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39
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
Noncommutative Counterparts of the Springer Resolution
, 2006
"... Springer resolution of the set of nilpotent elements in a semisimple Lie algebra plays a central role in geometric representation theory. A new structure on this variety has arisen in several representation theoretic constructions, such as the (local) geometric Langlands duality and modular represe ..."
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Cited by 45 (3 self)
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Springer resolution of the set of nilpotent elements in a semisimple Lie algebra plays a central role in geometric representation theory. A new structure on this variety has arisen in several representation theoretic constructions, such as the (local) geometric Langlands duality and modular representation theory. It is also related to some algebrogeometric problems, such as the derived equivalence conjecture and description of T. Bridgeland’s space of stability conditions. The structure can be described as a noncommutative counterpart of the resolution, or as a tstructure on the derived category of the resolution. The intriguing fact that the same tstructure appears in these seemingly disparate subjects has strong technical consequences for modular representation theory.
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.
Almostcommuting variety, Dmodules, and Cherednik
"... We study a scheme M closely related to the set of pairs of n × nmatrices with rank 1 commutator. We show that M is a reduced complete intersection with n+ 1 irreducible components, which we describe. There is a distinguished Lagrangian subvariety Mnil ⊂ M. We introduce a category, C, of Dmodules w ..."
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Cited by 28 (5 self)
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We study a scheme M closely related to the set of pairs of n × nmatrices with rank 1 commutator. We show that M is a reduced complete intersection with n+ 1 irreducible components, which we describe. There is a distinguished Lagrangian subvariety Mnil ⊂ M. We introduce a category, C, of Dmodules whose characteristic variety is contained in Mnil. Simple objects of that category are
Dualizing complexes and perverse sheaves on noncommutative ringed schemes
, 2002
"... Let (X, A) be a separated differential quasicoherent ringed scheme of finite type over a field k. We prove that there exists a rigid dualizing complex over A. The proof consists of two main parts. In the algebraic part we study differential filtrations on rings, and use the results obtained to sh ..."
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Cited by 22 (9 self)
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Let (X, A) be a separated differential quasicoherent ringed scheme of finite type over a field k. We prove that there exists a rigid dualizing complex over A. The proof consists of two main parts. In the algebraic part we study differential filtrations on rings, and use the results obtained to show that a rigid dualizing complex exists on every affine open set in X. In the geometric part of the proof we construct a perverse tstructure on the derived category of bimodules, and this allows us to glue the affine rigid dualizing complexes to
HarishChandra homomorphisms and symplectic reflection algebras for wreathproducts
, 2005
"... The main result of the paper is a natural construction of the spherical subalgebra in a symplectic reflection algebra associated with a wreathproduct in terms of quantum hamiltonian reduction of an algebra of differential operators on a representation space of an extended Dynkin quiver. The existen ..."
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Cited by 18 (2 self)
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The main result of the paper is a natural construction of the spherical subalgebra in a symplectic reflection algebra associated with a wreathproduct in terms of quantum hamiltonian reduction of an algebra of differential operators on a representation space of an extended Dynkin quiver. The existence of such a construction has been conjectured in [EG]. We also present a new approach to reflection functors and shift functors for generalized
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
Braided doubles and rational Cherednik algebras
 Adv. Math
"... Abstract. We introduce and study a large class of algebras with triangular decomposition which we call braided doubles. Braided doubles provide a unifying framework for classical and quantum universal enveloping algebras and rational Cherednik algebras. We classify braided doubles in terms of quasi ..."
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Cited by 13 (5 self)
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Abstract. We introduce and study a large class of algebras with triangular decomposition which we call braided doubles. Braided doubles provide a unifying framework for classical and quantum universal enveloping algebras and rational Cherednik algebras. We classify braided doubles in terms of quasiYetterDrinfeld (QYD) modules over Hopf algebras which turn out to be a generalisation of the ordinary YetterDrinfeld modules. To each braiding (a solution to the braid equation) we associate a QYDmodule and the corresponding braided Heisenberg double — this is a quantum deformation of the Weyl algebra where the role of polynomial algebras is played by NicholsWoronowicz algebras. Our main result is that any rational Cherednik algebra canonically embeds into the braided Heisenberg double attached to the corresponding complex reflection group.