Generic Hecke algebra for Renner monoids

TL;DR

This paper introduces a generic Hecke algebra for Renner monoids, generalizing classical Hecke algebras associated with finite reductive monoids, and answers a longstanding open question in the field.

Contribution

It constructs a universal Hecke algebra for Renner monoids over erstein rings, extending the theory of Hecke algebras to a broader class of algebraic structures.

Findings

Defines the generic Hecke algebra al{H}(R) for Renner monoids.

Shows how to recover classical Iwahori-Hecke algebras via specialization.

Provides a positive answer to L. Solomon's long-standing question.

Abstract

We associate with every Renner monoid $R$ a \emph{generic Hecke algebra} $\H(R)$ over $\mathbb{Z}[q]$ which is a deformation of the monoid $\mathbb{Z}$-algebra of $R$. If $M$ is a finite reductive monoid with Borel subgroup $B$ and associated Renner monoid $R$, then we obtain the associated Iwahori-Hecke algebra $\H(M,B)$ by specialising $q$ in $\H(R)$ and tensoring by $\mathbb{C}$ over $\mathbb{Z}$, as in the classical case of finite algebraic groups. This answers positively to a long-standing question of L. Solomon.