On the Reductive Monoid Associated to a Parabolic Subgroup

Let G be a connected reductive group over a perfect field k .We study a certain normal reductive monoid M associated to a parabolic ksubgroup P of G.The group of units of M is the Levi factor M of P .We show that M is a retract of the affine closure of the quasi-affine variety G/U (P ) .Fixing a parabolic P -opposite to P , we prove that the affine closure of G/U (P ) is a retract of the affine closure of the boundary degeneration (G G)/(P M P -) .Using idempotents, we relate M to the Vinberg semigroup of G.The monoid M is used implicitly in the study of stratifications of Drinfeld's compactifications of the moduli stacks Bun P and Bun G .