\rho \in \mathfrak{t}_\mathbb{C}^*\) denote the half-sum of positive roots. Let \(\sigma\) denote the algebra automorphism of \(\mathop{\mathrm{Sym}}(\mathfrak{t}_\mathbb{C})\) given on \(\mathfrak{t}_\mathbb{C}\) by \(t \mapsto t - \rho(t)\). The Harish–Chandra isomorphism is then the composition \[\mathfrak{Z}(\mathfrak{g}_\mathbb{C}) \xrightarrow{\mathop{\mathrm{HC}}} \mathop{\mathrm{Sym}}(\mathfrak{t}_\mathbb{C})^W \cong \mathop{\mathrm{Sym}}(\mathfrak{g}_{\mathbb{C}})^G \cong \{\text{regular functions on } [\mathfrak{g}_\mathbb{C}^*]\}\] \[\mathop{\mathrm{HC}}(z) := \sigma(z_T),\] where \(z_T \in \mathop{\mathrm{Sym}}(\mathfrak{g}_\mathbb{C})\) denotes the component of \(z\) with respect to the decomposition \((7.1)\).

§7.1.4. Infinitesimal characters

Let \(\pi\) be an irreducible admissible representation of \(G\). Then \(\mathfrak{Z}(\mathfrak{g}_{\mathbb{C}})\) acts on \(\pi\) by scalars. Using the Harish–Chandra isomorphism, we may thus associate to \(\pi\) an infinitesimal character \(\lambda_\pi \in [\mathfrak{g}^*_{\mathbb{C}}]\), characterized as follows: \(z \in \mathfrak{Z}(\mathfrak{g}_{\mathbb{C}})\) acts on \(\pi\) as multiplication by \(\mathop{\mathrm{HC}}(z)(\lambda_\pi)\), where we identify \(\mathop{\mathrm{HC}}(z)\) with a regular function on \([\mathfrak{g}^*_{\mathbb{C}}]\) as above. If \(\pi\) is unitary, then \(\lambda_\pi \in [\mathfrak{g}^\wedge]\) (see (P. D. Nelson and Venkatesh 2021, sec. 9.5)).

Dual groups

We write \(G^\vee\) for the complex dual group and \(\mathfrak{g}^\vee\) for its complex Lie algebra. For any Cartan subalgebras \(\mathfrak{t}_\mathbb{C} \leqslant\mathfrak{g}_\mathbb{C}\) and \(\mathfrak{t}^\vee \leqslant\mathfrak{g}^\vee\), we may canonically identify \[\label{eqn:inf-chars-via-langlands-params}\tag{7.2} \mathfrak{g}^\vee /\!\!/G^\vee \cong \mathfrak{t}^\vee / W \cong \mathfrak{t}_\mathbb{C}^*/W \cong [\mathfrak{g}_\mathbb{C}^*],\] where \(W\) denotes the Weyl group (see (P. D. Nelson and Venkatesh 2021, sec. 15.1)).

§7.2. General linear groups

We may specialize the above discussion to the case of a general linear group \(G\) over the reals, say \(G = {\mathop{\mathrm{GL}}}_n(\mathbb{R})\). Then \(\mathfrak{g}\) is the space \({\mathop{\mathrm{\mathfrak{g}\mathfrak{l}}}}_n(\mathbb{R})\) of \(n \times n\) real matrices. We have \(G^\vee = {\mathop{\mathrm{GL}}}_n(\mathbb{C})\) and \(\mathfrak{g}^\vee = {\mathop{\mathrm{\mathfrak{g}\mathfrak{l}}}}_n(\mathbb{C})\).

