Let $H_c$ be the rational Cherednik algebra of type $A_{n-1}$ with spherical subalgebra $U_c=eH_ce$ . Then $U_c$ is filtered by order of differential operators with associated graded ring ${\rm gr}U_c=\mathbb{C}[\mathfrak{h}\oplus\mathfrak{h}^*]^{W}$ , 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 $\widehat{\Phi}(M)$ 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 $\tau : {\rm Hilb(n)}\to \mathfrak{h}\oplus\mathfrak{h}^*/W$ . For example, we prove the following.
¶ • If $c=1/n$ so that $L_c(\ttt{triv})$ is the unique one-dimensional simple $H_c$ -module, then $\widehat{\Phi} (eL_c(\ttt{triv})) \cong \mathcal{O}_{Z_n}$ , 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 $L_c(\ttt{triv})$ , ${\rm gr} eL_{c}(\ttt{triv})$ has a natural bigraded structure that coincides with that on $\mathrm{H}^0(Z_n, \mathcal{L}^k)$ , where $\mathcal{L}\cong\mathcal{O}_{{\rm Hilb(n)}}(1)$ ; this confirms conjectures of Berest, Etingof, and Ginzburg [BEG2, Conjectures 7.2, 7.3].
¶ • Under mild restrictions on $c$ , the characteristic cycle of $\widehat{\Phi} (e\Delta_c(\mu))$ equals $\sum_{\lambda}K_{\mu\lambda}[Z_\lambda]$ , where $K_{\mu\lambda}$ are Kostka numbers and the $Z_{\lambda }$ are (known) irreducible components of $\tau^{-1}(\mathfrak{h}/W)$