Genuine pro-$p$ Iwahori–Hecke algebras, Gelfand–Graev representations, and some applications

Abstract

We study the Iwahori-component of the Gelfand–Graev representation of a central cover of a split linear reductive group and utilize our results for three applications. In fact, it is advantageous to begin at the pro-$p$ level. Thus to begin we study the structure of a genuine pro-$p$ Iwahori–Hecke algebra, establishing a Bernstein presentation. With this structure theory we first describe the pro-$p$ part of the Gelfand–Graev representation and then the more subtle Iwahori part. For the first application we relate the Gelfand–Graev representation to the metaplectic representation of Sahi–Stokman–Venkateswaran, which conceptually realizes the Chinta–Gunnells action from the theory of Weyl group multiple Dirichlet series. For the second we compute the Whittaker dimension of the constituents of regular unramified principal series representations; for the third we do the same for unitary unramified principal series representations.