Yuri Neretin | University of Vienna (original) (raw)

Papers by Yuri Neretin

$Research paper thumbnail of Some remarks on stable densities and operators of fractional differentiation$

arXiv: Classical Analysis and ODEs, 2004

Let D(s)D(s)D(s) be a fractional derivation of order sss. For a real pne0p\ne 0pne0, we construct an integral... more Let D(s)D(s)D(s) be a fractional derivation of order sss. For a real pne0p\ne 0pne0, we construct an integral operator A(p)A(p)A(p) in an appropriate functional space such that A(p)D(s)A(p)−1=D(ps)A(p) D(s) A(p)^{-1}=D(p s)A(p)D(s)A(p)−1=D(ps) for all sss. The kernel of the operator A(p)A(p)A(p) is expressed in terms of a function similar to the stable densities.

0.1. Berezin formulae. Apparently, first elements of a strange analogy between orthogonal and sym... more 0.1. Berezin formulae. Apparently, first elements of a strange analogy between orthogonal and symplectic spinors were observed by K. O. Friedrichs in early 1950s, see [10]. F. A. Berezin in early 1960s obtained explicit formulae [2] for both representations. We shortly recall his results. First of all, let us realize the real symplectic group Sp(2n, R) as the group of complex (n + n) × (n + n) matrices

1.1. History of the objects. The purpose of these notes is an elementary introduction to Stein–Sa... more 1.1. History of the objects. The purpose of these notes is an elementary introduction to Stein–Sahi complementary series and unipotent representation. All these objects are elementary in a certain sense, however they appeared in the representation theory relatively late. Theory of infinite dimensional representations of semi-simple groups was initiated by pioneer works of I. M. Gelfand and M. A. Naimark (1946–1950), V. Bargmann [2] (1947), and K. O. Friedrichs [11] (1951–1953). The book [13] of I. M. Gelfand and M. A. Naimark (1950) contained a well-developed theory for the complex classical groups GL(n,C), SO(n,C), Sp(2n,C) (the parabolic induction, complementary series, spherical functions, characters, Plancherel theorems). However, this classical book contained various statements and asseverations that were not actually proved. In modern terminology, some of chapters were ’mathematical physics’. The most of these statements were really proved by 1958–1962 in works of different au...

Moscow Mathematical Journal, 2006

We construct canonically defined central extensions of groups of symplectomorphisms. We show that... more We construct canonically defined central extensions of groups of symplectomorphisms. We show that this central extension is nontrivial in the case of a torus of dimension ≥ 6 and in the case of a two-dimensional surface of genus ≥ 3. 1 Formulation of results Central extensions of the groups of symplectomorphisms discussed in this paper appeared as a byproduct in [25]. Here we prove several nontrivality and triviality theorems concerning this cocycle.

Eprint Arxiv 0707 0570, Jul 4, 2007

We obtain explicit formulae for the spinor representation of the orthosymplectic supergroup OSp(2... more We obtain explicit formulae for the spinor representation of the orthosymplectic supergroup OSp(2p|2q) and the corresponding Olshanski super semigroup. We also extend this representation to the Lagrangian super-Grassmannian. 2

Sbornik: Mathematics, 2014

Let Gms be the group of transformations of a Lebesgue space leaving the measure quasiinvariant, l... more Let Gms be the group of transformations of a Lebesgue space leaving the measure quasiinvariant, let Ams be its subgroup consisting of transformations preserving the measure. We describe canonical forms of double cosets of Gms by the subgroup Ams and show that all continuous Ams-biinvariant functions on Gms are functionals on of the distribution of a Radon-Nikodym derivative.

Pour commencer, j'essaierai de raconter l'histoire des origines du livre de Berezin [16] ... more Pour commencer, j'essaierai de raconter l'histoire des origines du livre de Berezin [16] et de montrer son influence sur la vie mathématique. Je donnerai dans les paragraphes 2 et 3 un abrégé des deux constructions les plus compliquées de ce livre, que l'on peut maintenant décrire de façon suffisamment simple. Enfin, j'examine dans le dernier paragraphe quelques exemples (mais assez peu) d'applications que cette construction a pu connaître dans les années passées.

