Yuri Berest

Finite-dimensional representations of rational Cherednik algebras

A complete classification and character formulas for finite-dimensional irreducible representations of the rational Cherednik algebra of type A is given. Less complete results for other types are obtained. Links to the geometry of affine flag manifolds and Hilbert schemes are discussed.

Ideal classes of the Weyl algebra and noncommutative projective geometry (with an appendix by Michel Van den Bergh)

Cornell Univ, Dept Math, Ithaca, NY 14853 USA. Univ London Imperial Coll Sci Technol & Med, Dept Math, London SW7 2BZ, England. Limburgs Univ Ctr, Dept Math Phys & Informat, B-3590 Diepenbeek, Belgium.Berest, Y, Cornell Univ, Dept Math, White Hall, Ithaca, NY 14853 USA.

Quasi-invariants of complex reflection groups

We introduce quasi-invariant polynomials for an arbitrary finite complex reflection group W. Unlike in the Coxeter case, the space Q_k of quasi-invariants of a given multiplicity is not, in general, an algebra but a module over the coordinate ring of some (singular) affine variety X_k. We extend the main results of Etingof, Ginzburg and the first author (see [BEG]) to this setting: in particular, we show that the variety X_k and the module Q_k are Cohen-Macaulay, and the rings of differential operators on X_k and Q_k are simple rings, Morita equivalent to the Weyl algebra A_n(C), where n = dim X_k . Our approach relies on representation theory of complex Cherednik algebras and is parallel to that of [BEG]. As a by-product, we prove the existence of shift operators for an arbitrary complex reflection group, confirming a conjecture of Dunkl and Opdam. Another result is a proof of a conjecture of Opdam, concerning certain operations (KZ twists) on the set of irreducible representations of W.

A∞-modules and Calogero-Moser spaces

The relations between these objects are well known and almost immediate. Thus, (1) is essentially the definition of (closed) points of HilbðCÞ. The bijection (1)! (2) is given by taking the quotient M 7! A0=M modulo a given ideal and letting i be the image of 1 A A0 in A0=M. The inverse map (2)! (1) is then defined by assigning to a given cyclic module its annihilator in A0. The correspondence (1)

The problem of lacunas and analysis on root systems

(3) follows from the fact that every f. g. rank 1 torsion-free A0-module is isomorphic to a unique ideal of finite codimension in A0. Finally, the bijection (3)! (4) can be constructed geometrically by extending A0-modules to coherent sheaves on P, and its inverse by restricting such sheaves via the natural embedding C ,! P.

Double affine Hecke algebras and generalized Jones polynomials

A lacuna of a linear hyperbolic differential operator is a domain inside its propagation cone where a proper fundamental solution vanishes identically. Huygens’ principle for the classical wave equation is the simplest important example of such a phenomenon. The study of lacunas for hyperbolic equations of arbitrary order was initiated by I. G. Petrovsky (1945). Extending and clarifying his results, Atiyah, Bott and G̊arding (1970–73) developed a profound and complete theory for hyperbolic operators with constant coefficients. In contrast, much less is known about lacunas for operators with variable coefficients. In the present paper we study this problem for one remarkable class of partial differential operators with singular coefficients. These operators stem from the theory of special functions in several variables related to finite root systems (Coxeter groups). The underlying algebraic structure makes it possible to extend many results of the Atiyah-Bott-G̊arding theory. We give a generalization of the classical Herglotz-Petrovsky-Leray formulas expressing the fundamental solution in terms of Abelian integrals over properly constructed cycles in complex projective space. Such a representation allows us to employ the Petrovsky topological condition for testing regular (strong) lacunas for the operators under consideration. Some illustrative examples are constructed. A relation between the theory of lacunas and the problem of classification of commutative rings of partial differential operators is discussed.

Topology, Geometry and Quantum Field Theory: Differential isomorphism and equivalence of algebraic varieties

In this paper, we propose and discuss implications of a general conjecture that there is a canonical action of a rank 1 double affine Hecke algebra on the Kauffman bracket skein module of the complement of a knot

The Picard group of a noncommutative algebraic torus

K \subset S^3

Heat Kernel Coefficients for Two-Dimensional Schrödinger Operators

. We prove this in a number of nontrivial cases, including all

Dual Hodge decompositions and derived Poisson brackets

(2,2p+1)