**Antisymmetric exponential
functions and (anti)symmetric multivariate Fourier and trigonometric transforms **

**Abstract:**

The talk represents the results on multivariate symmetric and antisymmetric functions and transforms, obtained
together with Prof. J. Patera. Multivariate exponential functions, antisymmetric with respect to a Weyl group W of a simple
Lie algebra, are studied. They are eigenfunctions of the Laplace operator vanishing on the boundary of the fundamental
domain of the affine Weyl group, corresponding to the Weyl group W.
Multivariate symmetric and antisymmetric continuous and finite Fourier transforms are constructed. Multivariate symmetric
and antisymmetric sine and cosine functions are given. By means of these functions, multivariate symmetric and
antisymmetric continuous and finite sine and cosine transforms are obtained.

Joint work with *Jiri Patera*
(Centre de Recherches Mathématiques,
Université de Montréal, Canada).