Harish-Chandra modules and symmetric spaces

The geometric theory of symmetric spaces has a close relation with the Harish-Chandra modules, that is the representations of the pair consisting of a real semisimple group G of classical type and its maximal compact subgroup. Such representations are realized in the Frechet space of the sections of certain line bundles over the quotient of the complexification of G modulo a borel subgroup. We are going to examine the general theory and to discuss some examples. At the end we mention some work in progress on the extension of all of these results to supergeometry (joint collaboration with C. Carmeli, University of Genova, and V. S. Varadarajan, UCLA).