Wishart--Pickrell distributions and closures of group actions

Consider probabilistic distributions on the space of infinite Hermitian matrices $Herm(\infty)$ invariant with respect to the unitary group $U(\infty)$. We describe the closure of $U(\infty)$ in the space of spreading maps (polymorphisms) of $Herm(\infty)$, this closure is a semigroup isomorphic to the semigroup of all contractive operators.Comment: 8pp, typos were corrected, minor addition

Restriction of representations of GL(n+1,C)GL(n+1,C) to GL(n,C)GL(n,C) and action of the Lie overalgebra

Consider a restriction of an irreducible finite dimensional holomorphic representation of \GL(n+1,C) to the subgroup $GL(n,C)$ (it is determined by the Gelfand-Tsetlin branching rule). We write explicitly formulas for generators of the Lie algebra $gl(n+1)$ in the direct sum of representations of \GL(n,C). Nontrivial generators act as differential-difference operators, the differential part has order $(n-1)$, the difference part acts on the space of parameters (highest weights) of representations. We also formulate a conjecture about unitary principal series of $GL(n,C)$Comment: 34

On the Weil representation of infinite-dimensional symplectic group over a finite field

We extend the Weil representation of infinite-dimensional symplectic group to a representation a certain category of linear relations.Comment: 19p

On pp-adic colligations and 'rational maps' of Bruhat-Tits trees

Consider matrices of order $k+N$ over $p$-adic field determined up to conjugations by elements of $GL$ over $p$-adic integers. We define a product of such conjugacy classes and construct the analog of characteristic functions (transfer functions), they are maps from Bruhat-Tits trees to Bruhat-Tits buildings. We also examine categorical quotient for usual operator colligations.Comment: 20p

Hua type beta-integrals and projective systems of measures on flag spaces

We construct a family of measures on flag spaces (or, equivalently, on the spaces of upper-triangular matrices) compatible with respect to natural projections. We obtain an $n(n-1)/2$-parametric family of beta-integrals over space of upper-triangular matrices of size $n$.Comment: 9p

Radial parts of Haar measures and probability distributions on the space of rational matrix-valued functions

Consider the space $C$ of conjugacy classes of a unitary group $U(n+m)$ with respect to a smaller unitary group $U(m)$. It is known that for any element of the space $C$ we can assign canonically a matrix-valued rational function on the Riemann sphere (a Livshits characteristic function). In the paper we write an explicit expression for the natural measure on $C$ obtained as the pushforward of the Haar measure of the group $U(n+m)$ in the terms of characteristic functions.Comment: 14

Multiplication of conjugacy classes, colligations, and characteristic functions of matrix argument

We extend the classical construction of operator colligations and characteristic functions. Consider the group $G$ of finite block unitary matrices of size $\alpha+\infty+...+\infty$ ($k$ times). Consider the subgroup $K=U(\infty)$, which consists of block diagonal unitary matrices (with a block 1 of size $\alpha$ and a matrix $u\in U(\infty)$ repeated $k$ times). It appears that there is a natural multiplication on the conjugacy classes $G//K$. We construct 'spectral data' of conjugacy classes, which visualize the multiplication and are sufficient for a reconstruction of a conjugacy class.Comment: 20pp, extended versio

Hua measures on the space of pp-adic matrices and inverse limits of Grassmannians

We construct $p$-adic counterparts of Hua measures, measures on inverse limits of $p$-adic Grassmannians, and describe natural groups of symmetries of such measures.Comment: 15p

The subgroup PSL(2,R)PSL(2,R) is spherical in the group of diffeomorphisms of the circle

We show that the group $PSL(2,R)$ is a spherical subgroup in the group of $C^3$-diffeomorphisms of the circle. Also, the group of automorphisms of a Bruhat--Tits tree is a spherical subgroup in the group of hierarchomorphisms of the tree.Comment: 6pp, typos were correcte

An analog of the Dougall formula and of the de Branges--Wilson integral

We derive a beta-integral over $\mathbb{Z}\times \mathbb{R}$ , which is a counterpart of the Dougall $_5H_5$-formula and of the de Branges--Wilson integral, our integral includes $_{10}H_{10}$-summation. For a derivation we use a two-dimensional integral transform related to representations of the Lorentz group, this transform is a counterpart of the Olevskii index transform (a synonym: Jacobi transform).Comment: 11