Simple type theory for metaplectic covers of $\mathrm{GL}(r)$ over a non-archimedean local field

TL;DR

This paper develops a type-theoretic framework for understanding representations of metaplectic covers of GL(r) over non-archimedean fields, constructing types and describing associated Hecke algebras.

Contribution

It introduces maximal simple types for metaplectic covers and proves the explicit construction of cuspidal representations via compact induction, advancing the representation theory of these covers.

Findings

Construction of maximal simple types as inertial types.

Proof that all cuspidal representations can be explicitly constructed.

Description of Hecke algebras as affine Hecke algebras of type A.

Abstract

Let $F$ be a non-archimedean locally compact field of residual characteristic $p$, let $G=\mathrm{GL}_{r}(F)$ and let $\widetilde{G}$ be an $n$-fold metaplectic cover of $G$ with $\mathrm{gcd}(n,p)=1$. We study the category $\mathrm{Rep}_{\mathfrak{s}}(\widetilde{G})$ of complex smooth representations of $\widetilde{G}$ having inertial equivalence class $\mathfrak{s}=(\widetilde{M},\mathcal{O})$, which is a block of the category $\mathrm{Rep}(\widetilde{G})$, following the "type theoretical" strategy of Bushnell-Kutzko. Precisely, first we construct a "maximal simple type" $(\widetilde{J_{M}},\widetilde{\lambda}_{M})$ of $\widetilde{M}$ as an $\mathfrak{s}_{M}$-type, where $\mathfrak{s}_{M}=(\widetilde{M},\mathcal{O})$ is the related cuspidal inertial equivalence class of $\widetilde{M}$. Along the way, we prove the forklore conjecture that every cuspidal representation of $\widetilde{M}$ could be constructed explicitly by a compact induction. Secondly, we construct "simple types" $(\widetilde{J},\widetilde{\lambda})$ of $\widetilde{G}$, and prove that each of them is an $\mathfrak{s}$-type of a certain block $\mathrm{Rep}_{\mathfrak{s}}(\widetilde{G})$. When $\widetilde{G}$ is either a Kazhdan-Patterson cover or Savin's cover, the corresponding blocks turn out to be those containing discrete series representations of $\widetilde{G}$. Finally, for a simple type $(\widetilde{J},\widetilde{\lambda})$ of $\widetilde{G}$ we describe the related Hecke algebra $\mathcal{H}(\widetilde{G},\widetilde{\lambda})$, which turns out to be not far from an affine Hecke algebra of type A, and is exactly so if $\widetilde{G}$ is one of the two special covers mentioned above. We leave the construction of a "semi-simple type" related to a general block $\mathrm{Rep}_{\mathfrak{s}}(\widetilde{G})$ to a future phase of the work.