Infinite-dimensional groups and spectral problems

New standpoints for the project are recent progress in noncommutative harmonic analysis(symmetries of Plancherel decompositions and explicit constructions of separations of spectra) and geometric descriptions of categories of double cosets for infinite-dimensional groups; The research will follow several lines: 1. to develop operational calculus for Fourier transforms on groups and for Plancherel decompositions of unitary representations; 2. to investigate various problems related to separation of spectra into uniform pieces in non- commutative harmonic analysis; 3. to continue my works on categories of double cosets on infinite dimensional-groups (trains), their actions om measure spaces, applications for finite and finite-dimensional groups; 4. to examine infinite-dimensional groups over finite fields and to solve some classification problems; 5. to investigate some spectral problems of ergodic theory; 6. to examine combinatorial counterpart of the group of diffeomorphisms of the circle and Thompson groups; 7. to examine some explicit spectral problems for commuting differential operators and the corresponding integral transforms;

The topics of the project involve unitary representations of several types of groups and related spectral problems. One of our tools for Lie groups are difference operators acting in spaces of square integrable functions by shifts in imaginary directions and their expansions in eigenfunctions. Using operators of this kind, we obtain the operational calculus for the Fourier transform on the Lobachevsky plane. We define 'generalized hypergeometric functions of the complex field' of arbitrary level (they are real functions of two variables), and propose a flexible technology for operating with such functions. We show that most of the properties of the usual generalized hypergeometric functions have counterparts for 'hypergeometric functions of complex fields'. We examine infinite-dimensional groups and their unitary representations. It turns out that such representations admit extensions from groups to essentially wider structures (semigroups and categories). Such extensions became tools for investigation of representations, on the other hand, they are interesting by themselves. We use recent progress in investigations of Polish groups (which basically was related to descriptive set theory, general topology, model theory) and involve additional types of groups to representation theoretical considerations. Sometimes problems, which are well-undersood (or relatively well-understood) for infinite-dimensional groups, have classical counterparts, which seems totally incomprehensible. We apply progress in infinite-dimensional groups to classical objects and get non-standard results on finite symmetric groups. We investigate actions of groups on measure spaces (measure preserving actions and nonsingular actions) In particular, for a generic conjugacy classes of measure preserving transformations we get explicit decompositions in generalized eigenfunctions. In this case, spectral measures (they were obtained by Bourgain) are singular; moreover, they are singular in a very strong sense. On the other hand, we show that actions of infinite-dimensional groups on measure spaces automatically generate actions of corresponding semigroups and categories.