§7.2.1. Characteristic polynomials

We denote by \(e : {\mathop{\mathrm{\mathfrak{g}\mathfrak{l}}}}_n(\mathbb{R}) \rightarrow \mathop{\mathrm{Sym}}(\mathfrak{g})\) the “identity map” coming from the identity \({\mathop{\mathrm{\mathfrak{g}\mathfrak{l}}}}_n(\mathbb{R}) = \mathfrak{g}\) and the inclusion \(\mathfrak{g} \hookrightarrow \mathop{\mathrm{Sym}}(\mathfrak{g})\). We regard \(e\) as a matrix \((e_{i j})\), with entries \(e_{i j} \in \mathfrak{g} \hookrightarrow \mathop{\mathrm{Sym}}(\mathfrak{g})\), and form the characteristic polynomial \[\begin{align} \label{eq:detx-+-e}\tag{7.3} \det(X + e) &:= \sum _{\sigma \in S(n)} (-1)^\sigma \prod_{i=1}^n (1_{i=\sigma(i)} X + e_{i,\sigma(i)}) \\ \nonumber &= X^n + \mathfrak{c}_1 X^{n-1} + \dotsb + \mathfrak{c}_n, \end{align}\] which defines a polynomial in the variable \(X\) with coefficients \(\mathfrak{c}_j\) in the ring \(\mathop{\mathrm{Sym}}(\mathfrak{g})^G\) of \(G\)-invariant elements of \(\mathop{\mathrm{Sym}}(\mathfrak{g})\). We may identify each \(\mathfrak{c}_j\) with a \(G\)-invariant polynomial \(\mathfrak{g}_\mathbb{C}^* \rightarrow \mathbb{C}\), hence with a regular function \([\mathfrak{g}_\mathbb{C}^*] \rightarrow \mathbb{C}\). The resulting map \[\det(X + e) : [\mathfrak{g}_\mathbb{C}^*] \rightarrow \{ \text{monic degree $n$ polynomials over $\mathbb{C}$} \}\] is an isomorphism. For \(\lambda \in [\mathfrak{g}_{\mathbb{C}}^*]\), we denote by \(\mathcal{P}_\lambda(X) \in \mathbb{C}[X]\) the corresponding polynomial, i.e., \[\label{eq:polynomial-P-lambda}\tag{7.4} \mathcal{P}_\lambda(X) := \det(X+e)(\lambda) = X^n + \mathfrak{c}_1(\lambda) X^{n-1} + \dotsb + \mathfrak{c}_n(\lambda).\]

We have noted that the space \(\mathfrak{g}^\wedge\) identifies (canonically) with the imaginary dual of \(\mathfrak{g}\), hence, via the trace pairing, with the space \(i {\mathop{\mathrm{\mathfrak{g}\mathfrak{l}}}}_n(\mathbb{R})\) of \(n \times n\) imaginary matrices: \[i {\mathop{\mathrm{\mathfrak{g}\mathfrak{l}}}}_n(\mathbb{R}) \cong \mathfrak{g}^\wedge\] \[\xi \mapsto [x \mapsto \mathop{\mathrm{trace}}(x \xi)].\] With respect to this identification, we see that for \(\xi \in \mathfrak{g}^\wedge\) with image \([\xi] \in [\mathfrak{g}^\wedge]\), \[\det(X + \xi) = \mathcal{P}_{[\xi]}(X) = \det(X + e)(\xi).\] In other words, the map \(\xi \mapsto \mathcal{P}_{[\xi]}(X)\) associates to \(\xi\) its characteristic polynomial.

Eigenvalues

Let \(\lambda \in [\mathfrak{g}_\mathbb{C}^*]\). Under \((7.2)\), we may identify \(\lambda\) with an element of \(\mathfrak{t}^\vee/W\). We write \(t_\lambda \in \mathfrak{t}^\vee\) for any lift of that element, well-defined up to \(W\). We write \[\mathop{\mathrm{eval}}(\lambda) = \{\lambda_1,\dotsc,\lambda_n\}\] for the multiset of eigenvalues of \(t_\lambda\). (This definition is consistent with that of (P. D. Nelson and Venkatesh 2021, sec. 13.4.1).) The characteristic polynomial \(\det(X + t_\lambda) \in \mathbb{C}[X]\) then coincides with \(\mathcal{P}_\lambda(X)\), as defined above: \[\label{eq:detx-+-t_lambda}\tag{7.5} \det(X + t_\lambda) = \mathcal{P}_{\lambda}(X).\]

Characterizations of \([\mathfrak{g}^\wedge] \subseteq [\mathfrak{g}_{\mathbb{C}}^*]\)

We observe that the following conditions on \(\lambda \in [\mathfrak{g}_\mathbb{C}^*]\) are equivalent to each other.

