Parabolic subgroups and algebraic monoids

An (affine) algebraic monoid is an affine variety over an algebraically closed field K endowed with a monoid structure such that the product map is an algebraic variety morphism. Let M be an irreducible algebraic monoid, G its unit group, P a parabolic subgroup of G, and e ∈ M a minimal idempotent. We show that P = C P ( e ) R u ( G ) and that the assignment P ↦ C P ( e ) defines a one-to-one correspondence between parabolic subgroups of G and of C G ( e ) .