Hecke Algebras and Semisimplicity of Monoid Algebras

We begin by constructing Hecke algebras for arbitrary finite regular monoids M. We then show that the semisimplicity of the complex monoid algebra M is equivalent to the semisimplicity of the associated Hecke algebras and a condition on induced group characters. We apply these results to finite Lie-type monoids M. We show that the monoid algebra FM over a field F is semisimple if and only if the characteristic of F does not divide the order of the unit group G. This is accomplished by developing formulas for the unities of J, J a -class of M. The unity is explicitly given when G is a simply connected Chevalley group and J is associated with a Borel subgroup of G.