Generic Hecke algebra and theta correspondence over finite fields

TL;DR

This paper investigates the structure of Hecke algebra modules related to theta correspondence over finite fields, providing explicit descriptions and applications to conservation relations and generalizations of prior results.

Contribution

It introduces a generic Hecke algebra module framework that unifies various modules and extends previous work on theta correspondence for finite fields.

Findings

Explicit description of the generic Hecke algebra module at specialization 1

Proof of the conservation relation on first occurrence indices

Generalization of previous theta correspondence results

Abstract

We study the Hecke algebra modules arising from theta correspondence between certain Harish-Chandra series for type I dual pairs over finite fields. For the product of the pair of Hecke algebras under consideration, we show that there is a generic Hecke algebra module whose specializations at prime powers give the Hecke algebra modules and whose specialization at $1$ can be explicitly described. As an application, we prove the conservation relation on the first occurrence indices for all irreducible representations. As another application, we generalize the results of Aubert-Michel-Rouquier and Pan on theta correspondence between the Harish-Chandra series.