Iwahori's Question for Affine Hecke Algebras

TL;DR

This paper proves that affine Hecke algebras are not isomorphic to group algebras of extended affine Weyl groups under certain conditions related to the root system type and parameter order.

Contribution

It establishes specific non-isomorphism results for affine Hecke algebras with complex parameters, refining understanding of their algebraic structure.

Findings

Affine Hecke algebra $H_q$ is not isomorphic to the group algebra of the extended affine Weyl group under given conditions.

The non-isomorphism holds when the root system has no $A_1$ factors and $q$'s order is not 11 or 13 for $E_8$ factors.

Provides conditions distinguishing affine Hecke algebras from group algebras in complex settings.

Abstract

In this paper we show that an affine Hecke algebra $H_q$ over complex numbers field with parameter $q\ne 1$ is not isomorphic to the group algebra over complex numbers field of the corresponding extend affine Weyl group if the corresponding root system has no factors of type $A_1$ and the order of $q$ is different from 11 and 13 if the root system has factors of type $E_8$.