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Yu. A. Neretin

Publications and source records attributed to Yu. A. Neretin.

6 recordsLinked to original sources
arXiv1999

Plancherel formula for Berezin deformation of $L^2$ on Riemannian symmetric space

Consider the space B of complex $p\times q$ matrces with norm <1. There exists a standard one-parameter family $S_a$ of unitary representations of the pseudounitary group U(p,q) in the space of holomorphic functions on B (i.e. scalar highest weight representations). Consider the restriction $T_a$ of $S_a$ to the pseudoorthogonal group O(p,q). The representation of O(p,q) in $L^2$ on the symmetric space $O(p,q)/O(p)\times O(q)$ is a limit of the representations $T_a$ in some precise sence. Spectrum of a representation $T_a$ is comlicated and it depends on $α$. We obtain the complete Plancherel formula for the representations $T_a$ for all admissible values of the parameter $α$. We also extend this result to all classical noncompact and compact Riemannian symmetric spaces.

math.RT↗
arXiv1999

Separation of spectra in analysis of Berezin kernels

Consider an unitary highest weight representation of a group U(p,q) in holomorphic functions on the symmetric space U(p,q)/U(p)\times U(q). Consider its restriction ρto the subgroup O(p,q). This restriction has a complicated spectrum consisting of representations having different types. We construct a decomposition of ρto a finite direct sum of representations τ_j such that each summand τ_j has spectrum consisting of one-type representations. Our tool is theorems about restrictions of holomorphic functions on Cartan domain U(p,q)/U(p)\times U(q) to submanifolds of the boundary. We also obtain Plancherel formula for this restriction.

math.RT↗
arXiv1999

Matrix analogs of B-functions and Plancherel formula for Berezin kernel representations

We obtain a family of matrix integrals which decompose to a product of Gamma-functions (they have some relations with S.G.Gindikin 'Beta', but generally speaking essentially differ from it). We obtain Plancherel formula for Berezin representations for all series of classical groups (for large values of parameters of representations). The Berezin representations are deformations of L^2 on Riemann noncompact symmetric spaces G/K defined by Berezin kernels, i.e powers of |det(1-zu^*)| (in matrix ball models). These representations also can be obtained by restrictions of holomorphic representations of some group Q containing G as symmetric subgroup. We also discuss models of noncompact Riemann symmetric spaces: matrix balls, matrix cones, matrix wedges, and sections of wedges.

math.RT↗
arXiv1999

Pseudoriemannian symmetric spaces: one-type realizations and open embeddings to grassmannians

We show that each classical pseudoriemann symmetric space G/H can be realized as space of pairs of complementary subspaces in a linear space. For each classical symmetric space we construct an open embedding to a grassmannian or to a product of two grassmanianns. We also show that the representation of the group G in L^2 on G/H is equivalent to restriction of a degenerated principal series representation of some group Q containing G.

math.DG↗