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Vi lenkin. N. fA. (Naum fAkovlevichl Representation of Lie groups and special functions / by N.J. Vi lenk In and A.U. Kl1myk. p. cm. --(Mathematics and its applications. Soviet series v. 72) Translation from the Russian. lnc ludes index. Contentso v. 1. Simplest Lie groups. special funtions. and integral transforms.

Abstract. In the paper, properties of orbit functions are reviewed and further developed. Orbit functions on the Euclidean space E n are symmetrized exponential functions. The symmetrization is fulfilled by a Weyl group corresponding to a Coxeter-Dynkin diagram. Properties of such functions will be described. An orbit function is the contribution to an irreducible character of a compact semisimple Lie group G of rank n from one of its Weyl group orbits. It is shown that values of orbit functions are repeated on copies of the fundamental domain F of the affine Weyl group (determined by the initial Weyl group) in the entire Euclidean space E n . Orbit functions are solutions of the corresponding Laplace equation in E n , satisfying the Neumann condition on the boundary of F . Orbit functions determine a symmetrized Fourier transform and a transform on a finite set of points.

Abstract. In the paper, properties of antisymmetric orbit functions are reviewed and further developed. Antisymmetric orbit functions on the Euclidean space E n are antisymmetrized exponential functions. Antisymmetrization is fulfilled by a Weyl group, corresponding to a Coxeter-Dynkin diagram. Properties of such functions are described. These functions are closely related to irreducible characters of a compact semisimple Lie group G of rank n. Up to a sign, values of antisymmetric orbit functions are repeated on copies of the fundamental domain F of the affine Weyl group (determined by the initial Weyl group) in the entire Euclidean space E n . Antisymmetric orbit functions are solutions of the corresponding Laplace equation in E n , vanishing on the boundary of the fundamental domain F . Antisymmetric orbit functions determine a so-called antisymmetrized Fourier transform which is closely related to expansions of central functions in characters of irreducible representations of the group G. They also determine a transform on a finite set of points of F (the discrete antisymmetric orbit function transform). Symmetric and antisymmetric multivariate exponential, sine and cosine discrete transforms are given.

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