Derivatives and Asymptotics of Whittaker functions

TL;DR

This paper provides an asymptotic expansion of Whittaker functions for certain p-adic groups using derivative functors, characterizes representations in L2 spaces, and proves a conjecture for these groups.

Contribution

It introduces a new asymptotic expansion method for Whittaker functions and proves a conjecture relating generic representations to discrete series for specific p-adic groups.

Findings

Derived asymptotic expansion of Whittaker functions.

Characterized generic representations in L2 spaces.

Proved Lapid and Mao's conjecture for the considered groups.

Abstract

Let F be a p-adic field, and Gn one of the groups GL(n, F), GSO(2n-1, F), GSp(2n, F), or GSO(2(n - 1), F). Using the mirabolic subgroup or analogues of it, and related "derivative" functors, we give an asymptotic expansion of functions in the Whittaker model of generic representations of Gn, with respect to a minimal set of characters of subgroups of the maximal torus. Denoting by Zn the center of Gn, and by Nn the unipotent radical of its standard Borel subgroup, we characterize generic representations occurring in L2(ZnNn\Gn) in terms of these characters. This is related to a conjecture of Lapid and Mao for general split groups, asserting that the generic representations occurring in L2(ZnNn\Gn) are the generic discrete series; we prove it for the group Gn.