Homogeneous spaces

We investigate the following problems. A. Symmetric spaces, harmonic analysis, and special functions. 1. In the previous works there were constructed natural kernels on pseudo-Riemannian symmetric spaces. Now I intended to study their properties, and in particular Toeplitz operators related to these kernels 2. I plan to investigate various applications of Plancherel formula, and, in particular, intend to continue a search of new reasonable problems `after Plancherel formula` 3. I intend to investigate various applications of harmonic analysis to special functions and mathematical physics. 4. I want to examine conformal geometry of symmetric spaces and properties of integral manifolds. B. Infinite-dimensional groups and infinite-dimensional integral transforms. 1. I hope to investigate properties of some infinite-dimensional integral transforms and relations between different models of Fock spaces 2. I hope to investigate actions of mantles of infinite-dimensional groups by polymorphisms. 3. I intend to examine representations and quasiinvariant actions of the group of diffeomorphisms of the circle and some other infinite-dimensional groups, C. Wild invariant theory. 1. I intend to attack the complex of questions related to wild invariant theory in relations with the representation theory.

1.It is constructed a functor that produces topological field theories from representation of an infinite symmetric group. 2.It is constructed a multivariate characteristic function for multi-operator colligations (and, more generally, for products of double cosets on infinite-dimensional classical groups). 3.Consider the space of n-point configurations in 3-dimensional sphere. We write explicitly a generating function for all spherical functions on this space. We also get explicit formulas for actions of braid groups and the group of outer automorphisms of the free group on the space of configurations 4. Explicit construction of representation of ortho-symplectic (super)group by integral operators in super-Fock space is obtained. 5. It is obtained a canonical orthogonal decomposition of spaces of functions on some pseudo-Riemannian symmetric spaces 6. It is constructed a canonical `integral` operator from the space of real functions to the space of p-adic distributions 7. It is obtained a spectral decomposition of a pair of commuting hypergeometric operators 8. It is shown that developments of Coxeter polyhedra are unions of Coxeter polyhedra of smaller dimension. It is shown that matrix elements of infinite-dimensional representations of semisimple groups admit finite expressions in terms of Heckman-Opdam hypergeometric functions