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 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.

Metaplectic Covers of $p$-adic Groups and Quantum Groups at Roots of Unity

“…Building upon (what we understand were) ideas and suggestions by G. Savin and S. Lysenko, the recent work [38] approaches W ψ ( G, I − ) by working at the level of pro-p Iwahori subgroups. Using this work, one can extract an alternate proof of Proposition 6.2.1.…”

“…(4) Note that in[24,38], the conductor of the additive character is taken to be −1; this introduces a "ρ"-shift when comparing formulas with these sources. On the other hand, our choice of conductor is in alignment with the classical work of Casselman-Shalika[23] and the literature on its geometric version[32,33].…”