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

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 Iwahori-Matsumoto and Bernstein presentations. 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; for the third we do the same for unitary unramified principal series.