Minimal depth $K$-types for wild double covers and Shimura correspondences

TL;DR

This paper constructs specific types for double covers of Chevalley groups over 2-adic fields, establishing Hecke algebra presentations and Shimura correspondences, advancing understanding of representations in this setting.

Contribution

It introduces minimal depth Iwahori types for double covers of Chevalley groups, proves their Hecke algebras admit explicit IM presentations, and extends these types to maximal compact subgroups, enabling new Shimura correspondences.

Findings

Hecke algebras admit Iwahori-Matsumoto presentations for certain types.

Explicit Shimura correspondences are established for these double covers.

Minimal depth genuine representations are constructed for maximal compact subgroups.

Abstract

We construct some Iwahori types, in the sense of Bushnell-Kutzko, for the double cover of an almost simple simply-laced simply-connected Chevalley group $\widetilde{G}$ over any $2$-adic field. These types capture the covering group analog of the Bernstein block of unramified principal series. We also prove that the associated Hecke algebra essentially admits an Iwahori-Matsumoto (IM) presentation. The complete presentation is obtained for types $A_{r}$, $D_{2r+1}$, $E_{6}$, $E_{7}$; for the other types, some technical obstacles remain. Those Hecke algebras with the complete IM presentation are isomorphic to Iwahori-Hecke algebras of explicit linear Chevalley groups, giving rise to Shimura correspondences. Along the way, we show that the Iwahori type extends to a hyperspecial maximal compact subgroup $\widetilde{K}\subseteq \widetilde{G}$. This extension has minimal depth among the genuine $\widetilde{K}$-representations and allows us to construct a finite Shimura correspondence, generalizing a result of Savin.