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Yury A. Neretin

Publications and source records attributed to Yury A. Neretin.

At least 55 records · Page 3Linked to original sources
arXiv2018

Operational calculus for Fourier transform on the group $GL(2,R)$

Consider the Fourier transform on the group $GL(2,R)$ of real $2\times 2$-matrices. We show that Fourier-images of polynomial differential operators on $GL(2,R)$ are differential-difference operators with coefficients meromorphic in parameters of representations. Expressions for operators contain shifts in imaginary direction with respect to the integration contour in the Plancherel formula. We present explicit formulas for images of partial derivations and multiplications by coordinates.

math.RT↗
arXiv2018

After Plancherel formula

We discuss two topics related to Fourier transforms on Lie groups and on homogeneous spaces: the operational calculus and the Gelfand--Gindikin problem (program) about separation of non-uniform spectra. Our purpose is to indicate some non-solved problems of non-commutative harmonic analysis that definitely are solvable. This is a sketch of my talks on VI School "Geometry and Physics", Bialowieza, Poland, June 2017.

math.RT↗
arXiv2017

Symmetric groups and checker triangulated surfaces

We consider triangulations of surfaces with edges painted three colors so that edges of each triangle have different colors. Such structures arise as Belyi data (or Grothendieck dessins d'enfant), on the other hand they enumerate pairs of permutations determined up to a common conjugation. The topic of these notes is links of such combinatorial structures with infinite symmetric groups and their representations.

math.RT↗
arXiv2017

Nikolay Luzin, his students, adversaries, and defenders (notes on the history of Moscow mathematics, 1914-1936)

This is historical-mathematical and historical notes on Moscow mathematics 1914-1936. Nikolay Luzin was a central figure of that time. Pavel Alexandroff, Nina Bari, Alexandr Khinchin, Andrey Kolmogorov, Mikhail Lavrentiev, Lazar Lyusternik, Dmitry Menshov, Petr Novikov, Lev Schnirelman, Mikhail Suslin, and Pavel Urysohn were his students. We discuss the time of the great intellectual influence of Luzin (1915-1924), the time of decay of his school (1922-1930), a moment of his administrative power (1934-1936), and his fall in July 1936.

math.HO↗
arXiv2017

The Fourier transform on the group $GL_2(R)$ and the action of the overalgebra $gl_4$

We define a kind of 'operational calculus' for $GL_2(R)$. Namely, the group $GL_2(R)$ can be regarded as an open dense chart in the Grassmannian of 2-dimensional subspaces in $R^4$. Therefore the group $GL_4(R)$ acts in $L^2$ on $GL_2(R)$. We transfer the corresponding action of the Lie algebra $gl_4$ to the Plancherel decomposition of $GL_2(R)$, the Lie algebra acts by differential-difference operators with shifts in an imaginary direction. We also write similar formulas for the action of $gl_4\oplus gl_4$ in the Plancherel decomposition of $GL_2(C)$

math.RT↗
arXiv2017

Projectors separating spectra for $L^2$ on pseudounitary groups $U(p,q)$

The spectrum of $L^2$ on a pseudo-unitary group $U(p,q)$ (we assume $p\ge q$ naturally splits into $q+1$ types. We write explicitly orthogonal projectors in $L^2$ to subspaces with uniform spectra (this is an old question formulated by Gelfand and Gindikin). We also write two finer separations of $L^2$. In the first case pieces are enumerated by $r=0$, 1,..., $q$ and representations of discrete series of $U(p-r,q-r)$, where $r=0$, \dots, $q$. In the second case pieces are enumerated by all discrete parameters of the tempered spectrum of $U(p,q)$.

math.RT↗
arXiv2016

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.

math.RT↗
arXiv2016

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 $α+\infty+...+\infty$ ($k$ times). Consider the subgroup $K=U(\infty)$, which consists of block diagonal unitary matrices (with a block 1 of size $α$ 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.

math.RT↗
arXiv2016

The group of diffeomorphisms of the circle: reproducing kernels and analogs of spherical functions

The group $Diff$ of diffeomorphisms of the circle is an infinite dimensional analog of the real semisimple Lie groups $U(p,q)$, $Sp(2n,R)$, $SO^*(2n)$; the space $Ξ$ of univalent functions is an analog of the corresponding classical complex Cartan domains. We present explicit formulas for realizations of highest weight representations of $Diff$ in the space of holomorphic functionals on $Ξ$, reproducing kernels on $Ξ$ determining inner products, and expressions ('canonical cocycles') replacing spherical functions.

math.CV↗
arXiv2016

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.

math.GR↗
arXiv2015

Topological groups and invariant measures

Lecture notes in Russian. Topics: the Haar measure (abstract theorems and explicit descriptions for different groups), measures on infinite-dimensional spaces with large natural groups of symmetries (Gaussian measures, Poisson measures, virtual permutations, inverse limits of unitary groups), zoo of examples of topological groups, generalities for large (infinite-dimensional) groups, polymorphisms.

math.FA↗
arXiv2015

Infinite symmetric groups and combinatorial constructions of topological field theory type

The paper contains a survey of train constructions for infinite symmetric groups and related groups. For certain pairs (a group $G$, a subgroup $K$), we construct categories, whose morphisms are two-dimensional surfaces tiled by polygons and colored in a certain way. A product of morphisms is a gluing of combinatorial bordisms. For a unitary representation of $G$ we assign a functor from the category of bordisms to the category of Hilbert spaces and bounded operators. The construction has numerous variations, instead of surfaces there arise also one-dimensional objects of Brauer diagram type, multi-dimensional pseudomanifolds, bipartite graphs

math.RT↗
arXiv2015

Matrix beta-integrals: an overview

First examples of matrix beta-integrals were discovered on 1930-50s by Siegel and Hua, in 60s Gindikin obtained multi-parametric series of such integrals. We discuss beta-integrals related to symmetric spaces, their interpolation with respect to the dimension of a ground field, and adelic analogs; also we discuss beta-integrals related to flag spaces.

math.CA↗
arXiv2015

Infinite symmetric group and bordisms of pseudomanifolds

We consider a category whose morphisms are bordisms of $n$-dimensional pseudomanifolds equipped with a certain additional structure (coloring). On the other hand, we consider the product $G$ of $(n+1)$ copies of infinite symmetric group. We show that unitary representations of $G$ produce functors from the category of $(n-1)$-dimensional bordisms to the category of Hilbert spaces and bounded linear operators.

math.RT↗