25 may 2016 -- 15:45
Aula di Consiglio, Dip.Matematica, Università "La Sapienza", Roma
Abstract.
Gel'fand and Graev found that the horospherical transform is a perfect way to construct harmonic analysis on complex semisimple Lie groups or Riemannian symmetric spaces. However, for real semisimple Lie groups (starting from SL(2;R)) or, more generally, for pseudo-Riemannian symmetric spaces the horospherical transform has a kernel corresponding to the discrete series. Gelfand's problem is the following: is it possible to find a version of the horospherical transform that works for discrete series? We will consider the idea to extend the set of real horospheres by complex horospheres without real points. We define a horospherical Cauchy transform on a real symmetric space with singularities on complex horospheres and it completely reproduces the harmonic analysis on pseudo-Riemannian symmetric spaces.