Complex Representations of Matrix Semigroups

Let M be a finite monoid of Lie type (these are the finite analogues of linear algebraic monoids) with group of units G . The multiplicative semigroup ^AfF), where F is a finite field, is a particular example. Using HarishChandra's theory of cuspidal representations of finite groups of Lie type, we show that every complex representation of M is completely reducible. Using this we characterize the representations of G extending to irreducible representations of M as being those induced from the irreducible representations of certain parabolic subgroups of G . We go on to show that if F is any field and S any multiplicative subsemigroup of J?n (F), then the semigroup algebra of S over any field of characteristic zero has nilpotent Jacobson radical. If S = ^A„(F), then this algebra is Jacobson semisimple. Finally we show that the semigroup algebra of ^„(F) over a field of characteristic zero is regular if and only if ch(F) = p > 0 and F is algebraic over its prime field.