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=CP(e)Ru(G) and that the assignment P↦CP(e) defines a one-to-one correspondence between parabolic subgroups of G and of CG(e).