Geometric realizations of affine Hecke algebras with unequal parameters

TL;DR

This paper provides a geometric $K$-theoretic realization of affine Hecke algebras with two unequal parameters, extending previous equal-parameter results and offering a classification of simple modules for type $G_2$.

Contribution

It extends $K$-theoretic realizations to affine Hecke algebras with two unequal parameters, including exceptional types, and offers a geometric classification of simple modules.

Findings

Realization of affine Hecke algebras with two unequal parameters.

Extension of Kazhdan-Lusztig's equal parameter realization.

Full geometric classification of simple modules for $G_2$.

Abstract

We give a $K$-theoretic realization of all affine Hecke algebras with two unequal parameters including exceptional types. This extends the celebrated work of Kazhdan and Lusztig, who gave a $K$-theoretic realization of affine Hecke algebras with equal parameters, and complements results of Kato, who extended this construction to the three-parameter affine Hecke algebra of type $C$. A key idea behind our new construction is to exploit the reducibility of the adjoint representation in small characteristic. We also show that under suitable geometric conditions, our construction leads to a Deligne-Langlands style classification of simple modules. We verify these geometric conditions for $G_2$ thereby obtaining a full geometric classification of the simple modules for the affine Hecke algebra of $G_2$ with two parameters away from roots of unities.