A refinement of the complex convexity theorem via symplectic techniques
We apply techniques from symplectic geometry to extend and give a new proof of the complex convexity theorem of Gindikin-Kroetz.
math.SG↗
arXiv subjects
Bernhard Kroetz
Publications and source records attributed to Bernhard Kroetz.
A Convexity property for the SO(2,C)-double coset decomposition of SL(2,C) and applications to spherical functions
We make a fine study of the SO(2,C)-double coset decomposition in SL(2,C) and give a full description of the intersection of the various cells with the complex crown Xi of SL(2,R)/SO(2). A non-linear convexity theorem is proved and applications to analytically continued spherical functions are given.
math.RT↗
The c-function for non-compactly causal symmetric spaces and its relations to harmonic analysis and representation theory
We give an overview on the c-function of a non-compactly causal symmetric space G/H and explain its interplay with harmonic analysis and representation theory.
math.RT↗
A convexity property for real orbits in complexified Riemannian symmetric spaces
A convexity theorem for certain G-orbits in a complexified Riemannian symmetric space G_C/K_C is proved. Applications to analytically continued spherical functions will be given.
math.RT↗
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↗
Discrete group actions on Stein domains in complex Lie groups
This paper deals with the analytic continuation of holomorphic automorphic forms on a Lie group $G$. We prove that for any discrete subgroup $Γ$ of $G$ there always exists a non-trivial holomorphic automorphic form, i.e., there exists a $Γ$-spherical unitary highest weight representation of $G$. Holomorphic automorphic forms have the property that they analytically extend to holomorphic functions on a complex Ol'shanski\uı semigroup $S\subeq G_\C$. As an application we prove that the bounded holomorphic functions on $Γ\bs S\subseteq Γ\bs G_\C$ separate the points.
math.RT↗
Complex crowns of Riemannian symmetric spaces and non-compactly causal symmetric spaces
In this paper we define a distinguished boundary for the complex crowns $Ξ\subeq G_\C /K_\C$ 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.
math.RT↗
Invariant Stein domains in Stein symmetric spaces and a non-linear complex convexity theorem
We prove a complex version of Kostant's non-linear convexity theorem. Applications to the construction of G-invariant Grauert tubes of non-compact Riemannian symmetric G/K spaces are given.
math.RT↗
On the dual of complex Olshanskii semigroups
Let $G$ be a connected Lie group and $\hat G$ its unitary dual. We are interested in the part $Λ\subset\hat G$ which corresponds to the unitary highest weight representations of $G$. Then there are several topologies on $Λ$: The euclidean topology $T_E$ which comes from the identification of $Λ$ with the set of highest weights, the induced topology $T_I$ induced from the Fell topology on $\hat G$ and finally a natural topology $T_S$ which comes from the hull kernel topology of certain CCR C^*-algebras which are related to the holomorphic extemsion of unitary highest weight representations to complex Olshanskii semigroups $S$. One of the main results in this paper is the inclusion chain $T_S\subset T_I\subset T_E$. Further we exhibit very large interesting subspaces of $Λ$ where these topologies coincide. Finally we show that the Borel structures on $Λ$ induced from the three different topologies coincide.
math.RT↗
Unitary spherical highest weight representations
In this paper we give a classification of the $H$-spherical unitary highest weight representations of a hermitian Lie group $G$.
math.RT↗
Formal dimension for semisimple symmetric spaces
In this paper we generalize Harish Chandra's formula for the formal dimension of a representation of the holomorphic discrete series of a hermitian Lie group $G$ to semisimple symmetric spaces $G/H$.
math.RT↗
The c-function for non-compactly causal symmetric spaces
In this paper we prove the product formula for the c-function of non-compactly causal symmetric spaces.
math.RT↗