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96
Symplectic reflection algebras, CalogeroMoser space, and deformed HarishChandra homomorphism
 Invent. Math
"... To any finite group Γ ⊂ Sp(V) of automorphisms of a symplectic vector space V we associate a new multiparameter deformation, Hκ, of the algebra C[V]#Γ, smash product of Γ with the polynomial algebra on V. The parameter κ runs over points of CP r, where r = number of conjugacy classes of symplectic ..."
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Cited by 280 (39 self)
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To any finite group Γ ⊂ Sp(V) of automorphisms of a symplectic vector space V we associate a new multiparameter deformation, Hκ, of the algebra C[V]#Γ, smash product of Γ with the polynomial algebra on V. The parameter κ runs over points of CP r, where r = number of conjugacy classes of symplectic reflections in Γ. The algebra Hκ, called a symplectic reflection algebra, is expected to be related to the coordinate ring of a universal Poisson deformation of the quotient singularity V/Γ. If Γ is the Weyl group of a root system in a vector space h and V = h ⊕ h ∗ , then the algebras Hκ are certain ‘rational ’ degenerations of the double affine Hecke algebra introduced earlier by Cherednik. Let Γ = Sn, the Weyl group of g = gl n. We construct a 1parameter deformation of the HarishChandra homomorphism from D(g) g, the algebra of invariant polynomial differential operators on the Lie algebra g = gl n, to the algebra of Sninvariant differential operators on the Cartan subalgebra C n with rational coefficients. The second order Laplacian on g goes, under our deformed homomorphism, to the CalogeroMoser
Cluster algebra structures and semicanonical bases for unipotent groups
, 2008
"... Let Q be a finite quiver without oriented cycles, and let Λ be the associated preprojective algebra. To each terminal CQmodule M (these are certain preinjective CQmodules), we attach a natural subcategory CM of mod(Λ). We show that CM is a ..."
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Cited by 49 (7 self)
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Let Q be a finite quiver without oriented cycles, and let Λ be the associated preprojective algebra. To each terminal CQmodule M (these are certain preinjective CQmodules), we attach a natural subcategory CM of mod(Λ). We show that CM is a
Noncommutative Geometry and Quiver algebras
"... We develop a new framework for noncommutative differential geometry based on double derivations. This leads to the notion of moment map and of Hamiltonian reduction in noncommutative symplectic geometry. For any smooth associative algebra B, we define its noncommutative cotangent bundle T ∗ B, which ..."
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Cited by 46 (12 self)
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We develop a new framework for noncommutative differential geometry based on double derivations. This leads to the notion of moment map and of Hamiltonian reduction in noncommutative symplectic geometry. For any smooth associative algebra B, we define its noncommutative cotangent bundle T ∗ B, which is a basic example of noncommutative symplectic manifold. Applying Hamiltonian reduction to noncommutative cotangent bundles gives an interesting class of associative algebras, Π = Π(B), that includes preprojective algebras associated with quivers. Our formalism of noncommutative Hamiltonian reduction provides the space Π/[Π,Π] with a Lie algebra structure, analogous to the Poisson bracket on the zero fiber of the moment map. In the special case where Π is the preprojective algebra associated with a quiver of nonDynkin type, we give a complete description of the Gerstenhaber algebra structure on the Hochschild cohomology of Π in terms of the Lie algebra Π/[Π,Π].
On the exceptional fibres of Kleinian singularities
 Amer. J. Math
"... Abstract. We give a new proof, avoiding casebycase analysis, of a theorem of Y. Ito and I. Nakamura which provides a moduletheoretic interpretation of the bijection between the irreducible components of the exceptional fibre for a Kleinian singularity, and the nontrivial simple modules for the c ..."
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Cited by 42 (3 self)
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Abstract. We give a new proof, avoiding casebycase analysis, of a theorem of Y. Ito and I. Nakamura which provides a moduletheoretic interpretation of the bijection between the irreducible components of the exceptional fibre for a Kleinian singularity, and the nontrivial simple modules for the corresponding finite subgroup of SL (2, C). Our proof uses a classification of certain cyclic modules for preprojective algebras. Introduction. Let Γ be a finite subgroup of SL (2, C), let X = C 2=Γ be the corresponding Kleinian singularity and let : ˜X! X be its minimal resolution of singularities. The exceptional fibre E, the fibre of over the singular point of X, is known to be a union of projective lines meeting transversally, and the graph whose vertices correspond to the irreducible components of E, with two vertices
On matrices in prescribed conjugacy classes with no common invariant subspace and sum zero
 Duke Math. J
"... We determine those ktuples of conjugacy classes of matrices from which it is possible to choose matrices that have no common invariant subspace and have sum zero. This is an additive version of the DeligneSimpson problem. We deduce the result from earlier work of ours on preprojective algebras and ..."
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Cited by 37 (4 self)
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We determine those ktuples of conjugacy classes of matrices from which it is possible to choose matrices that have no common invariant subspace and have sum zero. This is an additive version of the DeligneSimpson problem. We deduce the result from earlier work of ours on preprojective algebras and the moment map for representations of quivers. Our answer depends on the root system for a KacMoody Lie algebra.
Absolutely indecomposable representations and KacMoody Lie algebras (with an appendix by Hiraku Nakajima)
 INVENT. MATH
, 2001
"... A conjecture of Kac states that the polynomial counting the number of absolutely indecomposable representations of a quiver over a finite field with given dimension vector has positive coefficients and furthermore that its constant term is equal to the multiplicity of the corresponding root in the ..."
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Cited by 36 (1 self)
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A conjecture of Kac states that the polynomial counting the number of absolutely indecomposable representations of a quiver over a finite field with given dimension vector has positive coefficients and furthermore that its constant term is equal to the multiplicity of the corresponding root in the associated KacMoody Lie algebra. In this paper we prove these conjectures for indivisible dimension vectors.
Decomposition of MarsdenWeinstein reductions for representations of quivers
, 2001
"... We prove that the MarsdenWeinstein reductions for the moment map associated to representations of a quiver are normal varieties. We give an application to conjugacy classes of matrices. ..."
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Cited by 34 (2 self)
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We prove that the MarsdenWeinstein reductions for the moment map associated to representations of a quiver are normal varieties. We give an application to conjugacy classes of matrices.
Indecomposable parabolic bundles and the existence of matrices in prescribed conjugacy class closures with product equal to the identity
, 2003
"... We study the possible dimension vectors of indecomposable parabolic bundles on the projective line, and use our answer to solve the problem of characterizing those collections of conjugacy classes of n × n matrices for which one can find matrices in their closures whose product is equal to the ide ..."
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Cited by 28 (5 self)
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We study the possible dimension vectors of indecomposable parabolic bundles on the projective line, and use our answer to solve the problem of characterizing those collections of conjugacy classes of n × n matrices for which one can find matrices in their closures whose product is equal to the identity matrix. Both answers depend on the root system of a KacMoody Lie algebra. Our proofs use Ringel’s theory of tubular algebras, work of Mihai on the existence of logarithmic connections, the RiemannHilbert correspondence and an algebraic version, due to Dettweiler and Reiter, of Katz’s middle convolution operation.
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
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,