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 ${\mathrm{Bun}}_G$.

15 pages