To any finite group Γ ⊂ Sp(V) of automorphisms of a symplectic vector space V we associate a new multi-parameter 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 1-parameter deformation of the Harish-Chandra homomorphism from D(g) g, the algebra of invariant polynomial differential operators on the Lie algebra g = gl n, to the algebra of Sn-invariant differential operators on the Cartan subalgebra C n with rational coefficients. The second order Laplacian on g goes, under our deformed homomorphism, to the Calogero-Moser

Abstract. Let A1 be the (first) Weyl algebra, and let G be its automorphism group. We study the natural action of G on the space of isomorphism classes of right ideals of A1 (equivalently, of finitely generated rank 1 torsion-free right A1-modules). We show that this space breaks up into a countable number of orbits each of which is a finite dimensional algebraic variety. Our results are strikingly similar to those for the commutative algebra of polynomials in two variables; however, we do not know of any general principle that would allow us to predict this in advance. As a key step in the proof, we obtain a new description of the bispectral involution of [W1]. We also make some comments on the group G from the viewpoint of Shafaravich’s theory of infinite dimensional algebraic groups.

Abstract. Symplectic reflection algebras arise in many different mathematical disciplines: integrable systems, Lie theory, representation theory, differential operators, symplectic geometry. In this paper, we introduce baby Verma modules for symplectic reflection algebras of complex reflection groups at parameter t = 0 (the so–called rational Cherednik algebras at parameter t = 0) and present their most basic properties. By analogy with the representation theory of reductive Lie algebras in positive characteristic, we believe these modules are fundamental to the understanding of the representation theory and associated geometry of the rational Cherednik algebras at parameter t = 0. As an example, we use baby Verma modules to solve one problem posed by Etingof and Ginzburg and partially solve another, [5], and give an elementary proof of a theorem of Finkelberg and Ginzburg, [6]. 1. Notation 1.1. Let W be a complex reflection group and h its reflection representation over C. Let S denote the set of complex reflections in W. Let ω be the canonical symplectic form on V = h ⊕ h ∗. For s ∈ S, let ωs by skew–symmetric form which coincides with ω on im(idV − s) and has ker(idV − s) as the radical. Let c: S − → C be a W–invariant function sending s to cs. The rational Cherednik

. Recently, V. Ginzburg proved that Calogero phase space is a coadjoint orbit for some infinite dimensional Lie algebra coming from noncommutative symplectic geometry, [12]. In this note we generalize his argument to specific quotient varieties of representations of (deformed) preprojective algebras. This result was also obtained independently by V. Ginzburg [ 13]. Using results of W. Crawley-Boevey and M. Holland [ 10], [8] and [9] we give a combinatorial description of all the relevant couples (ff# ) which are coadjoint orbits. We give a conjectural explanation for this coadjoint orbit result and relate it to different noncommutative notions of smoothness. 1.

. We give a new proof, avoiding case-by-case analysis, of a theorem of Y. Ito and I. Nakamura which provides a module-theoretic interpretation of the bijection between the irreducible components of the exceptional fibre for a Kleinian singularity, and the non-trivial 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 \Gamma be a finite subgroup of SL(2; C ), let X = C 2 =\Gamma be the corresponding Kleinian singularity and let ß : ~ X ! X be its minimal resolution of singularities. The exceptional fibre E, the fibre of ß 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 joined if and only if the components intersect, is a Dynkin diagram. If N 0 ; N 1 ; : : : ; Nn are a complete set of simple C \Gamma-modules, with N 0 the trivial...

Let R be the set of isomorphism classes of ideals in the Weyl algebra A = A1(C), and let C be the set of isomorphism classes of triples (V, X, Y), where V is a finite-dimensional (complex) vector space, and X, Y are endomorphisms of V such that [X, Y]+I has rank 1. Following a suggestion of L. Le Bruyn, we define a map θ: R → C by appropriately extending an ideal of A to a sheaf over a quantum projective plane, and then using standard methods of homological algebra. We prove that θ is inverse to a bijection ω: C → R constructed in [BW] by a completely different method. The main step in the proof is to show that θ is equivariant with respect to natural actions of the group G = Aut(A) on R and C: for that we have to study also the extensions of an ideal to certain weighted quantum projective planes. Along the way, we find an elementary description of θ.

Abstract. We prove that the Marsden-Weinstein reductions for the moment map associated to representations of a quiver are normal varieties. We give an application to conjugacy classes of matrices. 1.

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We study a scheme M closely related to the set of pairs of n × n-matrices 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 D-modules whose characteristic variety is contained in Mnil. Simple objects of that category are

We construct projective moduli spaces for torsion-free sheaves on noncommutative projective planes. These moduli spaces vary smoothly in the parameters describing the noncommutative plane and have good properties analogous to those of moduli spaces of sheaves over the usual (commutative) projective plane P 2. The generic noncommutative plane corresponds to the Sklyanin algebra S = Skl(E, σ) constructed from an automorphism σ of infinite order on an elliptic curve E ⊂ P 2. In this case, the fine moduli space of line bundles over S with first Chern class zero and Euler characteristic 1 − n provides a symplectic variety that is a deformation of the Hilbert scheme of n points on P² \ E.