Functional Analysis and Its Applications, 2007

We show that each K-finite matrix element of an irreducible Harish-Chandra module can be obtained... more We show that each K-finite matrix element of an irreducible Harish-Chandra module can be obtained from spherical functions by a finite collection of operations.

Bulletin des Sciences Mathématiques, 2008

Journal of Mathematical Sciences, 2011

We construct B-function of the Hermitian symmetric space O(n, 2)/O(n) × O(2) or equivalently of t... more We construct B-function of the Hermitian symmetric space O(n, 2)/O(n) × O(2) or equivalently of the tube (Re z0) 2 > (Re z1) 2 + • • • + (Re zn) 2 in C n+1. 1 Formulation of the result 1.1. Preliminary references. The beta-function of symmetric cones GL(n, R)/O(n), GL(n, C)/U(n), GL(n, H)/Sp(n) was constructed by Gindikin in [4], see also [6]. For the remaining series of classical symmetric spaces the beta-function was obtained in [7]. The subseries O(n, 2)/O(n) × O(2) has two beta-functions, the first one is a special case of the beta-function of O(p, q)/O(p) × O(q). The second beta-function is discussed here, it is related to the Hermitian structure of these spaces. 2 1.2. The tube of light cone. Consider the space C n+1 with coordinates z 0 , z 1 ,. .. , z n. By T n we denote the tube (Re z 0) 2 > (Re z 1) 2 + (Re z 2) 2 + • • • + (Re z n) 2 , Re z 0 > 0 The space T n is homogeneous with respect to the pseudo-orthogonal group O(n + 1, 2) (apparently, this was discovered by E.

American Mathematical Society Translations: Series 2, 2007

We construct a canonical embedding of the Schwartz space on R n to the space of distributions on ... more We construct a canonical embedding of the Schwartz space on R n to the space of distributions on the adelic product of all the p-adic numbers. This map is equivariant with respect to the action of the symplectic group Sp(2n, Q) over rational numbers and with respect to the action of rational Heisenberg group. These notes contain two elements. First, we give a funny realization of a space of complex functions of a real variable as a space of functions of p-adic variable. Secondly, we try to clarify classical contstruction of modular forms through θfunctions and Howe duality.

American Mathematical Society Translations: Series 2, 2007

In this note, we construct a canonical embedding of the space L 2 over a determinantal point proc... more In this note, we construct a canonical embedding of the space L 2 over a determinantal point process to the fermionic Fock space. Equivalently, we show that a determinantal process is the spectral measure for some explicit commutative group of Gaussian operators in the fermionic Fock space.

$Research paper thumbnail of Stable densities and operators of fractional differentiation$

American Mathematical Society Translations: Series 2, 2011

Consider the space of double cosets of the product of n copies of SU(2) with respect to the diago... more Consider the space of double cosets of the product of n copies of SU(2) with respect to the diagonal subgroup. We get a parametrization of this space, the radial part of the Haar measure, and explicit formulas for the actions of the group of outer automorphisms of the free group Fn−1 and of the braid group of n − 1 strings.

Analysis and Mathematical Physics, 2011

In the spectral theory of non-self-adjoint operators there is a well-known operation of product o... more In the spectral theory of non-self-adjoint operators there is a well-known operation of product of operator colligations. Many similar operations appear in the theory of infinite-dimensional groups as multiplications of double cosets. We construct characteristic functions for such double cosets and get semigroups of matrix-valued functions in matrix balls.

Journal of Mathematical Sciences, 2006

Consider an affine Bruhat-Tits building Latn of the type An−1 and the complex distance in Latn, i... more Consider an affine Bruhat-Tits building Latn of the type An−1 and the complex distance in Latn, i.e., the complete system of invariants of a pair of vertices of the building. An elements of the Nazarov semigroup is a lattice in the duplicated p-adic space Q n p ⊕ Q n p. We investigate behavior of the complex distance with respect to the natural action of the Nazarov semigroup on the building.

International Mathematics Research Notices, 2010