Koszul duality for generalised Steinberg representations of $p$-adic groups

TL;DR

This paper establishes a Koszul duality between the extension algebra of generalized Steinberg representations of a p-adic group and a category of equivariant perverse sheaves on the Langlands parameter variety, linking representation theory and geometric Langlands.

Contribution

It proves a new Koszul duality equivalence connecting module categories of extension algebras with categories of perverse sheaves on Langlands parameter spaces for p-adic groups.

Findings

Extension algebra of generalized Steinberg representations is Koszul dual to endomorphism algebra of perverse sheaves.

Establishes an equivalence between module categories and perverse sheaves related to Langlands parameters.

Demonstrates the duality concretely in the setting of split semisimple p-adic groups.

Abstract

Let $G$ be a semisimple group, split over a non-Archimedean field $F$. We prove that the category of modules over the extension algebra of generalised Steinberg representations of $G(F)$ is equivalent to a full subcategory of equivariant perverse sheaves on the variety of Langlands parameters for these representations. Specifically, we establish an equivalence \[ \textbf{Mod}(\text{Ext}_G^\bullet(\Sigma_\lambda, \Sigma_\lambda)) \simeq \textbf{Per}_{\widehat{G}}^\circ(X_\lambda), \] where $\Sigma_\lambda$ is the direct sum of generalised Steinberg representations and $\textbf{Per}_{\widehat{G}}^\circ(X_\lambda)$ is the subcategory of perverse sheaves on the variety of Langlands parameters $X_\lambda$ corresponding to these representations under Vogan's geometrisation of the Langlands correspondence. Furthermore, we demonstrate that this equivalence is a true Koszul duality by showing that the extension algebra of generalised Steinberg representations is Koszul dual to the endomorphism algebra of the direct sum of corresponding equivariant perverse sheaves, taken in the equivariant derived category $D_{\widehat{G}}^b(X_\lambda)$.