Tits Groups of Iwahori-Weyl Groups and Presentations of Hecke Algebras

Abstract

Let G be a connected reductive group over a non-archimedean local field F and I be an Iwahori subgroup of G(F). Let In is the n-th Moy-Prasad filtration subgroup of I. The purpose of this paper is two-fold: to give some nice presentations of the Hecke algebra of connected, reductive groups with In -level structure; and to introduce the Tits group of the Iwahori-Weyl group of groups G that split over an unramified extension of F. The first main result of this paper is a presentation of the Hecke algebra H(G(F) , In) , generalizing the previous work of Iwahori-Matsumoto on the affine Hecke algebras. For split GLn , Howe gave a refined presentation of the Hecke algebra H(G(F) , In) . To generalize such a refined presentation to other groups requires the existence of some nice lifting of the Iwahori-Weyl group W to G(F). The study of a certain nice lifting of W is the second main motivation of this paper, which we discuss below. In 1966, Tits introduced a certain subgroup of G(k) for any algebraically closed field k , which is an extension of the finite Weyl group W by an elementary abelian 2-group. This group is called the Tits group and provides a nice lifting of the elements in the finite Weyl group. The “Tits group” T for the Iwahori-Weyl group W is a certain subgroup of G(F), which is an extension of the Iwahori-Weyl group W by an elementary abelian 2-group. The second main result of this paper is a construction of Tits group T for W when G splits over an unramified extension of F. As a consequence, we generalize Howe’s presentation to such groups. We also show that when G is ramified over F, such a group T of W may not exist.