• \(\lambda \in [\mathfrak{g}^\wedge]\).

• \(- \overline{t_\lambda} \in W \cdot t_\lambda\).

• \(\mathfrak{c}_1(\lambda), \mathfrak{c}_3(\lambda), \mathfrak{c}_5(\lambda), \dotsc \in i \mathbb{R}\), \(\mathfrak{c}_2(\lambda), \mathfrak{c}_4(\lambda), \mathfrak{c}_6(\lambda), \dotsc \in \mathbb{R}\).

Thus the map \(\lambda \mapsto \mathcal{P}_\lambda\) defines a bijection between \([\mathfrak{g}^\wedge]\) and the space of polynomials \(X^n + c_1 X^{n-1} + \dotsb + c_n\) with \(c_{2 j} \in \mathbb{R}\) and \(c_{2j - 1} \in i \mathbb{R}\).

Scaling

The scaling action of \(\mathbb{R}\) on \([\mathfrak{g}^\wedge]\) (or of \(\mathbb{C}\) on \([\mathfrak{g}_{\mathbb{C}}^*]\)) may be described explicitly as follows. For \((t,\lambda) \in \mathbb{R} \times [\mathfrak{g}^\wedge]\), the scaled element \(t \lambda \in [\mathfrak{g}^\wedge]\) is characterized by \[\label{eq:cal-P-t-lambda}\tag{7.6} \mathcal{P}_{t \lambda}(X) = t^{n} \mathcal{P}_{\lambda}(t^{-1} X) = X^n + t \mathfrak{c}_1(\lambda) X^{n-1} + \dotsb + t^n \mathfrak{c}_n(\lambda).\] Equivalently, the roots of \(\mathcal{P}_{t \lambda}\) are obtained by multiplying those of \(\mathcal{P}_{\lambda}\) by \(t\).

Infinitesimal characters and parameters

Let \(\pi\) be an irreducible admissible representation of \(G\). It has an infinitesimal character \(\lambda_\pi \in [\mathfrak{g}_{\mathbb{C}}^*]\), which, as we have noted (§7.1.4), belongs to \([\mathfrak{g}^\wedge]\) when \(\pi\) is unitary. We may in any case associate to \(\pi\) a semisimple element \(t_{\lambda_\pi}\) and a multiset \(\mathop{\mathrm{eval}}(\lambda_{\pi}) = \{\lambda_{\pi,1},\dotsc, \lambda_{\pi,n}\}\) of complex numbers. We refer to these numbers as the “archimedean Satake parameters” of \(\pi\).

By the discussion of (P. D. Nelson and Venkatesh 2021, sec. 15.4), the standard local \(L\)-factor for \(\pi\) is given in terms of the infinitesimal character by \[L(\pi,s) = \prod_j \Gamma_\mathbb{R}(s + \lambda_{\pi,j}^+ + a_j),\] where the elements \(a_j \in \{0, 1\}\) depend upon \(\pi\) and we set \[(x + i y)^+ := |x| + i y, \quad \Gamma_\mathbb{R}(s) := \pi^{-s/2} \Gamma(s/2), \quad \Gamma_{\mathbb{C}}(s) := 2 (2 \pi)^{-s} \Gamma(s).\] (Strictly speaking, it is assumed in parts of loc. cit. that \(\pi\) is tempered, but the same argument applies in general, using the Langlands decomposition (Knapp 1986, Theorem 8.54) and the compatibility of infinitesimal characters with induction (Knapp 1986, Proposition 8.22) to reduce to the tempered case. See also (Rudnick and Sarnak 1996, Appendix).)

