The based ring of the lowest two-sided cell of an affine Weyl group, III

TL;DR

This paper explores the relationship between Lusztig's homomorphism and Chriss-Ginzburg's construction in the context of affine Hecke algebras, revealing a matrix connection over the representation ring.

Contribution

It establishes a link between Lusztig's homomorphism and Chriss-Ginzburg's equivariant K-theory approach via a matrix over the representation ring.

Findings

Lusztig's homomorphism relates to Chriss-Ginzburg's construction.

The relationship is expressed through a matrix over the representation ring.

Provides new insights into the structure of affine Hecke algebras.

Abstract

We show that Lusztig's homomorphism from an affine Hecke algebra to the direct summand of its asymptotic Hecke algebra corresponding to the lowest two-sided cell is related to the homomorphism constructed by Chriss and Ginzburg using equivariant K-theory by a matrix over the representation ring of the associated algebraic group.