Representations of the elliptic affine Hecke algebras

TL;DR

This paper establishes a correspondence between irreducible representations of elliptic affine Hecke algebras and certain nilpotent Higgs bundles on elliptic curves, using equivariant elliptic cohomology and exploring implications in gauge theory.

Contribution

It provides a novel link between algebraic representations and geometric objects, introduces elliptic Demazure-Lusztig operators, and studies representations at roots of unity.

Findings

Irreducible representations correspond to nilpotent Higgs bundles.

Defined elliptic Demazure-Lusztig operators with dynamical parameters.

Explored implications for 4d N=2 gauge theory.

Abstract

We prove that irreducible representations of the elliptic affine Hecke algebras of Ginzburg, Kapranov, and Vasserot are in one-to-one correspondence with certain nilpotent Higgs bundles on the elliptic curve. The main tool we use is the equivariant elliptic cohomology of the Steinberg variety of the Springer resolution. As a by-product, we study representations at roots of unity in type-$A$. As another by-product, we define a version of elliptic Demazure-Lusztig operators with dynamical parameters that satisfy the braid relations. We discuss speculative indications of this correspondence in 4d $\mathcal N=2$ gauge theory.