In §1.2, we referred to the numbers \(t_j = \lambda_{\pi,j}^+ + a_j\) as the “archimedean \(L\)-function parameters” of \(\pi\) (or more precisely, of a global automorphic representation having \(\pi\) as its local component at \(\infty\)). These are not quite the same as the “archimedean Satake parameters” \(\lambda_{\pi,j}\) described above, but they have very similar asymptotic behavior. For instance:

General linear pairs

Let \((G,H)\) be a general linear pair over \(\mathbb{R}\), say \((G,H) = ({\mathop{\mathrm{GL}}}_n(\mathbb{R}), {\mathop{\mathrm{GL}}}_{n-1}(\mathbb{R}))\). The above discussion then applies both to \(\mathfrak{g} = \mathop{\mathrm{Lie}}(G)\) and to \(\mathfrak{h} = \mathop{\mathrm{Lie}}(H)\). For \(\xi \in \mathfrak{g}^\wedge\), we let \(\xi_H \in \mathfrak{h}^\wedge\) denote its restriction.

§7.3.1. Stability

The two definitions are related as follows (see, e.g., (P. D. Nelson 2023, Lemma 11.4)):

In particular, \(\mathfrak{g}^\wedge_{\mathop{\mathrm{stab}}}\) is the dense open subset of \(\mathfrak{g}^\wedge\) given by the nonvanishing of the polynomial \[\mathfrak{g}^\wedge \ni \xi \mapsto \mathcal{R}(\xi) := \operatorname{Resultant} (\mathcal{P}_{[\xi]}, \mathcal{P}_{[\xi_H]}).\] Here \([\xi] \in [\mathfrak{g}^\wedge] \subseteq [\mathfrak{g}_{\mathbb{C}}^*]\) and \([\xi_H] \in [\mathfrak{h}^\wedge] \subseteq [\mathfrak{h}_{\mathbb{C}}^*]\) give rise to polynomials \(\mathcal{P}_{[\xi]}, \mathcal{P}_{[\xi_H]}\) via \((7.4)\).

We obtain the following equivalence:

§7.3.2. The parameter \(\theta\)

Let \(P\) denote the mirabolic subgroup of \(G\), consisting of elements with bottom row \((0,\dotsc,0,1)\). Recall that \(\bar{B}_H\) denotes the lower-triangular Borel subgroup of \(H\). The Lie algebra of the mirabolic subgroup then decomposes as \[\mathop{\mathrm{Lie}}(P) = \mathop{\mathrm{Lie}}(N) \oplus \mathop{\mathrm{Lie}}(\bar{B}_H).\]

Let \(\psi\) be a nondegenerate unitary character of \(N\). We denote by \[\theta_P(\psi) \in \mathop{\mathrm{Lie}}(P)^\wedge\] the unique element with trivial restriction to \(\mathop{\mathrm{Lie}}(\bar{B}_H)\) that satisfies, for all \(x \in \mathop{\mathrm{Lie}}(N)\), \[\psi(\exp(x)) = \exp (\langle x, \theta_P(\psi) \rangle).\] We recall that, by Definition 6.33, we may associate to each \(\xi \in \mathop{\mathrm{Lie}}(P)^\wedge\) and chart \(c : (P,1) % \mathrel{% \mathpalette{\da@xarrow{}{}{}\mathchar"0\hexnumber@\symAMSa 4B {\,}{}}{}% }% (\mathop{\mathrm{Lie}}(P),0)\) a function \(\psi_{\xi, \log} : P % \mathrel{% \mathpalette{\da@xarrow{}{}{}\mathchar"0\hexnumber@\symAMSa 4B {\,}{}}{}% }% \mathop{\mathrm{U}}(1)\), defined near the identity element. (The use of “\(\psi\)” in this notation is purely symbolic, unrelated to the character \(\psi\) under consideration.) In particular, we may define \(\psi_{\theta_P(\psi),\log}\) with respect to the chart \(\log : (P, 1) % \mathrel{% \mathpalette{\da@xarrow{}{}{}\mathchar"0\hexnumber@\symAMSa 4B {\,}{}}{}% }% (\mathop{\mathrm{Lie}}(P), 0)\). Then \[\psi(\exp(x)) = \psi_{\theta_P(\psi), \log}(x) \quad \text{ for } x \in N,\] while \(\psi_{\theta_P(\psi), \log}(x) = 1\) for \(x \in \bar{B}_H\).

