Harmonic Analysis and Gamma Functions on Symplectic Groups

TL;DR

This paper develops a new harmonic analysis framework on symplectic groups over p-adic fields, linking it to gamma functions and confirming conjectures in the local Langlands program, extending classical methods to broader groups.

Contribution

It introduces a novel harmonic analysis approach on symplectic groups and GL(1), connecting to gamma functions and local zeta integrals, extending the Godement-Jacquet method to classical groups.

Findings

Established harmonic analysis associated with Langlands gamma functions on symplectic groups.

Connected new harmonic analysis on GL(1) to abelian gamma functions.

Confirmed conjectures in the local theory of the Braverman-Kazhdan proposal.

Abstract

Over a $p$-adic local field $F$ of characteristic zero, we develop a new type of harmonic analysis on an extended symplectic group $G={\mathbb G}_m\times{\mathrm Sp}_{2n}$. It is associated to the Langlands $\gamma$-functions attached to any irreducible admissible representations $\chi\otimes\pi$ of $G(F)$ and the standard representation $\rho$ of the dual group $G^\vee({\mathbb C})$, and confirms a series of the conjectures in the local theory of the Braverman-Kazhdan proposal for the case under consideration. Meanwhile, we develop a new type of harmonic analysis on ${\rm GL}_1(F)$, which is associated to a $\gamma$-function $\beta_\psi(\chi_s)$ (a product of $n+1$ certain abelian $\gamma$-functions). Our work on ${\rm GL}_1(F)$ plays an indispensable role in the development of our work on $G(F)$. These two types of harmonic analyses both specialize to the well-known local theory developed in Tate's thesis when $n=0$. The approach is to use the compactification of ${\rm Sp}_{2n}$ in the Grassmannian variety of ${\rm Sp}_{4n}$, with which we are able to utilize the well developed local theory of Piatetski-Shapiro and Rallis and many other works) on the doubling local zeta integrals for the standard $L$-functions of ${\rm Sp}_{2n}$. The method can be viewed as an extension of the work of Godement-Jacquet for the standard $L$-function of ${\rm GL}_n$ and is expected to work for all classical groups. We will consider the archimedean local theory and the global theory in our future work.