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

TL;DR

This paper constructs genuine Iwahori-type data for the wild 2-fold cover $\tilde{G}$ of an almost simple simply-connected Chevalley group over a 2-adic field, aiming to mirror the Bernstein block for unramified principal series in the linear case. The authors develop a pseudo-spherical, Weyl-invariant theory for the covering torus $\tilde{T}$, and use parahoric induction to build a Wakimoto-like Iwahori type $\sigma$ that extends to a hyperspecial subgroup $\tilde{K}$, enabling a finite Shimura correspondence and significant control over corresponding Hecke algebras. They prove a (weak) IM presentation in general and a complete IM presentation in types $A_r$, $D_{2r+1}$, $E_6$, and $E_7$, with partial results in $D_{2r}$ and $E_8$, yielding a local Shimura-type equivalence to Iwahori-Hecke algebras of explicit linear groups in the complete cases. The work leverages a detailed analysis of the $2$-adic Hilbert symbol, a refined description of the central structure of the covering torus, and the parahoric-induction framework to translate representation-theoretic questions for $\tilde{G}$ into tractable questions about linear groups, thereby producing a Shimura correspondence at the level of blocks. Overall, the results provide a robust framework for understanding wild 2-fold covers via explicit types, Hecke algebras, and finite-group correspondences, with potential applications to constructing minimal-depth representations and enabling transfer of Hecke-algebra data to linear groups."

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.

Theorems & Definitions (228)

• Theorem 1.2: Propositions \ref{['Gamma12eSplitting']}, \ref{['P:SplitNormIm']}
• Theorem 1.3: Theorem \ref{['KRep']}
• Theorem 1.4: Theorem \ref{['FiniteShimCor']}
• Theorem 1.5: Theorem \ref{['IType']}
• Theorem 1.6: Corollary \ref{['C:local_Shimura_Correspondence']}
• Proposition 2.2
• proof
• Corollary 2.3
• proof
• Proposition 2.4