Simon Gindikin

Invariant Stein domains in Stein symmetric spaces anda nonlinear complex convexity theorem

Let X = G/K be a semisimple noncompact Riemannian symmetric space. We may assume that G is algebraic. By our assumption, G sits in its universal complexification GC and so X ⊆ XC := GC/KC. Note that XC is a Stein symmetric space. Observe that the group G does not act properly (i.e., with compact isotropy subgroups) on GC/KC since KC is not compact. In our considerations, a central role is played by certain G-invariant neighborhood Ξ of X in XC which was first studied in [1]. We recall its definition. Write G = NAK for an Iwasawa decomposition of G. We consider G-orbits in XC. Those have a quite complicated parametrization. We restrict ourselves to G-orbits which intersect exp(ia)KC/KC where a: =Lie(A). Among those G-orbits define Ξ as the maximal connected union of Gorbits with compact isotropy subgroups. In [1], it was shown that

Pseudo-Hermitian symmetric spaces of tube type

Let Ω be an open connected cone in a real vector space V ≃ ℝn. One defines G(Ω) = {g ∈ GL(n,∝) ∣ gΩ = Ω}. The cone Ω is said to be homogeneous if the group G(Ω) acts transitively on Ω. For the beginning let us assume that Ω is convex and that \(\bar \Omega \) is pointed (this means that \(\bar \Omega \) ∩ (}\(\bar \Omega \)) = {0}). The convex cone Ω is said to be selfdual if there exists a positive inner product on V such that Ω✶ = Ω, where the open dual cone Ω✶ is defined by

On the automorphism group of the generalized conformal structure of a symmetric R-space

G(\Omega ) = \{ g \in GL(n,\mathbb{R}|g\Omega = \Omega \} .

Holomorphic H-spherical distribution vectors in principal series representations

The open convex cone Ω is said to be symmetric if it is homogenous and selfdaul. Let us recall the connection between symmetric convex cones and Jordan algebras. A Jordan algebra V is a vector space equipped with a product, i.e., a bilinear map V × V → V such that (J1) xy = xy, (J2) x(x 2 ) = x 2 (xy).

Conformal harmonic analysis on hyperboloids

Abstract A language is developed for ∂-cohomology, which is different from both the Dolbeault and the Cech descriptions, and involves only holomorphic objects. This language is then illustrated in certain cases of interest to representation theory. This makes possible a new geometric construction of the ladder representations for SU(2, p) and the non-holomorphic discrete series representations of SU(2, 1). The constructions are closely related to Penrose transforms.

Horospherical model for holomorphic discrete series and horospherical Cauchy transform

Abstract In this paper we define a canonical locally flat generalized conformal structure on a symmetric R -space of the rank greater than 1. We prove that the group of automorphisms of this structure coincides with the noncompact group of automorphisms of the symmetric space.

SO(1 ; n)-twistors

In this paper we define a distinguished boundary for the complex crowns Ξ ⊆ G C /K C of non-compact Riemannian symmetric spaces G/K. The basic result is that affine symmetric spaces of G can appear as a component of this boundary if and only if they are non-compactly causal symmetric spaces.

Holomorphic horospherical transform on noncompactly causal spaces

Let G/H be a semisimple symmetric space. The main tool to embed a principal series representation of G into L2(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 ΞH 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 ΞH, 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 L2(G/H)mc in a space of holomorphic functions.