55 research outputs found
Holomorphic horospherical duality "sphere-cone"
We describe a construction of complex geometrical analysis which corresponds
to the classical theory of spherical harmonics.Comment: 9 page
Complex crowns of Riemannian symmetric spaces and non-compactly causal symmetric spaces
In this paper we define a distinguished boundary for the complex crowns
\Xi\subeq G_\C /K_\C of non-compact Riemannian symmetric spaces . The
basic result is that affine symmetric spaces of can appear as a component
of this boundary if and only if they are non-compactly causal symmetric spaces.Comment: 29 page
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.Comment: 9 page
Holomorphic horospherical transform on non-compactly causal spaces
We develop integral geometry for non-compactly causal symmetric spaces. We
define a complex horospherical transform and, for some cases, identify it with
a Cauchy type integral.Comment: Revised, final version; to appear in IMRN, 38
HOROSPHERICAL CAUCHY TRANSFORM ON QUADRICS
Abstract. We describe a construction of complex geometrical analysis which corresponds
to the classical theory of spherical harmonics
I believe that the connection of harmonic analysis and complex analysis has an
universal character and is not restricted by the case of complex homogeneous manifolds.
It looks as a surprise that such a connection exists and though it is quite
natural for finite dimensional representations and compact Lie groups [Gi00,Gi02].
In this note we describe the complex picture which corresponds to harmonic analysis
on the real sphere. The basic construction is a version of horospherical transform
which in this case is a holomorphic integral transform between holomorphic functions
on the complex sphere and the complex spherical cone. This situation looks
quite unusual from the point of view of complex analysis and I believe presents a
serious interest also in this setting. It can be considered as a version of the Penrose
transform), but in a purely holomorphic situation when there is neither cohomology
nor complex cycles
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