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

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 $\mathcal{H}_R(0, c_{\tilde{s}})$.