Center of pro-p-Iwahori-Hecke algebra

Abstract

Let G be a connected reductive group over a p-adic eld F. The study of representations of G(F) naturally involves the pro-p-Iwahori-Heche algebra of G(F). The pro-p-Iwahori-Hecke algebra is a deformation of the group algebra of the pro-p-Iwahori Weyl group of G(F) with generic parameters. The pro- p-Iwahori-Hecke algebra with zero parameters plays an important role in the study of mod-p representations of G(F). In a series of paper, Vigneras introduced a generic algebra HR(q~s; c~s) which generalizes the pro-p-Iwahori-Hecke algebra of a reductive p-adic group. Vign- eras also gave a basis of the center of HR(q~s; c~s) when HR(q~s; c~s) is associated with a pro-p-Iwahori Weyl group. This basis is dened by using the Bernstein presentation of HR(q~s; c~s) and the alcove walk. In this article, we restrict to the case where q~s = 0 and give an explicit description of the center of HR(0; c~s) using the Iwahori-Matsumoto presentation. First, we introduce the generic algebra. Let W be the semidirect product of a Coxeter group and a group acting on the Coxeter group and stabilizing the generating set of the Coxeter group. Let W(1) be an extension of W with a commutative group. Let R be a commutative ring. We give the denition of the R-algebra HR(q~s; c~s) of W(1) with parameters (q~s; c~s). Then for any pair (v;w) in W W with v w, we dene a linear operator rv;w between R-submodules of HR(q~s; c~s). It takes some work to show that rv;w is well dened. Next, we restrict W to be an IwahoriWeyl group. We show that the maximal length terms of a central element in HR(q~s; c~s) is given by a union of nite conjugacy classes in W(1). Then we prove some techical results regarding rv;w acting on the maximal length terms of a central element in HR(q~s; c~s). In the last part, we restrict to the case when q~s = 0 and give a explicit basis of the center of HR(0; c~s) in the Iwahori-Matsumoto presentation by using the operator rv;w. Two examples are given to help understand how this basis looks like.