Zhengyu Mao

On the asymptotics of Whittaker functions

Whittaker models are ubiquitous in the representation theory of quasi-split reductive groups over local fields. They comprise the bedrock for whole families of local zeta integrals of Rankin-Selberg type. In the analysis of the latter, it is imperative to know the asymptotics of Whittaker functions, in order to control the domain of convergence of the integrals. This question was studied extensively in the literature, especially in the context of GLn (e.g. [JS90b], [JS90a], [JPSS79], [CPS]). In the Archimedean case a fairly complete answer is given in [Wal92]. On the other hand, to the best of the authors’ knowledge, the connection between the asymptotics of the Whittaker functions and the exponents of the representation in the p-adic case is not made explicit in the literature. The purpose of this short note is to partially fill this gap. The precise statement is given in Theorem 1 below, and is motivated by the results above. (Cf. [JS90b, Proposition 2.2], [JS90a, §2], [JPSS79]). As a consequence we realize the inner product of generic square-integrable and more generally, tempered representations, on the Whittaker model. For simplicity we work with split groups. After completing an early version of this note we learned that Yiannis Sakellaridis and Akshay Venkatesh have launched an ambitious program to study the decomposition of the L-space of spherical varieties. In particular, they obtained asymptotic results of the kind which appear in this paper, in a very general setup. Although strictly speaking their current setup does not include the Whittaker case, there is no doubt that this can be eventually incorporated to the general scheme. In particular, Conjectures 1 and 2 below are probably within reach. Nevertheless, we believe that the Whittaker case is both sufficiently important and elementary to merit its own exposition.

A conjecture on Whittaker–Fourier coefficients of cusp forms

Abstract We formulate an analogue of the Ichino–Ikeda conjectures for the Whittaker–Fourier coefficients of automorphic forms on quasi-split reductive groups. This sharpens the conjectures of Sakellaridis–Venkatesh in the case at hand.

BESSEL IDENTITIES IN THE WALDSPURGER CORRESPONDENCE OVER THE REAL NUMBERS

We prove certain identities between Bessel functions attached to irreducible unitary representations ofPGL2(R) and Bessel functions attached to irreducible unitary representations of the double cover ofSL2(R). These identities give a correspondence between such representations which turns out to be the Waldspurger correspondence. In the process we prove several regularity theorems for Bessel distributions which appear in the relative trace formula. In the heart of the proof lies a classical result of Weber and Hardy on a Fourier transform of classical Bessel functions. This paper constitutes the local (real) spectral theory of the relative trace formula for the Waldspurger correspondence for which the global part was developed by Jacquet.

On an analogue of the Ichino–Ikeda conjecture for Whittaker coefficients on the metaplectic group

In previous papers we formulated an analogue of the Ichino--Ikeda conjectures for Whittaker--Fourier coefficients of automorphic forms on classical group and the metaplectic group. In the latter case we reduced the conjecture to a local identity. In this paper we will prove the local identity in the

On the formal degrees of square-integrable representations of odd special orthogonal and metaplectic groups

p

Whittaker-Fourier coefficients of cusp forms on

-adic case, and hence the global conjecture under simplifying conditions at the archimedean places.

\widetilde{\rm Sp}_n

The formal degree conjecture relates the formal degree of an irreducible square-integrable representation of a reductive group over a local field to the special value of the adjoint

: reduction to a Local Statement

\gamma

Model Transition for Representations of Metaplectic Type

-factor of its

Jacquet modules of the Weil representations and families of relative trace identities

L