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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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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
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 algebras and Hilbert schemes in characteristic p
, 2006
"... We prove a localization theorem for the type An−1 rational Cherednik algebra Hc = H1,c(An−1) overFp, an algebraic closure of the finite field. In the most interesting special case where c ∈ Fp, we construct an Azumaya algebra Hc on Hilbn A2, the Hilbert scheme of n points in the plane, such that Γ(H ..."
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Cited by 39 (8 self)
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We prove a localization theorem for the type An−1 rational Cherednik algebra Hc = H1,c(An−1) overFp, an algebraic closure of the finite field. In the most interesting special case where c ∈ Fp, we construct an Azumaya algebra Hc on Hilbn A2, the Hilbert scheme of n points in the plane, such that Γ(Hilbn A2, Hc) = Hc. Our localization theorem provides an equivalence between the bounded derived categories of Hcmodules and sheaves of coherent Hcmodules on Hilbn A2, respectively. Furthermore, we show that the Azumaya algebra splits on the formal neighborhood of each fiber of the HilbertChow morphism. This provides a link between our results and those of Bridgeland, King and Reid, and Haiman.
Orbit closures in the enhanced nilpotent cone
 Advances in Mathematics 219
, 2008
"... Abstract. We study the orbits of G = GL(V) in the enhanced nilpotent cone V × N, where N is the variety of nilpotent endomorphisms of V. These orbits are parametrized by bipartitions of n = dim V, and we prove that the closure ordering corresponds to a natural partial order on bipartitions. Moreover ..."
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Cited by 28 (7 self)
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Abstract. We study the orbits of G = GL(V) in the enhanced nilpotent cone V × N, where N is the variety of nilpotent endomorphisms of V. These orbits are parametrized by bipartitions of n = dim V, and we prove that the closure ordering corresponds to a natural partial order on bipartitions. Moreover, we prove that the local intersection cohomology of the orbit closures is given by certain bipartition analogues of Kostka polynomials, defined by Shoji. Finally, we make a connection with Kato’s exotic nilpotent cone in type C, proving that the closure ordering is the same, and conjecturing that the intersection cohomology is the same but with degrees doubled. 1.
Quiver varieties, category O for rational Cherednik algebras, and Hecke algebras
, 2007
"... We relate the representations of the rational Cherednik algebras associated with the complex reflection group µℓ ≀ Sn to sheaves on Nakajima quiver varieties associated with extended Dynkin graphs via a Zalgebra construction. This is done so that as the parameters defining the Cherednik algebra va ..."
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Cited by 28 (2 self)
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We relate the representations of the rational Cherednik algebras associated with the complex reflection group µℓ ≀ Sn to sheaves on Nakajima quiver varieties associated with extended Dynkin graphs via a Zalgebra construction. This is done so that as the parameters defining the Cherednik algebra vary, the stability conditions defining the quiver variety change. This construction motivates us to use the geometry of the quiver varieties to interpret the ordering function (the cfunction) used to define a highest weight structure on category O of the Cherednik algebra. This interpretation provides a natural partial ordering on O which we expect will respect the highest weight structure. This partial ordering has appeared in a conjecture of Yvonne on the composition factors in O and so our results provide a small step towards a geometric picture for that. We also interpret geometrically another ordering function (the afunction) used in the study of Hecke algebras. (The connection between Cherednik algebras and Hecke algebras is provided by the KZfunctor.) This is related to a conjecture of Bonnafé and Geck on equivalence classes of weight functions for Hecke algebras with unequal parameters since the classes should (and do for type B) correspond to the G.I.T. chambers defining the quiver varieties. As a result anything that can be defined via the quiver varieties,
Isomorphism of quantizations via quantization of resolutions
 Advances in Math. 231 (2012), 1216–1270. 34 KEVIN MCGERTY AND
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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
Linear representations, symmetric products and the commuting scheme
, 2007
"... We show that the ring of multisymmetric functions over a commutative ring is isomorphic to the ring generated by the coefficients of the characteristic polynomial of polynomials in commuting generic matrices. As a consequence we give a surjection from the ring of invariants of several matrices to th ..."
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Cited by 11 (7 self)
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We show that the ring of multisymmetric functions over a commutative ring is isomorphic to the ring generated by the coefficients of the characteristic polynomial of polynomials in commuting generic matrices. As a consequence we give a surjection from the ring of invariants of several matrices to the ring of multisymmetric functions generalizing a classical result of H.Weyl and F.Junker. We also find a surjection from the ring of invariants over the commuting scheme to the ring of multisymmetric functions. This surjection is an isomophism over a characteristic zero field and induces an isomorphism at the level of reduced structures over an infinite field of positive characteristic. 1
Deformed Harish–Chandra homomorphism for the cyclic quiver, RT:0504395
, 2005
"... Abstract. In the case of cyclic quiver we prove that the deformed HarishChandra map whose existence was conjectured by Etingof and Ginzburg is well defined. As an application we prove a Kirillovtype formula for the cyclotomic Bessel function. 1. ..."
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Cited by 6 (0 self)
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Abstract. In the case of cyclic quiver we prove that the deformed HarishChandra map whose existence was conjectured by Etingof and Ginzburg is well defined. As an application we prove a Kirillovtype formula for the cyclotomic Bessel function. 1.