Godement-Jacquet L-functions and full theta lifts

TL;DR

This paper explores the relationship between poles of local Godement-Jacquet L-functions and distributions on matrix spaces, applying this to prove the irreducibility of full theta lifts for generic representations over local fields.

Contribution

It establishes a connection between L-function poles and distributions, and demonstrates the irreducibility of full theta lifts for generic irreducible representations over local fields.

Findings

Poles of local Godement-Jacquet L-functions relate to distributions with singular supports.

Full theta lifts of generic irreducible representations are irreducible.

Results hold over arbitrary local fields.

Abstract

We relate poles of local Godement-Jacquet L-functions to distributions on matrix spaces with singular supports. As an application, we show the irreducibility of the full theta lifts to $GL_n(F)$ of generic irreducible representations of $GL_n(F)$, where $F$ is an arbitrary local field.