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- level. Thus to begin we study the structure of a genuine pro- Iwahori-Hecke algebra, establishing Iwahori-Matsumoto and Bernstein presentations. With this structure theory we first describe the pro- 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.
Keywords
Cite
@article{arxiv.2204.13053,
title = {Genuine pro-$p$ Iwahori--Hecke algebras, Gelfand--Graev representations, and some applications},
author = {Fan Gao and Nadya Gurevich and Edmund Karasiewicz},
journal= {arXiv preprint arXiv:2204.13053},
year = {2022}
}