HOROSPHERICAL CAUCHY TRANSFORM ON QUADRICS
Simon, Gindikin HOROSPHERICAL CAUCHY TRANSFORM ON QUADRICS. In: 表現論シンポジウム, 2005/11/15-18. (In Press)
November 2005: 表現論シンポジウム
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.