ИСТИНА

The research will follows several lines: 1. I intend to investigate multplication of double cosets on infinite-dimensional groups. 2. I want to investigate representations of infinite symmetric groups. 3. I hope to apply multiplication of double cosets to geometry and amaysis on finite-dimensional spaces. 4. I wish to investigate quasiinvariant actions of groups diffeomorphisms and hierarchomorphisms and corrersponding representations. 5. I intend to investigate polymorphisms related to infinite-dimensional groups. 6. I want to examine difference operators in imaginary direction and related multi-variate differential-difference operators. 7. I intend to continue my work on analysis on symmetric spaces. 8. I want to examine the analysis on inverse limits of symmetric spaces. 9. I want to examine braid groups and the corresponding Lie groups.

1. Sufficient spectral data for double cosets of infinite dimensional classical groups were obtained 2. There were obtained a big family of functors producing topological field theory type constructions from representations of infinite symmetric groups. In particular, where were obtained simplicial 'topological field theories' of arbitrary dimension. 3. There were obtained explicit formulas for action of the general linear Lie algebra in spectral decompositions of restriction of representations to a smaller subalgebra/ 4. Analogs of Hua measures and Hua integrals for flag spaces were constructed. 5. Mapping class groups of diffeomorphisms preserving a Cantor set were examined. 6. It is proved that the subgroup SL(2) is spherical in the group of diffeomorphisms of circles 7. A family of spherical representations of a combinatorial analog of the group of diffeomorphisms of the cirle is constructed. 8. Closures of some actions of infinite orthogonal group were obtained