Invariant Algebra and Cuspidal Representations of Finite Monoids

Motivated by the theories of Hecke algebras and Schur algebras, we consider in this paper the algebra MGofG-invariants of a finite monoidMwith unit groupG. IfMis a regular “balanced” monoid, we show that MGis a quasi-hereditary algebra. In such a case, we find the blocks of MGto be the “sections” of the blocks of M. We go on to develop a theory of cuspidal representations for balanced monoids. We then apply our results to the full transformation semigroup and the multiplicative monoid of triangular matrices over a finite field.