Using the trace form, we may identify \(\mathop{\mathrm{Lie}}(P)^\wedge\) with a quotient of \(\mathfrak{g}^\wedge = i {\mathop{\mathrm{\mathfrak{g}\mathfrak{l}}}}_n(\mathbb{R})\), e.g., when \(n=4\), \[\mathop{\mathrm{Lie}}(P)^\wedge \leftrightarrow \begin{pmatrix} \ast & \ast & \ast & ? \\ \ast & \ast & \ast & ? \\ \ast & \ast & \ast & ? \\ \ast & \ast & \ast & ? \end{pmatrix}.\] The meaning of this notation is that \(\mathop{\mathrm{Lie}}(P)^\wedge\) identifies with the space of equivalence clases of \(n \times n\) imaginary matrices, where two such matrices are declared equivalent if they have the same entries except possibly in their rightmost columns. We may write \[\label{eq:psiu-=-sum_i=1n}\tag{7.7} \psi(u) = \sum_{i=1}^{n-1} \exp(\eta_i u_{i,i+1})\] for some nonzero imaginary numbers \(\eta_1,\dotsc,\eta_{n-1}\) (e.g., each \(\eta_j = 2 \pi \sqrt{-1}\) for a suitable sign). Then, under the above identification, \[\label{eq:theta-leftr-beginpm}\tag{7.8} \theta_P(\psi) \leftrightarrow \begin{pmatrix} 0 & 0 & 0 & ? \\ \eta_1 & 0 & 0 & ? \\ 0 & \eta_2 & 0 & ? \\ 0 & 0 & \eta_3 & ? \end{pmatrix}.\]

§7.3.3. The parameter \(\tau\)

For \(\xi \in \mathfrak{g}^\wedge\), we denote by \(\xi_P \in \mathop{\mathrm{Lie}}(P)^\wedge\) its restriction. We have the following simple linear algebra lemma, closely related to “rational canonical form:”

For example, when \(n=4\) and \(\psi\) is as in \((7.7)\), the element \(\tau\) is given by \[\label{eqn:tau-three-by-three-theta-lambda}\tag{7.9} \tau = \begin{pmatrix} 0 & 0 & 0 & - \mathfrak{c}_4(\lambda) / \eta_1 \eta_2 \eta_3 \\ \eta_1 & 0 & 0 & \mathfrak{c}_3(\lambda) / \eta_2 \eta_3\\ 0 & \eta_2 & 0 & -\mathfrak{c}_2(\lambda)/ \eta_3 \\ 0 & 0 & \eta_3 & \mathfrak{c}_1(\lambda) \end{pmatrix}.\]

Lemma 7.6 allows us to make the following definition.

§8. Localized elements of the Kirillov model

Let \((G,H)\) be a fixed general linear pair over \(\mathbb{R}\). Recall that we write \(N \leqslant G\) and \(N_H \leqslant H\) for the standard upper-triangular maximal unipotent subgroups. We denote as in §7.3.2 by \(P \leqslant G\) the mirabolic subgroup.

Let \(\pi\) be a generic irreducible unitary representation of \(G\).

The main result of this section is Theorem 8.7, stated below in §8.3. Informally speaking, that result supplies us with a class of microlocalized vectors \(v\) in \(\pi\) that concentrate in the Kirillov model near a specific point, together with some quantitative control over how the group \(G\) acts on such vectors.

§8.1. The Kirillov model

Let \(\psi\) be a nondegenerate unitary character of \(N\). We denote also simply by \(\psi\) the restriction of \(\psi\) to \(N_H\), which is again nondegenerate.

