Introduction By spherical functions one usually means functions on the double coset space K\G/K, where G is a group and K is a subgroup of G. This is equivalent to considering functions on the homogeneous space G/K left invariant with respect to K. More generally, if V is a fixed irreducible representation of K, for example, finite-dimensional, one can look… (More)
This is a very unusual book, and I think it will be appropriate to put things into perspective and to provide some background information. V. I. Arnold is a famous mathematician who belongs to the generation of Russian mathematicians, now approaching their 70-th anniversaries. I will not list here Arnold's numerous awards and prizes: such lists are easily… (More)
This survey is the expanded version of my talk at the AMS meeting in April 1997. I explain to non-experts how to use the orbit method, discuss its strong and weak points and advertise some open problems.
Introduction Recently I.Macdonald defined a family of systems of orthogonal symmetric poly-nomials depending of two parameters q, k which interpolate between Schur's symmetric functions and certain spherical functions on SL(n) over the real and p-adic fields [M]. These polynomials are labeled by dominant integral weights of SL(n), and (as was shown by… (More)
We consider correlation functions for the Wess-Zumino-Witten model on the torus with the insertion of a Cartan element; mathematically this means that we consider the function of the form F = Tr(Φ 1 (z 1). .. Φ n (z n)q −∂ e h) where Φ i are intertwiners between Verma modules and evaluation modules over an affine Lie algebrâ g, ∂ is the grading operator in… (More)
Introduction. Jack's and Macdonald's polynomials are an important class of symmetric functions associated to root systems. In this paper we define and study an analogue of Jack's and Macdonald's polynomials for affine root systems. Our approach is based on representation theory of affine Lie algebras and quantum affine algebras, and follows the ideas of our… (More)
Dedicated to I.M.Gel'fand on the occasion of his 80th birthday A representation-theoretic approach to special functions was developed in the 40-s and 50-s in the works of I. and their collaborators (see [V],[VK]). The essence of this approach is the fact that most classical special functions can be obtained as suitable specializations of matrix elements or… (More)
Let G be a finite subgroup in SU(2), and Q the corresponding affine Dynkin diagram. In this paper, we review the relation between the categories of G-equivariant sheaves on P 1 and Rep Q h , where h is an orientation of Q, constructing an explicit equivalence of corresponding derived categories.
This paper gives a review of Cherednik's results on the representation-theoretic approach to Macdonald polynomials and related special functions. Macdonald polynomials are a remarkable 2-parameter family of polynomials which can be associated to every root system. As special cases, they include the Schur functions, the q-Jacobi polynomials, and certain… (More)