Speciality:
01.01.01 (Real analysis, complex analysis, and functional analysis)
Birth date:
27.02.1939
E-mail: ,
Keywords: theory of group representations; symmetric spaces; harmonic analysis on homogeneous spaces; quantization; canonical representations; boundary representations.
Subject:
In a series of my works (the 60 80 ies) the construction of harmonic analysis on semisimple symmetric spaces $G/H$ (non-Riemannian) of rank one was begun and completed. A description of the corresponding principal non-unitary series of representations was given. Principal notions of the theory were introduced ($H$-invariants, Fourier transform, Poisson transform, spherical functions) and corresponding methods were worked out. Plancherel formula was obtained explicitly ( in different variants, one of them is expansion of the delta function in terms of spherical functions). The Berezin quantization was transferred from Hermitian symmetric spaces to symplectic semisimple symmetric spaces. In particular, an important case of quantizations was described — the so–called polynomial quantization. A new form of the deformation decomposition (the decomposition of the Berezin transform) was offered using "generalized powers" (generalized Pochhammer symbols) instead of usual powers of a parameter. This form makes the decomposition natural and apparent and allows to compute it explicitly. Canonical representations on these symplectic spaces were studied - in connection with the construction of quantizations (decompositions into irreducible constitutients — right up to explicit formulae for one rank spaces). The canonical representations (sometimes called the Berezin representations) on Hermitian symmetric spaces were introduced by Berezin and Vershik–Gelfand–Graev. They are unitary representations. We consider the canonical representations in a much wider sense: we give up the condition of unitarity, they act on sufficiently extensive function spaces, in paricular, on spaces of distributions. Also boundary representations generated by canonical representations were studied. In particular, appearance of Jordan blocks in the decomposition of these representations was discovered. It is found that the decomposition of boundary representations is intimately connected with the meromorphic structure of Poisson and Fourier transforms associated with the canonical representations. These results (quantizations, canonical and boundary representations) can be transferred to a certain extent to some semisimple symmetric spaces which are not symplectic, for example, to hyperboloids of arbitrary signature. This work (quantizations, canonical and boundary representations etc.) is a part of what I call a non-unitary version of harmonic analysis, a new and promising field of research. For hyperboloids of Hermitian type, the holomorphic discrete series was investigated, Cauchy-Szego kernels were computed, projection operators on analytic and antianalytic series of irreducible unitary reresentations were explicitly found, an analogue of the Hilbert transform was introduced and computed. One of results — separation of series — was carried over to hyperboloids of arbitrary signature. For finite reflection groups, Poincare polynomials and series were explicitly computed.
Main publications:
Molchanov V. F. Quantization on para-Hermitian symmetric spaces // Amer. Math. Soc. Transl. Ser. 2, vol. 175 (Adv. Math. Sci., 31), 1996, 81–96.
Dijk. G. van, Molchanov V. F. The Berezin form for rank one para-Hermitian symmetric spaces // J. Math. Pures Appl., 1998, 77, no. 8, 747–799.
Dijk. G. van, Molchanov V. F. Tensor products of maximal degenerate series representations of the group $SL(n,\Bbb R)$ // J. Math. Pures Appl., 1999, 78, no. 1, 99–119.
