An isomorphism theorem for Lusztig algebras

TL;DR

This paper proves an explicit isomorphism between Lusztig's algebra components and matrix algebras over tensor products of Hecke and group algebras, providing new structural insights.

Contribution

It establishes a concrete algebra isomorphism for Lusztig's algebras, linking them to matrix algebras over tensor products of Hecke and group algebras.

Findings

Explicit algebra isomorphisms are constructed.

Structural understanding of Lusztig algebras is enhanced.

Applications of the isomorphism are demonstrated.

Abstract

In [Lu6] Lusztig defined a certain algebra $H,$ which is a direct sum of various algebras $H_{\mathfrak{o}}.$ We establish an explicit algebra isomorphism between each algebra $H_{\mathfrak{o}}$ and some matrix algebra with coefficients in the tensor product of an (affine or finite) Hecke algebra and a group algebra. We give an application.