Infinite-dimensional groups and harmonic analysis on G-spaces

Our new standpoint is a recent progress in the representation theory of infinite-dimensional classical groups and symmetric groups. There were obtained geometric and combinatorial descriptions of double coset spaces with respect to small subgroups and train constructions (representations of categories enveloping infinite-dimensional groups). Our purposes are following: 1) to continue investigation of categories of double cosets for infinite-dimensional classical groups and for groups of infinite matrices over finite fields; 2) to search projective systems of measures on homogeneous spaces to obtain new possibilities for harmonic analysis related to infinite-dimensional groups; 3) to investigate representations of the group of diffeomorphisms of the circle related to fractional diffusions; to investigate representations and spherical functions for the combinatorial analog of this group; 4) for quasi-regular representations on some pseudo-Riemannian symmetric spaces, we wish to obtain the explicit orthogonal decomposition into subspaces with uniform spectra; 5) to construct orthonormal bases in unitary representations of Lie groups, which are permuted by actions of lattices; 6) to investigate some restriction problems for unitary representations and actions of overalgebras in spectral decompositions.

The main results of the project are: --- Construction of an operational calculus for the Fourier transform on the group of matrices of order 2. --- Explicit spectral decomposition of unitary operators corresponding to generic measure preserving transformations. --- A solution of the Gelfand-Gindikin problem about separation of spectra for the space $L^2$ on a pseudo-unitary group. --- Examination of restrictions of representations of a general linear group to a smaller subgroup and construction of an action of large Lie algebra by differential-difference operators. --- Joint spectral decomposition of a pair of commuting hypergeometric operators on complex plane. --- Classification of unitary representations of the group of p-adic infinite integer matrices. --- Investigations of spherical representations of groups of spheromorphisms of Bruhat--Tits trees.