Let be the rational Cherednik algebra of type with spherical subalgebra . Then is filtered by order of differential operators with associated graded ring , where is the nth symmetric group. Using the -algebra construction from [GS], it is also possible to associate to a filtered - or -module a coherent sheaf on the Hilbert scheme Hilb(n). Using this technique, we study the representation theory of and , and we relate it to Hilb(n) and to the resolution of singularities . For example, we prove the following.
• If so that is the unique one-dimensional simple -module, then , where is the punctual Hilbert scheme.
• If for , then under a canonical filtration on the finite-dimensional module , has a natural bigraded structure that coincides with that on , where ; this confirms conjectures of Berest, Etingof, and Ginzburg [BEG2, Conjectures 7.2, 7.3].
• Under mild restrictions on , the characteristic cycle of equals , where are Kostka numbers and the are (known) irreducible components of
"Rational Cherednik algebras and Hilbert schemes, II: Representations and sheaves." Duke Math. J. 132 (1) 73 - 135, 15 March 2006. https://doi.org/10.1215/S0012-7094-06-13213-1