Center of $\mathcal{H}_R(0,c_{\tilde{s}})$ associated to a pro-$p$-Iwahori Weyl group

TL;DR

This paper investigates the structure of the center of a specialized Hecke algebra associated with a pro-$p$-Iwahori Weyl group, providing a new basis in the case where a parameter is zero, using the Iwahori-Matsumoto presentation.

Contribution

It introduces a basis for the center of the Hecke algebra $ ext{H}_R(0, c_{ ilde{s}})$ using the Iwahori-Matsumoto presentation, focusing on the case where $q_{ ilde{s}}=0$.

Findings

Established a basis for the center of $ ext{H}_R(0, c_{ ilde{s}})$.

Utilized the Iwahori-Matsumoto presentation for the algebra.

Focused on the case $q_{ ilde{s}}=0$.

Abstract

Let $W$ be an Iwahori Weyl group and $W(1)$ be an extension of $W$ by an abelian group. Vigneras gave a description of the $R$-algebra $\mathcal{H}_R(q_{\tilde{s}}, c_{\tilde{s}})$ associated to $W(1)$, and also gave a basis of the center of $\mathcal{H}_R(q_{\tilde{s}}, c_{\tilde{s}})$ using the Bernstein presentation of $\mathcal{H}_R(q_{\tilde{s}}, c_{\tilde{s}})$. In this paper, we restrict to the case where $q_{\tilde{s}}=0$ and use the Iwahori-Matsumoto presentation to give a basis of the center of $\mathcal{H}_R(0, c_{\tilde{s}})$.