Complex representations of matrix semigroups

Abstract

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