Abstract
A representation-theoretic approach to special functions was developed in the 40s and 50s in the works of I. M. Gel'land, M. A. Naimark, N. Ya. Vilenkin, and their collaborators (see [1, 10]). This approach is based on the fact that most of classical special functions c~m be obtained as suitable specializations of matrix elements or characters of group representations. Another rich source of special functions is the theory of Clebsch-Gordan coefficients, which describes the geometric juxtaposition of irreducible components in the tensor product of two representations. Finally, in recent works on representations of (quantum) affine Lie algebras it was shown that matrix elements of intertwining operators between certain representations of these algebras are interesting special functions, namely (q-)hypergeometric functions and their generalizations [5, 9]. In this paper we propose a general method of obtaining special functions from representation theory that unifies the three methods mentioned above and allows one to define and study many new special functions. We illustrate this method by a number of examples, namely, Macdonald's polynomials, eigenfunctions of the Sutherland operator, and Lam~ functions. Other examples will be described in our future papers.
Published Version
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