Journal of Lie Theory

no. 3
On the Reductive Monoid Associated to a Parabolic Subgroup
Journal of Lie Theory, Volume 27 (2017) no. 3, pp. 637-655
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 ${\rm Bun}_G$.
DOI: 10.5802/jolt.963
Classification: 14M17, 14R20, 20M32
Keywords: Reductive monoid, affine embedding of homogeneous space, boundary degeneration, Vinberg semigroup 
@article{JOLT_2017_27_3_a1,
     author = {J. Wang},
     title = {On the {Reductive} {Monoid} {Associated} to a {Parabolic} {Subgroup}},
     journal = {Journal of Lie Theory},
     pages = {637--655},
     year = {2017},
     volume = {27},
     number = {3},
     doi = {10.5802/jolt.963},
     zbl = {1400.14132},
     url = {https://jolt.centre-mersenne.org/articles/10.5802/jolt.963/}
}
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J. Wang. On the Reductive Monoid Associated to a Parabolic Subgroup. Journal of Lie Theory, Volume 27 (2017) no. 3, pp. 637-655. doi: 10.5802/jolt.963

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