Representations in conjunction with symmetric spaces G_k /H_k occur in many places. First, there is harmonic analysis on G_k /H_k , whose main aim is to decompose into irreducible components the regular representation of G_k on the Hilbert space L²(G_k /H_k) of square integrable functions on G_k /H_k. This is also called the Plancherel decomposition. In the case of real symmetric spaces this has been a major area of research in the last few decades and was only recently completed by Delorme. Before that, many specific cases and subclasses were solved by various people, but a crucial and central role was played here by Harish Chandra.

In the real case the first general subclass for which the Plancherel formula was determined was that of Riemannian symmetric spaces. In view of this fact we looked at the p-adic analogue, namely the case where H is anisotropic over k. Using the specific geometry of this situation, we proved that one has a multiplicity free decomposition in this case. Encouraged by this result we showed additional algebraic properties of the space G_k /H_k, necessary for the determination of the structure of the Hecke algebra of spherical functions and its characters. We hope our continuing work will finish the Plancherel formula for this subclass.

In the non compact situation, concrete examples show that in general, we no longer have a multiplicity free decomposition. We showed how the estimate of the multiplicity can be reduced to the supercuspidal case. Presently we investigate how estimates for this case can be further improved.

In the real case, the H_k-fixed distribution vectors played a central role in the determination of the Plancherel formula. We introduced the notion of H_k-fixed distribution vectors in the p-adic case and obtained a number of results. First, we found several equivalent descriptions of them and gave different procedures to construct them. We also introduced the class of s-split parabolic k-subgroups, where s is the involution related to the symmetric space, and showed that representations induced from parabolic k-subgroups not in this class have generically no H_k-fixed distribution vectors. Therefore, we concentrated our attention on representations induced from this class of parabolic k-subgroups. For the minimal s-split parabolic k-subgroups, we constructed the H_k-fixed distribution vectors on the open orbits by meromorphic continuation. This analysis we intend to extend to the other relevant parabolic k-subgroups.

Another area of representation theory where symmetric varieties play a role is that of the so-called canonical representations. They were originally introduced by Vershik, Gelfand and Graev. Recently Hille gave an extension including all previous ones. A similar notion can be introduced in the p-adic case. The first step is to show that there exists a non-degenerate Berezin form on the representation space. The next challenge is to see which unitary representations can be produced with this procedure. This is under current investigation.