I. Gordon, J. T. Stafford

Duke Math. J. 132 (1), 73-135, (15 March 2006) DOI: 10.1215/S0012-7094-06-13213-1
KEYWORDS: 14C05, 16D90, 32S45, 16S80, 05E10

Let ${H}_{c}$ be the rational Cherednik algebra of type ${A}_{n-1}$ with spherical subalgebra ${U}_{c}={\mathrm{eH}}_{c}e$. Then ${U}_{c}$ is filtered by order of differential operators with associated graded ring $\mathrm{gr}{U}_{c}=\mathbb{C}[\mathfrak{h}\oplus {\mathfrak{h}}^{*}]{}^{W}$, where $W$ is the nth symmetric group. Using the $\mathbb{Z}$-algebra construction from [GS], it is also possible to associate to a filtered ${H}_{c}$- or ${U}_{c}$-module $M$ a coherent sheaf $\stackrel{\wedge}{\Phi}(M)$ on the Hilbert scheme *Hilb(n)*. Using this technique, we study the representation theory of ${U}_{c}$ and ${H}_{c}$, and we relate it to *Hilb(n)* and to the resolution of singularities $\tau :\mathrm{Hilb}(n)\to \mathfrak{h}\oplus {\mathfrak{h}}^{*}/W$. For example, we prove the following.

• If $c=1/n$ so that ${L}_{c}(\mathsf{triv})$ is the unique one-dimensional simple ${H}_{c}$-module, then $\stackrel{\wedge}{\Phi}({\mathrm{eL}}_{c}(\mathsf{triv}))\cong {\mathcal{O}}_{{Z}_{n}}$, where ${Z}_{n}={\tau}^{-1}(0)$ is the *punctual Hilbert scheme*.

• If $c=1/n+k$ for $k\in \mathbb{N}$, then under a canonical filtration on the finite-dimensional module ${L}_{c}(\mathsf{triv})$, $\mathrm{gr}{\mathrm{eL}}_{c}(\mathsf{triv})$ has a natural bigraded structure that coincides with that on ${H}^{0}({Z}_{n},{\mathcal{L}}^{k})$, where $\mathcal{L}\cong {O}_{\mathrm{Hilb}(n)}(1)$; this confirms conjectures of Berest, Etingof, and Ginzburg [BEG2, Conjectures 7.2, 7.3].

• Under mild restrictions on $c$, the characteristic cycle of $\stackrel{\wedge}{\Phi}(e{\Delta}_{c}(\mu ))$ equals $\sum _{\lambda}{K}_{\mu \lambda}[{Z}_{\lambda}]$, where ${K}_{\mu \lambda}$ are Kostka numbers and the ${Z}_{\lambda}$ are (known) irreducible components of ${\tau}^{-1}(\mathfrak{h}/W)$