Canonical representations on Lobachevsky spaces: An interaction with an overalgebra (Q2464062)

This is part of a series of papers where the author considers the following setting: \(G\) is a real semisimple Lie group, \(K\subset G\) is a maximal compact subgroup and \(G\) is realized as a subgroup of another \({\widetilde G} \supset G\) real semisimple Lie group, the ``overgroup''. Let \({\widetilde P}\) be a parabolic subgroup of \({\widetilde G}\) associated to the symmetric space \(G/K\)-- this means that \(G/K\) appears as an orbit of \(G\) in \({\widetilde G}/{\widetilde P}\). Consider a representation \({\widetilde R}_{\lambda}\) of \({\widetilde G}\) associated to a character of \({\widetilde P}\) on some space of functions on \({\widetilde G}/{\widetilde P}\) and restrict the representation to \(G\) and the functions to \(G/K\), to obtain a representation \(R_{\lambda}\). The goal is to compare \(R_{\lambda}\) with another representation \(T_{\sigma}\) constructed directly from \(G\). In the present paper, \(G = SO_0(n-1,1)\), \(K = SO(n)\) and \({\widetilde G} = SL(N, \mathbb R)\). The representation \({\widetilde R}_{\lambda}\) is realized in the Schwartz space of suitable functions on \(\mathbb R^{n-1}\) and \(R_{\lambda}\) is a representation of \(G\) on a space of functions on the closed ball \(\overline B = B \cap S\), where \(B\) is the open ball and \(S\) the unit sphere. In turn the representation \(T_{\sigma}\) is on the space of functions on \(S\). The author exhibits explicitly intertwining operators between \(R_{\lambda}\) and \(T_{\sigma}\), called Poisson and Fourier transforms. The main results of the paper is the description of the behaviour of these Poisson and Fourier transforms with respect to the action of the Lie algebra of \({\widetilde G}\).