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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.

Travis Scrimshaw
School of Mathematics and Physics, University of Queensland, St. Lucia, QLD 4072, Australia


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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