Two invariant subalgebras of rational Cherednik algebras

TL;DR

The paper develops a uniform, invariant-theoretic framework for two natural subalgebras of rational Cherednik algebras, realizing them as rings of invariants under reductive subgroups of $ m SL_2$. It establishes centres, Auslander--Gorenstein and Cohen--Macaulay properties, and PI behaviour at $t=0$, and describes how quantum Hamiltonian reduction connects to deformations of the minimal nilpotent orbit quotient by $W$. A key tool is the invariant-theoretic realization that allows a decomposition into twisted summands via primitive idempotents, enabling a controlled analysis of centres, simple modules, and double-centralizer phenomena. The work also links graded/filtered structures to Poisson geometry, showing that reductions yield (graded) deformations of symplectic singularities such as $\overline{\mathcal O}_{\min}/W$, with implications for representation theory and geometric representation theory. Overall, it provides a cohesive, structurally uniform treatment of the two invariant subalgebras and their reductions, with concrete centre computations and a robust framework for further extensions in Hamiltonian reduction and deformations.

Abstract

Originally motivated by connections to integrable systems, two natural subalgebras of the rational Cherednik algebra have been considered in the literature. The first is the subalgebra generated by all degree zero elements and the second is the Dunkl angular momentum subalgebra. In this article, we study the ring-theoretic and homological properties of these algebras. Our approach is to realise them as rings of invariants under the action of certain reductive subgroups of $\rm SL_2$. This allows us to describe their centres. Moreover, we show that they are Auslander-Gorenstein and Cohen-Macaulay and, at $t = 0$, give rise to prime PI-algebras whose PI-degree we compute. Since the degree zero subalgebra can be realized as the ring of invariants for the maximal torus $\rm T \subset SL_2$ and the action of this torus on the rational Cherednik algebra is Hamiltonian, we also consider its (quantum) Hamiltonian reduction with respect to $\rm T$. At $t = 1$, the quantum Hamiltonian reduction of the spherical subalgebra is a filtered quantization of the quotient of the minimal nilpotent orbit closure $\overline{\mathcal O}_{\min}$ in ${\mathfrak gl}(n)$ by the reflection group $W$. At $t = 0$, we get a graded Poisson deformation of the symplectic singularity $\overline{\mathcal O}_{\min}/W$.

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