On compression of Bruhat-Tits buildings

We obtain an analog of the compression of angles theorem in symmetric spaces for Bruhat--Tits buildings of the type $A$. More precisely, consider a $p$-adic linear space $V$ and the set $Lat(V)$ of all lattices in $V$. The complex distance in $Lat(V)$ is a complete system of invariants of a pair of points of $Lat(V)$ under the action of the complete linear group. An element of a Nazarov semigroup is a lattice in the duplicated linear space $V\oplus V$. We investigate behavior of the complex distance under the action of the Nazarov semigroup on the set $Lat(V)$.Comment: 6 page

Notes on Stein-Sahi representations and some problems of non L2L^2 harmonic analysis

We discuss one natural class of kernels on pseudo-Riemannian symmetric spaces.Comment: 40p

On the Ado Theorem for finite Lie conformal algebras with Levi decomposition

We prove that a finite torsion-free conformal Lie algebra with a splitting solvable radical has a finite faithful conformal representation.Comment: 11 page

Infinite-dimensional pp-adic groups, semigroups of double cosets, and inner functions on Bruhat--Tits builldings

We construct $p$-adic analogs of operator colligations and their characteristic functions. Consider a $p$-adic group $G=GL(\alpha+k\infty, Q_p)$, its subgroup $L=O(k\infty,Z_p)$, and the subgroup $K=O(\infty,Z_p)$ embedded to $L$ diagonally. We show that double cosets $\Gamma= K\setminus G/K$ admit a structure of a semigroup, $\Gamma$ acts naturally in $K$-fixed vectors of unitary representations of $G$. For each double coset we assign a 'characteristic function', which sends a certain Bruhat--Tits building to another building (buildings are finite-dimensional); image of the distinguished boundary is contained in the distinguished boundary. The latter building admits a structure of (Nazarov) semigroup, the product in $\Gamma$ corresponds to a point-wise product of characteristic functions.Comment: new version of the paper, 47pp, 3 figure

On beta-function of tube of light cone

We construct $B$-function of the Hermitian symmetric space \OO(n,2)/\OO(n)\times \OO(2) or equivalently of the tube $(Re z_0)^2> (Re z_1)^2+...+ (Re z_n)^2$ in $C^{n+1}Comment: 7 page

Contractions of Low-Dimensional Lie Algebras

Theoretical background of continuous contractions of finite-dimensional Lie algebras is rigorously formulated and developed. In particular, known necessary criteria of contractions are collected and new criteria are proposed. A number of requisite invariant and semi-invariant quantities are calculated for wide classes of Lie algebras including all low-dimensional Lie algebras. An algorithm that allows one to handle one-parametric contractions is presented and applied to low-dimensional Lie algebras. As a result, all one-parametric continuous contractions for the both complex and real Lie algebras of dimensions not greater than four are constructed with intensive usage of necessary criteria of contractions and with studying correspondence between real and complex cases. Levels and co-levels of low-dimensional Lie algebras are discussed in detail. Properties of multi-parametric and repeated contractions are also investigated.Comment: 47 pages, 4 figures, revised versio