We denote in what follows by \(\pi^0\) the underlying Hilbert space, by \(\pi^\infty\) the space of smooth vectors, and by \(\pi^{-\infty}\) the space of distributional vectors, i.e., the continuous dual of \(\pi^\infty\); these spaces are related via natural inclusions \[\pi^\infty \subseteq \pi^0 \subseteq \pi^{-\infty}.\]

By the definition of “generic,” there is a nonzero distributional vector \(\Theta_{\pi,\psi} \in \pi^{-\infty}\), unique up to scalars (Shalika 1974), for which \(\pi(n) \Theta_{\pi,\psi} = \psi(n) \Theta_{\pi,\psi}\) for all \(n \in N\). The map \(v \mapsto [g \mapsto \langle \pi(g) v, \Theta_{\pi,\psi} \rangle]\) defines the \(\psi\)-Whittaker model \(\mathcal{W}(\pi,\psi)\) of \(\pi\), which consists of \(W : G \rightarrow \mathbb{C}\) satisfying \(W(n g) = \psi(n) W(g)\) for all \((n,g) \in N \times G\). It follows from the results of (Baruch 2003a) (see also (J. N. Bernstein 1984) and (I. N. Bernstein and Zelevinsky 1977, Ch. III)) that by restricting elements of the model \(\mathcal{W}(\pi,\psi)\) to the subgroup \(H\), we may realize \(\pi^0\) as the Hilbert space of all measurable functions \(W : H \rightarrow \mathbb{C}\) satisfying \[\label{eqn:kirillov-model-defn}\tag{8.1} W(n h) = \psi(n) W(h) \text{ for all } (n,h) \in N_H \times H,\] \[\label{eq:int_n_h-backslash-h}\tag{8.2} \int_{N_H \backslash H} |W|^2 < \infty,\] with an invariant inner product on \(\pi\) given integration over \(N_H \backslash H\).

By analogy to §2.12, we denote by \(C_c^\infty(N_H \backslash H,\psi)\) the space of smooth compactly-supported functions \(H \rightarrow \mathbb{C}\) satisfying \((8.1)\). It is known, and relatively straightforward to see, that the space \(C_c^\infty(N_H \backslash H, \psi)\) is contained in the Hilbert space \(\pi^0\) and is smooth under the action of \(H\). Jacquet (Hervé Jacquet 2010, Prop 5) established the deeper fact that it is smooth under the action of \(G\), i.e., that \[C_c^\infty(N_H \backslash H, \psi) \subseteq \pi^\infty.\] Jacquet’s proof shows in particular that the inclusion is continuous and that for any bounded open subset \(E\) of \(N_H \backslash H\), the subspace \(C_c^\infty(E, \psi)\) (consisting of elements supported on the preimage of \(E\)) is invariant by \(\mathfrak{U}(G)\).

In more detail, by the proof of (Hervé Jacquet 2010, Prop 5), we may write any element of \(C_c^\infty(N_H \backslash H, \psi)\) in the form \(\pi(f) \Theta_{\pi,\psi}\) for some \(f \in C_c^\infty(H)\). We recall (Hervé Jacquet 2010, Lem 7): \[\label{eq:pif-theta_pi-h}\tag{8.3} \pi(f) \Theta_{\pi,\psi} (h) = \int_{n \in N_H} f(h^{-1} n) \psi(n) \, d n\]

We note that the above discussion may be reformulated using the bijection \(N_H \backslash H = N \backslash P\). In particular:

§8.2. A class of bump functions in the Kirillov model

We now take the nondegenerate unitary character \(\psi\) of \(N\) to be fixed. We define \(\theta_P(\psi) \in \mathop{\mathrm{Lie}}(P)^\wedge\) as in §7.3.2. We denote by \(\theta_H(\psi) \in \mathop{\mathrm{Lie}}(H)^\wedge\) the restriction of \(\theta_P(\psi)\).

Let \(T\) be a positive parameter with \(T \ggg 1\). We define the rescaled nondegenerate character \(\psi_T\) of \(N\) by requiring that, for \(x \in \mathop{\mathrm{Lie}}(N)\), \[\psi_T(\exp(x)) = \psi(\exp(T x)).\] Thus \(\psi_T(\exp(x)) = \psi_{T \theta_P(\psi), \log}(x)\). Equivalently, \(\psi_T(n) = \psi (T^{\rho_N^\vee} n T^{-\rho_N^\vee})\). In the notation of §7.3.2, we pass from \(\psi\) to \(\psi_T\) by replacing each \(\eta_i\) by its multiple \(T \eta_i\).