Duke Mathematical Journal
- Duke Math. J.
- Volume 132, Number 1 (2006), 73-135.
Rational Cherednik algebras and Hilbert schemes, II: Representations and sheaves
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
Duke Math. J., Volume 132, Number 1 (2006), 73-135.
First available in Project Euclid: 28 February 2006
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Primary: 14C05: Parametrization (Chow and Hilbert schemes) 16D90: Module categories [See also 16Gxx, 16S90]; module theory in a category-theoretic context; Morita equivalence and duality 32S45: Modifications; resolution of singularities [See also 14E15]
Secondary: 16S80: Deformations of rings [See also 13D10, 14D15] 05E10: Combinatorial aspects of representation theory [See also 20C30]
Gordon, I.; Stafford, J. T. Rational Cherednik algebras and Hilbert schemes, II: Representations and sheaves. Duke Math. J. 132 (2006), no. 1, 73--135. doi:10.1215/S0012-7094-06-13213-1. https://projecteuclid.org/euclid.dmj/1141136437