Complex representations of matrix semigroups

Okniński, Jan; Putcha, Mohan S.

doi:10.1090/S0002-9947-1991-1020044-8

Complex representations of matrix semigroups
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by Jan Okniński and Mohan S. Putcha PDF

Trans. Amer. Math. Soc. 323 (1991), 563-581 Request permission

Abstract:

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 ${\mathcal {M}_n}(F)$, where $F$ 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 $M$ 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 ${\mathcal {M}_n}(F)$, then the semigroup algebra of $S$ over any field of characteristic zero has nilpotent Jacobson radical. If $S = {\mathcal {M}_n}(F)$, then this algebra is Jacobson semisimple. Finally we show that the semigroup algebra of ${\mathcal {M}_n}(F)$ over a field of characteristic zero is regular if and only if $\operatorname {ch} (F) = p > 0$ and $F$ is algebraic over its prime field.

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