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SCIENCE CHINA Mathematics, https://doi.org/10.1007/s11425-018-1616-0

A refined Poisson summation formula for certain Braverman-Kazhdan spaces

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  • ReceivedFeb 7, 2018
  • AcceptedNov 5, 2019

Abstract

Braverman and Kazhdan (2000) introduced influential conjectures aimed at generalizing the Fourier transform and the Poisson summation formula. Their conjectures should imply that quite general Langlands $L$-functions have meromorphic continuations and functional equations as predicted by Langlands' functoriality conjecture. As evidence for their conjectures, Braverman and Kazhdan (2002) considered a setting related to the so-called doubling method in a later paper and proved the corresponding Poisson summation formula under restrictive assumptions on the functions involved. The connection between the two papers is made explicit in work of Li (2018). In this paper, we consider a special case of the setting in Braverman and Kazhdan's later paper and prove a refined Poisson summation formula that eliminates the restrictive assumptions of that paper. Along the way we provide analytic control on the Schwartz space we construct; this analytic control was conjectured to hold (in a slightly different setting) in the work of Braverman and Kazhdan (2002).


Acknowledgment

This work was supported by the National Science Foundation of the USA (Grant Nos. DMS-1405708 and DMS-1901883). The second author was supported by the National Science Foundation of the USA (Grant Nos. DMS-1702218 and DMS-1848058), and by a start-up fund from the Department of Mathematics at Purdue University. The authors thank Herve Jacquet and Aaron Pollack for their interest in the results in this paper and for helpful comments and suggestions. The authors thank Wen-Wei Li, Yiannis Sakellaridis, and Freydoon Shahidi for useful conversations, and thank Freydoon Shahidi and Wen-Wei Li for sharing [24,25] with them. The authors also thank Heekyoung Hahn for the help with editing and for her constant encouragement. Thanks are due to the anonymous referees for several useful comments. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.


Supplement

Appendix

Proof of Proposition 4.7 in the Archimedean case

Assume that $F$ is an Archimedean local field. Recall that $C_c^\infty(X(F),K)~<~C_c^\infty(X(F))$ is the subset of functions that are right $K$-finite. Our goal here is to prove Proposition 4.7 in the current Archimedean setting. We recall that Proposition 4.7 simply states that $C_c^\infty(X(F),K)~\leq~\mathcal{S}(X(F),K)$.

It is convenient to begin with a few convergence lemmas.

Lemma 21. Let $\Phi~\in~C_c^\infty(X(F),K)$. Then for $\mathrm{Re}(s)~\geq~n^2/2$ one has $$ |M_{w_0}\Phi_{\chi_s}(I_{2n})| \ll_{\Phi} 1. $$

Proof. Consider the map \begin{align} \mathrm{Pl}:X(F) \rightarrow \wedge^n F^{2n}-0. \tag{72} \end{align} Since $X$ is a homogeneous space for $\mathrm{Sp}_{2n}$ it is smooth (as a scheme over $F$). The map 72 is an injective diffeomorphism onto its image, a closed submanifold of $\wedge^n~F^{2n}-0$. In particular, there is a function $\Psi~\in~C_c^\infty(\wedge^n~F^{2n})$ such that $\Phi=\Psi~\circ~\mathrm{Pl}$. For $\mathrm{Re}(s)$ sufficiently large we have \begin{align*}M_{w_0} \Phi_{\chi_s}(I_{2n})&=\int_{N(F)} \Phi_{\chi_s}(w_0^{-1}n)dn \\ &=\int_{N(F)}\bigg(\int_{M^{\mathrm{ab}}(F)}\delta_P^{1/2}(m)\chi_s(\omega(m))\Psi( \mathrm{Pl}(m^{-1}w_0^{-1}n))dm^\times \bigg)dn. \end{align*} Temporarily denote \begin{align*}&\mathrm{Pl}_0:M_{n \times n}^{\oplus 2}(F)\rightarrow \wedge^{n}F^{2n}, \\ &(X,Y) \mapsto \mathrm{Pl}\left( \begin{matrix} * & * \\ X & Y \end{matrix}\right). \end{align*} This is just taking the wedge product of the $n$ rows of the $n~\times~2n$ matrix $(X\,~Y)$, going from top to bottom. Then the integral above can be written as \begin{align*}&\int_{\mathrm{Sym}^n(F)}\left(\int_{F^\times}\chi_s(a)|a|^{(n+1)/2}\Psi\left( \mathrm{Pl}_0\left(-\left(\begin{matrix} a & \\ & I_{n-1}\end{matrix} \right)J',-\left(\begin{matrix} a & \\ & I_{n-1}\end{matrix} \right)J'z \right)\right)da^\times \right)dz, \end{align*} where $\mathrm{Sym}^n(F)$ is the $F$-vector space of symmetric $n~\times~n$ matrices and $$ J'=\left(\begin{smallmatrix} & & 1 \\ & \reflectbox{$\ddots$} & \\ 1 & & \end{smallmatrix}\right). $$ We note that $\mathrm{Pl}_0$ is invariant under multiplication by $\mathrm{SL}_n$ on the left to see that the above is equal to \begin{align*}&\int_{\mathrm{Sym}^n(F)}\left(\int_{F^\times}\chi_s(a)|a|^{(n+1)/2}\Psi\left( -\mathrm{Pl}_0\left(\left(\begin{matrix} I_{n-1}& \\ & a\end{matrix} \right),\left(\begin{matrix} I_{n-1}& \\ & a\end{matrix} \right)z \right)\right)da^\times \right)dz. \end{align*} Take a change of variables $z~\mapsto~a^{-1}z$ to arrive at \begin{align*}&\int_{\mathrm{Sym}^n(F)}\left(\int_{F^\times}\chi_{s+(1-n^2)/2}(a)\Psi\left(- \mathrm{Pl}_0\left(\left(\begin{matrix} I_{n-1} & \\ & a\end{matrix} \right),\left(\begin{matrix} I_{n-1} & \\ & a\end{matrix} \right)a^{-1}z \right)\right)da^\times \right)dz. \end{align*} By inspection the integrand is rapidly decreasing as a function of $z~\in~\mathrm{Sym}^n(F)$ and $a~\in~F$, so this integral converges absolutely and is bounded by a constant depending only on $\Psi$ for $\mathrm{Re}(s)\geq~n^2/2$.

