Journal of Lie Theory 31 (2021), No. 1, 127--148
Copyright Heldermann Verlag 2021
The Regularity of Almost-Commuting Partial Grothendieck-Springer Resolutions and Parabolic Analogs of Calogero-Moser Varieties
Mee Seong Im
Dept. of Mathematical Sciences, United States Military Academy, West Point, NY 10996, U.S.A.
im@usna.edu
Travis Scrimshaw
School of Mathematics and Physics, University of Queensland, St. Lucia, QLD 4072, Australia
t.scrimshaw@uq.edu.au
[Abstract-pdf]
Consider the moment map $\mu \colon T^*(\mathfrak{p} \times \mathbb{C}^n) \to \mathfrak{p}^*$ for a parabolic subalgebra $\mathfrak{p}$ of $\mathfrak{gl}_n(\mathbb{C})$. We prove that the preimage of $0$ under $\mu$ is a complete intersection when $\mathfrak{p}$ has finitely many $P$-orbits, where $P\subseteq \operatorname{GL}_n(\mathbb{C})$ is a parabolic subgroup such that $\operatorname{Lie}(P) = \mathfrak{p}$, and give an explicit description of the irreducible components. This allows us to study nearby fibers of $\mu$ as they are equidimensional, and one may also construct GIT quotients $\mu^{-1}(0) /\!\!/_{\chi} P$ by varying the stability condition $\chi$. Finally, we study a variety analogous to the scheme studied by Wilson with connections to a Calogero-Moser phase space where only some of particles interact.
Keywords: Grothendieck-Springer resolution, moment map, complete intersection.
MSC: 14M10, 53D20, 17B08, 14L30; 14L24, 20G20.
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