2007.188: Rational Cherednik algebras and Hilbert schemes

2007.188: I. Gordon and J.T. Stafford (2005) Rational Cherednik algebras and Hilbert schemes. Advances in Mathematics, 198 (1). pp. 222-274. ISSN 0001-8708

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DOI: 10.1016/j.aim.2004.12.005

Abstract

Let $H_c$ be the rational Cherednik algebra of type $A_{n-1}$ with spherical subalgebra $U_c = e H_c e$. Then $U_c$ is filtered by order of differential operators, with associated graded ring $\mbox{gr} U_c = \mathbb{C} [ \mathfrak{h} \oplus \mathfrak{h}^*]$ where $W$ is the $n$th symmetric group. We construct a filtered $\mathbb{Z}$-algebra $b$ such that, under mild conditions on $c$:

• the category $B$-qgr of graded noetherian $B$-modules modulo torsion is equivalent to $U_c$-mod;

• the associated graded $\mathbb{Z}$-algebra $\mbox{gr}B$ has $\mbox{gr}B-lqgr \simeq \coh \Hilb(n)$

This can be regarded as saying that $U_c$ simultaneously gives a non-commutative deformation of $\mathfrak{h} \oplus \mathfrak{h}^* / W$ and of its resolution of singularities $\Hilb(n) \rightarrow \mathfrak{h} \oplus \mathfrak{h}^*$. As we show elsewhere, this result is a powerful tool for studying the representation theory of $H_c$ and its relationship to $\Hilb(n)$.

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