Convolution Algebras for Finite Reductive Monoids

TL;DR

This paper establishes a correspondence between irreducible representations of finite monoids and their convolution algebras, extending to reductive monoids and providing a proof of Frobenius Reciprocity for monoids.

Contribution

It introduces a bijection between irreducible monoid representations with fixed vectors and convolution algebra representations, linking monoid and Renner monoid representations.

Findings

Bijection between monoid and convolution algebra irreducible representations

Connection between monoid representations and Renner monoid representations in reductive cases

Quick proof of Frobenius Reciprocity for monoids

Abstract

For an arbitrary finite monoid $M$ and subgroup $K$ of the unit group of $M$, we prove that there is a bijection between irreducible representations of $M$ with nontrivial $K$-fixed space and irreducible representations of $\mathcal{H}_K$, the convolution algebra of $K\times K$-invariant functions from $M$ to $F$, where $F$ is a field of characteristic not dividing $|K|$. When $M$ is reductive and $K = B$ is a Borel subgroup of the group of units, this indirectly provides a connection between irreducible representations of $M$ and those of $F[R]$, where $R$ is the Renner monoid of $M$. We conclude with a quick proof of Frobenius Reciprocity for monoids for reference in future papers.