Hecke algebras for tame genuine principal series and local Shimura correspondence

TL;DR

This paper extends the local Shimura correspondence to covering groups, analyzing the structure of associated Hecke algebras and constructing types, revealing similarities and differences with endoscopy groups.

Contribution

It develops a framework for Hecke algebras of genuine principal series in covering groups and explores their relation to endoscopy groups, extending prior theories.

Findings

Hecke algebras share the same affine part as endoscopy groups

Construction of types for genuine principal series

Counterexample showing non-isomorphism in general

Abstract

In this paper, we extend the local Shimura correspondence building upon the groundwork laid by Gordan Savin. As preparations, we review part of the type theory of Bushnell and Kutzko which equally applies to covering groups. By adapting the method of linear algebraic groups, we describe the structure of Hecke algebras associated with genuine principal series components and construct the types. In particular, we show each of them shares the same affine Hecke algebra part as one corresponding Hecke algebra of the principal endoscopy group. However, we give a counter example to show they are not isomorphic in general.