The Gelfand--Graev representation of SO$(2n+1)$ in terms of Hecke algebras

TL;DR

This paper describes the Bernstein components of the Gelfand--Graev representation for the p-adic group SO(2n+1) using Hecke algebras, providing a new algebraic perspective on these representations.

Contribution

It explicitly characterizes the Bernstein components of the Gelfand--Graev representation for SO(2n+1) via Hecke algebra constructions, extending previous understanding.

Findings

Bernstein components are described in terms of Hecke algebras.

Utilizes Heiermann's construction to analyze the Gelfand--Graev representation.

Provides a new algebraic framework for studying representations of SO(2n+1).

Abstract

Let $G$ be a $p$-adic classical group. The representations in a given Bernstein component can be viewed as modules for the corresponding Hecke algebra---the endomorphism algebra of a pro-generator of the given component. Using Heiermann's construction of these algebras, we describe the Bernstein components of the Gelfand--Graev representation for $G=$SO$(2n+1)$.