Lemma 22. Let $\Phi~\in~C_c^\infty(X(F),K)$ and $A~\in~\RR$. Then for $\mathrm{Re}(s)~\geq~A$ one has \begin{align*}|M_{w_0}\Phi_{\chi_s}(I_{2n})| \ll_{\Phi,A} 1. \end{align*}

Proof. Let $\alpha~\in~\ZZ_{\geq~0}$. Notice that $$ \widetilde{\Phi}:=(\omega\bar{\omega})^{\alpha}\Phi \in C_c^\infty(X(F),K), $$ where $\omega\bar{\omega}$ is defined as in 43. As in the proof of Lemma 5.5 we have \begin{align*}((\omega \bar{\omega})^{-\alpha}\widetilde{\Phi})_{\chi_s} &=\widetilde{\Phi}_{\chi_{s+2\alpha[F:\RR]^{-1}}}. \end{align*} By Lemma 21 we therefore have $$ |M_{w_0}\Phi_{\chi_s}(I_{2n})|= |M_{w_0}((\omega \bar{\omega})^{-\alpha}\widetilde{\Phi})_{\chi_s}(I_{2n})|=|M_{w_0} \widetilde{\Phi}_{\chi_{s+2\alpha[F:\RR]^{-1}}}(I_{2n})| \ll_{\Phi,\alpha}\! 1, $$ for $ \mathrm{Re}(s)+2\alpha~[F:\RR]^{-1}\geq~n^2/2. $ Taking $\alpha$ sufficiently large we deduce the lemma.

ProofProof of Proposition $\ref{prop:compact}$ in the Archimedean case. If $\Phi~\in~C_c^\infty(X(F),K)$ then it is easy to see that $\Phi_{\chi_s}$ is holomorphic for all $\chi$. Hence by [8] we deduce that $\Phi_{\chi_s}$ is a good section. We thus have to verify that for all $g~\in~\mathrm{Sp}_{2n}(F)$, all characters $\chi$, $A<B$, and all $P_w$ as in the definition of an excellent section that \begin{align} |\Phi_{\chi_s}(g)|_{A,B,P_{\mathrm{Id}}} \tag{73} \end{align} and \begin{align} |M_{w_0}\Phi_{\chi_s}(g)|_{A,B,P_{w_0}} \tag{74} \end{align} are finite. In fact this is enough to complete the proof since the space $C_c^\infty(X(F),K)$ is preserved under the differential operators $D$ and $\bar{D}$ of 32 and 33.

Write $\chi$ as in 28. By Lemma 5.9 for any $N,N'~\in~\ZZ_{~\geq~0}$ with $N'=0$ if $F$ is real, one has \begin{align*}|D^N\Phi_{\chi_s}|_{A,B,1}=\bigg|\bigg({\rm i}t+s+\frac{n+1}{2} \bigg)^N\Phi_{\chi_s}(g)\bigg|_{A,B,1}, \end{align*} if $F$ is real, and \begin{align*}|D^N\bar{D}^{N'}\Phi_{\chi_s}(g)|_{A,B,1}= \bigg|\bigg(\frac{\alpha}{2}+{\rm i}t+s+\frac{n+1}{2} \bigg)^N\bigg(-\frac{\alpha}{2}+{\rm i}t+s+\frac{n+1}{2} \bigg)^{N'}\Phi_{\chi_s}(g)\bigg|_{A,B,1}, \end{align*} if $F$ is complex. Since $\Phi~\in~C_c^\infty(X(F),K)$ the left-hand sides here are bounded by a constant depending on $A,B$ $\Phi,N$ and $N'$. We deduce that 73 is finite for all $P~\in~\CC[x]$. An analogous argument, using Lemma 22, allows us to deduce that 74 is finite for all $P~\in~\CC[x]$.


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    $F$ $~~~dx$
    $\RR$ Lebesgue measure
    $\CC$ Twice Lebesgue measure
    Non-Archimedean $dx(\OO)=|\mathfrak{d}|^{1/2}$

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