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2007.206: Rational Cherednik algebras and Hilbert schemes. II: representations and sheaves

2007.206: I. Gordon and J. T. Stafford (2006) Rational Cherednik algebras and Hilbert schemes. II: representations and sheaves. Duke Mathematical Journal, 132 (1). pp. 73-135. ISSN 0012-7094

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DOI: 10.1215/S0012-7094-06-13213-1

Abstract

Let H_c be the rational Cherednik algebra of type A_{n-1} with spherical subalgebra U_c = e H_c e. Then U_c is filtered by order of differential operators with associated graded ring gr U_c = \mathbb{C} [ \mathfrac{h} ⊕ \mathfrac{h}^* ]^W, where is the n-th symmetric group. Using the Z-algebra construction from [GS], it is also possible to associate to a filtered H_c - or U_c - module \hat{Φ}(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 τ : Hilb(n) → \mathfrac{h} ⊕ \mathfrac{h}^* / W. For example, we prove the following.

• If c=1/n so that L_c(triv) is the unique one-dimensional simple H_c-module, then \hat{Φ}(e L_c(triv)) ≅ \mathcal{O}_{Z_n}, where Z_n = τ^{-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(triv), gr e L_c(triv) has a natural bigraded structure that coincides with that on H^0( Z_n, \mathscr{L}^k), where \mathscr{L} ≅ \mathcal{O}_{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 \hat{Φ}(e Δ_c(μ)) equals \sum_λ K_{μλ}[Z_λ], where K_{μλ} are Kostka numbers and the Z_λ are (known) irreducible components of τ^{-1}(\mathfrak{h}/W)

Item Type:Article
Subjects:MSC 2000 > 14 Algebraic geometry
MSC 2000 > 16 Associative rings and algebras
MSC 2000 > 32 Several complex variables and analytic spaces
MIMS number:2007.206
Deposited By:Mrs Louise Healey
Deposited On:22 November 2007

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