Gelfand--Graev representation as a Hecke algebra module of simple types of a finite central cover of $\mathrm{GL}(r)$

TL;DR

This paper expresses the Gelfand--Graev representation as a Hecke algebra module for certain covers of , enabling explicit calculations of Whittaker dimensions for all discrete series and irreducible representations.

Contribution

It provides a new realization of the Gelfand--Graev representation as a Hecke algebra module for specific covers of , facilitating explicit Whittaker dimension computations.

Findings

Explicit expression of Gelfand--Graev representation as a Hecke algebra module

Calculation of Whittaker dimensions for all discrete series representations

Extension of results to all irreducible representations via Zelevinsky's classification

Abstract

For an $n$-fold Kazhdan--Patterson cover or Savin's cover of a general linear group over a non-archimedean local field of residual characteristic $p$ with $\mathrm{gcd}(n,p)=1$, we realize the Gelfand--Graev representation as a Hecke algebra module of a simple type and study its explicit expression. As a main corollary, we calculate the Whittaker dimension of every discrete series representation of such a cover. Using Zelevinsky's classification, this theoretically gives the Whittaker dimension of every irreducible representation.