A Lie-Theoretic Construction of Spherical Symplectic Reflection Algebras

Abstract

We propose a construction of the spherical subalgebra of a symplectic reflection algebra of an arbitrary rank corresponding to a star-shaped affine Dynkin diagram. Namely, it is obtained from the universal enveloping algebra of a certain semisimple Lie algebra by the process of quantum Hamiltonian reduction. As an application, we propose a construction of finite-dimensional representations of the spherical subalgebra. IntroductionThe main result of this paper is the realization of the spherical subalgebra of the wreath product symplectic reflection algebra of rank n of types D 4 , E 6 , E 7 , E 8 , introduced in [EG], as a quantum Hamiltonian reduction of the tensor product of m quotients of the enveloping algebra U (sl n ), where is 2, 3, 4, and 6, and m is 4, 3, 3, and 3, respectively. This allows one to define a functor which attaches a representation of the spherical symplectic reflection algebra to a collection of m representations of sl n , which are annihilated by certain ideals. In particular, this gives an explicit Lie-theoretic construction of many finite-dimensional representations of spherical symplectic reflection algebras, most of which appear to be new. In the rank 1 case, all finite-dimensional representations of spherical symplectic reflection algebras are classified by Crawley-Boevey and Holland [CBH] and our construction yields several explicit Lie-theoretic realizations of all of them.The proof of the main result is based on the previous work [EGGO], in which the spherical subalgebra of a wreath product symplectic reflection algebra (of any type) is realized as the quantum Hamiltonian reduction from the algebra of differential operators on representations (with a certain dimension vector) of the Calogero-Moser quiver, obtained from the corresponding extended Dynkin quiver by adjoining an auxiliary vertex, linked to the extending vertex. Namely, in the case when the extended Dynkin graph is star-shaped (which happens in the cases D 4 , E 6 , E 7 , E 8 ), this reduction can be performed in two steps: first the reduction with respect to the groups of basis changes at the noncentral vertices, and then with respect to the group of basis changes at the central (branching) vertex of the star. By the localization theorem for partial flag varieties, after the first step we obtain a tensor product of m quotients of enveloping algebras (the factors correspond to the branches of the extended Dynkin diagram), which yields the result.Recall that by the results of [EG], a wreath product spherical symplectic reflection algebra can be viewed as a quantization of the wreath product CalogeroMoser space, which is a deformation of the nth Hilbert scheme of the resolution of a Kleinian singularity. Our main result is a quantum analog of the statement from classical symplectic geometry, stating that this wreath product Calogero-Moser space may be obtained by classical Hamiltonian reduction from a product of m coadjoint orbits of the Lie algebra sl n , or, equivalently, as the space of solutions of a special kind of ...

Representations of Gan–Ginzburg algebras

“…For more details about this construction and its properties, we refer to [7], Sect. 1.1, and [3], Chapter 4.…”

“…As observed in [7], the group G(α) acts freely on X , and the quotient is the flag variety F (α 1 , . .…”

“…In other words, we show that some of these representations are in the image of the functor e ⊗n 0 F n,χ . Our main tool is a result from [7], establishing an isomorphism between the algebra of twisted differential operators on a space of representations for the quiver of type A (where all arrows are oriented in the same direction and the coordinates of the dimension vector are non-zero and strictly increasing in the direction of the arrows) and the algebra of twisted differential operators on a corresponding partial flag variety. Namely, let D be a star-shaped diagram, that is not finite Dynkin, and let Q be the quiver obtained from D by assigning to all edges the orientation toward the node.…”

“…By the abovementioned result from [7] applied to the legs of Q (that are quivers of type A), and for an appropriate choice of character χ , after the first step we get a space of the form V 1 ⊗· · ·⊗V m ⊗(C N ) ⊗n , where each vector space V i is a finite dimensional irreducible i -stepped gl N -module whose highest weight is an explicit function of χ and α. We expect our construction for rational GDAHA to have a generalization to the case of non-degenerate GDAHA, involving quantum D-modules and quantum groups.…”

“…In Sect. 6, we apply the results of [7] to obtain a Borel-Weil type construction of some finite dimensional irreducible gl N -modules, starting from modules over algebras of differential operators on spaces of representations of quivers of type A. In Sect.…”