Representation Theory of Reductive Normal Algebraic Monoids

New results in the representation theory of ``semisimple' algebraic monoids are obtained, based on Renner's monoid version of Chevalley's big cell. (The semisimple algebraic monoids have been classified by Renner.) The rational representations of such a monoid are the same thing as ``polynomial' representations of the associated reductive group of units in the monoid, and this representation category splits into a direct sum of subcategories by ``homogeneous' degree. We show that each of these homogeneous subcategories is a highest weight category, in the sense of Cline, Parshall, and Scott, and so equivalent with the module category of a certain finite-dimensional quasihereditary algebra, which we show is a generalized Schur algebra in S. Donkin's sense.