Iwahori Matsumoto presentation for modules of Iwahori fixed functions on symmetric spaces

TL;DR

This paper explores the structure of Iwahori-invariant functions on symmetric spaces as modules over the Iwahori Hecke algebra, extending classical descriptions to a broader setting.

Contribution

It generalizes the Iwahori-Matsumoto description of modules over the Iwahori Hecke algebra to functions on symmetric spaces.

Findings

Provides a new module description for Iwahori-invariant functions on symmetric spaces.

Extends classical Iwahori-Matsumoto results to a broader class of spaces.

Enhances understanding of the representation theory of p-adic groups.

Abstract

We study the space $S(X)^I$ of smooth functions on a symmetric space $X=G/H$ invariant to the action of an Iwahori subgroup $I$, as a module over $\mathcal{H}(G,I)$, the Iwahori Hecke algebra of a p-adic group $G$. We present a description of this module that generalizes the description given to $\mathcal{H}(G,I)$ by Iwahori and Matsumoto.