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Simon Gindikin

Publications and source records attributed to Simon Gindikin.

14 recordsLinked to original sources
arXiv2020

Horospherical Cauchy Transform on Some Pseudo-Hyperbolic Spaces

We consider the horospherical transform and its inversion in 3 examples of hyperboloids. We want to illustrate via these examples the fact that the horospherical inversion formulas can be directly extracted from the classical Radon inversion formula. In a more broad context, this possibility reflects the fact that the harmonic analysis on symmetric spaces (Riemannian as well as pseudo-Riemannian ones) is equivalent (homologous), up to the Abelian Fourier transform, to the similar problem in the flat model. On the technical level it is important that we work not with the usual horospherical transform, but with its Cauchy modification.

math.DG↗
arXiv2016

Cohomological Laplace transform on non-convex cones and Hardy spaces of $\bar{\partial}$-cohomology on non-convex tube domains

We consider a class of non-convex cones $V$ in $\mathbb{R}^n$ which can be presented as (not unique) union of convex cones of some codimension $q$ which we call the index of non-convexity. This class contains non-convex symmetric homogeneous cones studied by the first author and his collaborators. For these cones we consider a construction of dual non-convex cones $V^*$ and corresponding non-convex tubes $T$ and define a cohomological Laplace transform from functions at $V$ to $q$-dimensional cohomology of $T$ using the language of smoothly parameterized uCech cohomology. We give a construction of Hardy space of $q$-dimensional cohomolgy at $T$.

math.FA↗
arXiv2012

Restricted Roots and Restricted Form of Weyl Dimension Formula for Spherical Varieties

We study in this paper the restricted roots for a class of spherical homogeneous spaces of semisimple groups which includes simply connected symmetric spaces. For these spaces we give a detailed description (case by case) of the set of roots of the group associated with each restricted root of the space (the nest of the restricted root). As an application, we obtain a refinement of the Weyl dimension formula in the case of spherical representations, expressing the dimension as a product over the set of indivisible positive restricted roots.

math.RT↗
arXiv2005

Horospherical Cauchy-Radon transform on compact symmetric spaces

Harmonic analysis on noncompact Riemannian symmetric spaces is in a sense equivalent to the theory of the horospherical transform. There are no horospheres on compact symmetric spaces, but we define a complex version of horospherical transform which plays a similar role for the harmonic analysis on them.

math.RT↗
arXiv2005

Complex horospherical transform on real sphere

We define a new integral transform on the real sphere which is invariant relative to the orthogonal group and similar to the horospherical Radon transform for the hyperbolic space. This transform involves complex geometry associated with the sphere.

math.RT↗
arXiv2004

Smoothly Parameterised Cech Cohomology of Complex Manifolds

A Stein covering of a complex manifold may be used to realise its analytic cohomology in accordance with the Cech theory. If, however, the Stein covering is parameterised by a smooth manifold rather than just a discrete set, then we construct a cohomology theory in which an exterior derivative replaces the usual combinatorial Cech differential. Our construction is motivated by integral geometry and the representation theory of Lie groups.

math.CV↗
arXiv2004

Holomorphic H-spherical distribution vectors in principal series representations

Let G/H be a semisimple symmetric space. The main tool to embed a principal series representation of G into L^2(G/H) are the H-invariant distribution vectors. If G/H is a non-compactly causal symmetric space, then G/H can be realized as a boundary component of the complex crown $Ξ$. In this article we construct a minimal G-invariant subdomain $Ξ_H$ of $Ξ$ with G/H as Shilov boundary. Let $π$ be a spherical principal series representation of G. We show that the space of H-invariant distribution vectors of $π$, which admit a holomorphic extension to $Ξ_H$, is one dimensional. Furthermore we give a spectral definition of a Hardy space corresponding to those distribution vectors. In particular we achieve a geometric realization of a multiplicity free subspace of L^2(G/H)_mc in a space of holomorphic functions.

math.RT↗
arXiv2002

Hardy spaces for non-compactly causal symmetric spaces and the most continuous spectrum

Let $G/H$ be a semisimple symmetric space. Then the space $L^2(G/H)$ can be decomposed into a finite sum of series representations induced from parabolic subgroups of $G$. The most continuous part of the spectrum of $L^2(G/H)$ is the part induced from the smallest possible parabolic subgroup. In this paper we introduce Hardy spaces canonically related to this part of the spectrum for a class of non-compactly causal symmetric spaces. The Hardy space is a reproducing Hilbert space of holomorphic functions living on a tube type bounded symmetric domain, containing $G/H$ as a boundary component. A boundary value map is constructed and we show that it induces an $G$-isomorphism onto a multiplicity free subspace of full spectrum in the most continuous part $L_{\rm mc}^2(G/H)$ of $L^2(G/H)$. We also relate our Hardy space with the classical Hardy space on the tube domain.

math.RT↗
arXiv2002

A remark on Schubert cells and duality of orbits on flag manifolds

It is known that the closure of an arbitrary K_c-orbit on a flag manifold is expressed as a product of a closed K_c-orbit and a Schubert cell ([M2], [Sp]). We already applied this fact to the duality of orbits on flag manifolds ([GM]). We refine here this result and and give its new applications to the study of domains arising from the duality.

math.RT↗