Homogeneous spaces

The research will follows several lines: 1. I intend to attack the harmonic analysis on some non-symmetric homogeneous spaces. 2. I want to investigate holomorphic continuations of representations of semisimple groups. 3. In the previous works there were constructed natural kernels on pseudo-Riemannian symmetric spaces. Now I intended to study their properties; in particular, the Plancherel formula and relations with the Gelfand-Gindikin program. 4. I plan to investigate various applications of Plancherel formula, and, in particular, intend to investigate actions of overalgebras in Plancherel decompositions. 5. I intend to investigate various applications of harmonic analysis to special functions and mathematical physics. 6. I want to examine conformal geometry of symmetric spaces and properties of integral manifolds 7. I want to apply the super-spinor representation to representations of super-groups and super-symmetric conformal field theories. 8. I hope to investigate properties of some infinite-dimensional integral transforms and relations between different models of Fock spaces. 9. I intend to examine the analysis on the space of univalent functions and models of the conformal field theory.

1. There were constructed multiplications of double coset on various infinite-dimensional groups. Explicit descriptions of such multiplications were obtained. In particular: a) There was constructed a family of functors, which produces 2-dimensional polygonal topological field theories from representations of infinite symmetric groups. b) There constructed multivariate characteristic functions for families of operators defined up to a common conjugation in a subspace. It is proved sufficiency of spectral data produced in this way. c) For p-adic groups there were constructed 'characteristic functions' as simplicial maps of Bruhat-Tits buildings sending a distinguished boundary to a distinguished boundary d) There constructed the action of a semigroup of operator colligations by polymorphisms (spreading maps with spreaded Radon-Nikodym-derivative) on the space with Gaussian measure. 2. Explicit spectral decompositions for several Sturm-Liouville difference operators in imaginary direction were obtained. The structure of the problem was clarified.3. It is obtained a decomposition of the space functions on a semi-infinite Grassmannian over a finite field.4. A p-adic analogs of Hua-Pickrell measures was constructed.5. It is proved that actions of Kohno's Lie algebra of the braid group on compact real analytic manifolds admit integration to the corresponding Lie group. 6. Models of the natural representation of affine symplectic group were examined.