Rational Cherednik algebras and Hilbert schemes, II: Representations and sheaves

Abstract

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 ${\tau }^{-1}(\mathfrak{h}/W)$