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Journal of Lie Theory 27 (2017), No. 3, 637--655
Copyright Heldermann Verlag 2017

On the Reductive Monoid Associated to a Parabolic Subgroup

Jonathan Wang
Dept. of Mathematics, University of Chicago, Chicago, IL 60637, U.S.A.


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 ${\rm Bun}_P$ and ${\rm Bun}_G$.

Keywords: Reductive monoid, affine embedding of homogeneous space, boundary degeneration, Vinberg semigroup.

MSC: 14M17, 14R20, 20M32

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