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We investigate the representations of the hyperalgebras associated to the map algebras $\mathfrak g\otimes \mathcal A$, where $\mathfrak g$ is any finite-dimensional complex simple Lie algebra and $\mathcal A$ is any associative commutative…

In algebraic geometry, one studies the solutions to polynomial equations, or, equivalently, to linear partial differential equations with constant coefficients. These lecture notes address the more general case when the coefficients are…

We study the growth of representations of the Lie algebra of vector fields on the affine space that admit a compatible action of the polynomial algebra. We establish the Bernstein inequality for these representations, enabling us to focus…

We apply the ideas of derived algebraic geometry and topological field theory to the representation theory of reductive groups. Our focus is the Hecke category of Borel-equivariant D-modules on the flag variety of a complex reductive group…

This note extends some results of a previous paper (math.RT/0403250) about finite dimensional representations of the wreath product symplectic reflection algebra H(k,c,N,G) of rank N attached to a finite subgroup G of SL(2,C) (here k is a…

Skew algebroid is a natural generalization of the concept of Lie algebroid. In this paper, for a skew algebroid E, its modular class mod(E) is defined in the classical as well as in the supergeometric formulation. It is proved that there is…

In this article, we describe all two sided ideals of a cyclotomic rational Cherednik algebra $H_\mathbf{c}$ and its spherical subalgebra $eH_\mathbf{c} e$ with a Weil generic aspherical parameter $\mathbf{c}$, and further describe the…

This paper is an expanded and updated version of the preprint arXiv:math/0406499. It includes a more detailed description of the basics of the theory of Cherednik and Hecke algebras of varieties started in arXiv:math/0406499, as well as a…

Let $H$ be a finite dimensional quasi-Hopf algebra over a field $k$ and ${\mathfrak A}$ a right $H$-comodule algebra in the sense of Hausser and Nill. We first show that on the $k$-vector space ${\mathfrak A}\ot H^*$ we can define an…

An algebraic classification is given for spaces of holomorphic vector-valued modular forms of arbitrary real weight and multiplier system, associated to irreducible, T-unitarizable representations of the full modular group, of dimension…

For a finite dimensional Lie algebra $\mathfrak{g}$, the Duflo map $S\mathfrak{g}\rightarrow U\mathfrak{g}$ defines an isomorphism of $\mathfrak{g}$-modules. On $\mathfrak{g}$-invariant elements it gives an isomorphism of algebras.…

We classify the irreducible unitary modules in category O for the rational Cherednik algebras of type G(r,1,n) and give explicit combinatorial formulas for their graded characters. More precisely, we produce a combinatorial algorithm…

The Poincar\'e polynomial of a Weyl group calculates the Betti numbers of the projective homogeneous space $G/B$, while the $h$-vector of a simple polytope calculates the Betti numbers of the corresponding rationally smooth toric variety.…

Let E be an operator algebra on a Hilbert space with finite-dimensional generated C*-algebra. A classification is given of the locally finite algebras and the operator algebras obtained as limits of direct sums of matrix algebras over E…

In this paper we study a correspondence between cyclic modules over the first Weyl algebra and planar algebraic curves in positive characteristic. In particular, we show that any such curve has a preimage under a morphism of certain…

We consider irreducible lowest-weight representations of Cherednik algebras associated to certain classes of complex reflection groups in characteristic p. In particular, we study maximal graded submodules of Verma modules associated to…

Let $D$ be a division ring, $\mathcal V$ and $ \mathcal W$ vector spaces over $D$, and ${\mathcal L(\mathcal V,\mathcal W)}$ the ${\mathcal L(\mathcal W)}$-${\mathcal L(\mathcal V)}$ bimodule of all linear transformations from $\mathcal V$…

We develop an elementary method to compute spaces of equivariant maps from a homogeneous space $G/H$ of a Lie group $G$ to a module of this group. The Lie group is not required to be compact. More generally, we study spaces of invariant…

In a paper by the authors, the associative and the Lie algebras of Weyl type $A[D]=A\otimes F[D]$ were introduced, where $A$ is a commutative associative algebra with an identity element over a field $F$ of any characteristic, and $F[D]$ is…

Let H be a connected Hopf k-algebra of finite Gel'fand-Kirillov dimension over an algebraically closed field k of characteristic 0. The objects of study in this paper are the left or right coideal subalgebras T of H. They are shown to be…