The regularity of almost-commuting partial Grothendieck--Springer resolutions and parabolic analogs of Calogero--Moser varieties

Abstract

Consider the moment map $\mu :T^{\ast }(\mathfrak{p}\times {\mathbb{C}}^n)\to {\mathfrak{p}}^{\ast }$ 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 $P\subseteq {\mathrm{GL}}_n(\mathbb{C})$ is a parabolic subgroup such that $\mathrm{Lie}(P)=\mathfrak{p}$, 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.

18 pages, to appear in Journal of Lie Theory