Affine Hecke algebras and their representations

TL;DR

This survey explores the algebraic aspects of affine Hecke algebras and their representations, culminating in a new bijection linking irreducible representations of the algebra to those of the affine Weyl group, generalizing Springer correspondence.

Contribution

The paper provides a comprehensive overview of affine Hecke algebra representations and introduces a novel bijection between their irreducible representations and those of the affine Weyl group.

Findings

Established a bijection between irreducible representations of affine Hecke algebras and affine Weyl groups.

Extended Springer correspondence to affine Hecke algebras with real parameters ≥ 1.

Reviewed algebraic aspects of affine Hecke algebra representation theory.

Abstract

This is a survey paper about affine Hecke algebras. We start from scratch and discuss some algebraic aspects of their representation theory, referring to the literature for proofs. We aim in particular at the classification of irreducible representations. Only at the end we establish a new result: a natural bijection between the set of irreducible representations of an affine Hecke algebra with real parameters $\geq 1$, and the set of irreducible representations of the affine Weyl group underlying the algebra. This can be regarded as a generalized Springer correspondence with affine Hecke algebras.