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

TL;DR

This work analyzes the space $S(X)^I$ of $I$-fixed smooth functions on a p-adic symmetric space $X=G/H$ as a module over the Iwahori Hecke algebra $\mathcal{H}(G,I)$. It introduces a natural affine Weyl group action $w\times x$ on the finite set of $I$-orbits in $X$, parameterized by $igcup_{T\in\mathcal{T}_\sigma/H}W^H_{aff}(T)$, and derives explicit action formulas for the Hecke generators $T_s$ and $T_\omega$ on basis elements $1_x$. A key geometric input is the Bruhat–Tits building and a length function $l_\sigma$ measuring chamber distances to their $\sigma$-images; together with a rank-one reduction, this yields the structure constants $\gamma_{f,g}$ controlling the $(T_s+1)$ action and establishes a finite generating set for $S(X)^I$ as an $\mathcal{H}(G,I)$-module. The paper also constructs a generic Hecke module $M_t$ over the generic Hecke algebra $\mathcal{H}_t$, with a specialization at $t=q$ recovering $S(X)^I$, and provides explicit examples for $G=SL_2$ and $G=SL_{2n}$ with $H=\text{Sp}_{2n}$ to illustrate the theory. This framework advances the understanding of distinguished representations and relative harmonic analysis on p-adic symmetric spaces by connecting orbit combinatorics, building geometry, and Hecke-module structures.

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.

Theorems & Definitions (117)

• Theorem 1.1
• Theorem 1.2
• Remark 1.3
• Theorem 1.4
• Theorem 1.6
• Proposition 1.7
• Lemma 1.8
• Definition 2.1: Coxeter group
• Theorem 2.2
• Proposition 2.3