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Parabolic induction and restriction functors for rational Cherednik algebras
, 2009
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Rational Cherednik algebras and Hilbert schemes
"... Abstract. Let Hc be the rational Cherednik algebra of type An−1 with spherical subalgebra Uc = eHce. Then Uc is filtered by order of differential operators with associated graded ring gr Uc = C[h ⊕ h ∗ ] W, where W is the nth symmetric group. Using the Zalgebra construction from [GS] it is also po ..."
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Cited by 58 (6 self)
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Abstract. Let Hc be the rational Cherednik algebra of type An−1 with spherical subalgebra Uc = eHce. Then Uc is filtered by order of differential operators with associated graded ring gr Uc = C[h ⊕ h ∗ ] W, where W is the nth symmetric group. Using the Zalgebra construction from [GS] it is also possible to associate to a filtered Hc or Ucmodule M a coherent sheaf Φ(M) on the Hilbert scheme Hilb(n). Using this technique, we study the representation theory of Uc and Hc, and relate it to Hilb(n) and to the resolution of singularities τ: Hilb(n) → h ⊕ h ∗ /W. For example, we prove: • If c = 1/n, so that Lc(triv) is the unique onedimensional simple Hcmodule, then Φ(eLc(triv)) ∼ = OZn, where Zn = τ −1 (0) is the punctual Hilbert scheme. • If c = 1/n + k for k ∈ N then, under a canonical filtration on the finite dimensional module Lc(triv), gr eLc(triv) has a natural bigraded structure which coincides with that on H 0 (Zn, L k), where L ∼ = O Hilb(n)(1); this confirms conjectures of Berest, Etingof and Ginzburg. • Under mild restrictions on c, the characteristic cycle of Φ(e∆c(µ)) equals ∑ λ Kµλ[Zλ], where Kµλ are Kostka numbers and the Zλ are (known) irreducible components of τ −1 (h/W). Contents
On the quotient ring by diagonal invariants
 Invent. Math
"... Abstract. For a Weyl group, W, and its reflection representation h, we find the character and Hilbert series for a quotient ring of C[h ⊕ h ∗ ] by an ideal containing the W–invariant polynomials without constant term. This confirms conjectures of Haiman. 1. ..."
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Cited by 50 (4 self)
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Abstract. For a Weyl group, W, and its reflection representation h, we find the character and Hilbert series for a quotient ring of C[h ⊕ h ∗ ] by an ideal containing the W–invariant polynomials without constant term. This confirms conjectures of Haiman. 1.
Representations of rational Cherednik algebras
 CONTEMPORARY MATHEMATICS
"... This paper surveys the representation theory of rational Cherednik algebras. We also discuss the representations of the spherical subalgebras. We describe in particular the results on category O. For type A, we explain relations with the Hilbert scheme of points on C². We insist on the analogy with ..."
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Cited by 44 (4 self)
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This paper surveys the representation theory of rational Cherednik algebras. We also discuss the representations of the spherical subalgebras. We describe in particular the results on category O. For type A, we explain relations with the Hilbert scheme of points on C². We insist on the analogy with the representation theory of complex semisimple Lie algebras.
Cherednik and Hecke algebras of varieties with a finite group action
"... Let h be a finite dimensional complex vector space, and G be a finite subgroup of GL(h). To this data one can attach a family of algebras Ht,c(h, G), called the rational Cherednik algebras (see [EG]); for t = 1 it provides the universal deformation of G ⋉ D(h) (where D(h) is the algebra of different ..."
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Cited by 26 (7 self)
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Let h be a finite dimensional complex vector space, and G be a finite subgroup of GL(h). To this data one can attach a family of algebras Ht,c(h, G), called the rational Cherednik algebras (see [EG]); for t = 1 it provides the universal deformation of G ⋉ D(h) (where D(h) is the algebra of differential operators on h).
Purity of Equivalued Affine Springer Fibers
"... this paper, we need two more definitions. We normalize the valuation on F so that val # = 1. We say that regular u t is equivalued with valuation s Q if val #(u)=s for every root # of T over F and X # (T ). (For adjoint groups G the second part of the condition is of course redundant.) ..."
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Cited by 19 (3 self)
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this paper, we need two more definitions. We normalize the valuation on F so that val # = 1. We say that regular u t is equivalued with valuation s Q if val #(u)=s for every root # of T over F and X # (T ). (For adjoint groups G the second part of the condition is of course redundant.) When s plays no role, we say simply that u is equivalued
UNITARY REPRESENTATIONS OF RATIONAL CHEREDNIK ALGEBRAS
, 2009
"... We study unitarity of lowest weight irreducible representations of rational Cherednik algebras. We prove several general results, and use them to determine which lowest weight representations are unitary in a number of cases. In particular, in type A, we give a full description of the unitarity lo ..."
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Cited by 16 (2 self)
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We study unitarity of lowest weight irreducible representations of rational Cherednik algebras. We prove several general results, and use them to determine which lowest weight representations are unitary in a number of cases. In particular, in type A, we give a full description of the unitarity locus (justified in Subsection 5.1 and the appendix written by S. Griffeth), and resolve a question by Cherednik on the unitarity of the irreducible subrepresentation of the polynomial representation. Also, as a byproduct, we establish Kasatani’s conjecture in full generality (the previous proof by Enomoto assumes that the parameter c is not a halfinteger).
A uniform bijection between nonnesting and noncrossing partitions
, 2011
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Differential operators and Cherednik algebras
, 2007
"... Abstract. We establish a link between two geometric approaches to the representation theory of rational Cherednik algebras of type A: one based on a noncommutative Proj construction [GS1]; the other involving quantum hamiltonian reduction of an algebra of differential operators [GG]. In the present ..."
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Cited by 14 (4 self)
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Abstract. We establish a link between two geometric approaches to the representation theory of rational Cherednik algebras of type A: one based on a noncommutative Proj construction [GS1]; the other involving quantum hamiltonian reduction of an algebra of differential operators [GG]. In the present paper, we combine these two points of view by showing that the process of hamiltonian reduction intertwines a naturally defined geometric twist functor on Dmodules with the shift functor for the Cherednik algebra. That enables us to give a direct and relatively short proof of the key result [GS1, Theorem 1.4] without recourse to Haiman’s deep results on the n! theorem [Ha1]. We also show that the characteristic cycles defined independently in these two approaches are equal, thereby confirming a conjecture from [GG]. Contents