Shalika models for general linear groups

TL;DR

This paper introduces a generalized concept of Shalika models for $GL_{n+m}(F)$, proving their multiplicity-free property over various fields, and explores related multiplicity-free phenomena in representation theory.

Contribution

It generalizes Shalika models to broader settings and establishes their multiplicity-free nature, providing new proofs and conjectures linking different multiplicity-free results.

Findings

Shalika models are multiplicity-free for $GL_{n+m}(F)$ over non-Archimedean and finite fields.

Bernstein-Zelevinsky product of an irreducible $GL_n(F)$ representation with a trivial $GL_m(F)$ is multiplicity-free.

Proposes a conjecture connecting twisted parabolic induction and Gelfand pairs.

Abstract

We define a generalization of Shalika models for $GL_{n+m}(F)$ and prove that they are multiplicity-free, where $F$ is either a non-Archimedean local field or a finite field and $n,m$ are any natural numbers. In particular, we give new proof for the case of $n=m$. We also show that the Bernstein-Zelevinsky product of an irreducible representation of $GL_n(F)$ and the trivial representation of $GL_m(F)$ is multiplicity-free. We relate the two results by a conjecture about twisted parabolic induction of Gelfand pairs.