Convolution Algebras for Finite Reductive 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 H K , the convolution algebra of K × 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.