On two modular geometric realizations of an affine Hecke algebra

TL;DR

This paper establishes equivalences between three geometric and representation-theoretic models of the affine Hecke algebra for a reductive algebraic group over a field of positive characteristic, extending localization theory and confirming a conjecture by Finkelberg-Mirkovi7.

Contribution

It constructs new equivalences of monoidal categories linking different geometric and algebraic realizations of the affine Hecke algebra in positive characteristic.

Findings

Proves a conjecture by Finkelberg-Mirkovi7 on geometric realization of the principal block.

Develops a positive-characteristic analogue of existing localization constructions.

Provides a unified categorical framework for affine Hecke algebra representations.

Abstract

In this paper we construct equivalences of monoidal categories relating three geometric or representation-theoretic categorical incarnations of the affine Hecke algebra of a connected reductive algebraic group $G$ over a field of positive characteristic: a category of Harish-Chandra bimodules for the Lie algebra of $G$; the derived category of equivariant coherent sheaves on (a completed version of) the Steinberg variety of the Frobenius twist $G^{(1)}$ of $G$; a derived category of constructible sheaves on the affine flag variety of reductive group which is Langlands dual to $G^{(1)}$. These constructions build on the localization theory developed by the first author with Mirkovi\'c and Rumynin and previous work of ours (partly joint with L. Rider), and provide an analogue for positive-characteristic coefficients of a construction of the first author. As an application, we prove a conjecture by Finkelberg-Mirkovi\'c giving a geometric realization of the principal block of algebraic representations of $G$.