On the reductive monoid associated to a parabolic subgroup

Abstract

Let $G$ be a connected reductive group over a perfect field $k$. We study a certain normal reductive monoid $\overline M$ associated to a parabolic $k$-subgroup $P$ of $G$. The group of units of $\overline M$ is the Levi factor $M$ of $P$. We show that $\overline M$ 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 $(G \times G)/(P \times_M P^-)$. Using idempotents, we relate $\overline M$ to the Vinberg semigroup of $G$. The monoid $\overline M$ is used implicitly in the study of stratifications of Drinfeld's compactifications of the moduli stacks $\mathrm{Bun}_P$ and $\mathrm{Bun}_G$.