Linear and Shalika local periods for the mirabolic group, and some consequences

TL;DR

This paper explores the use of linear periods and Shalika models in the representation theory of GL(n,F), establishing new criteria for distinguishing representations and deriving functional equations for local L-functions.

Contribution

It provides a classification of Levi subgroups that distinguish discrete series and generic representations, and links Shalika models with local models for GL(2m,F).

Findings

Maximal Levi subgroups that distinguish representations identified.

Functional equation for local exterior-square L-function derived.

Necessary and sufficient conditions for local models of Steinberg representations established.

Abstract

Using linear periods on the mirabolic subgroup of $GL(n,F)$, for $F$ a non archimedean local field, we give a list of the maximal Levi subgroups of $GL(n,F)$ which can distinguish a discrete series, and a generic representation. We also obtain the functional equation of the local exterior-square $L$-function of a generic representation of $GL(n,F)$ when $n=2m$ is even. Then we discuss the relation between Shalika models and local models for representations of $GL(2m,F)$. Finally we give a necessary and sufficient condition on the cuspidal representation $\rho$ for a generalised Steinberg representation $St_k(\rho)$ of $GL(2m,F)$ to have a local model.