(1) Invariant theory:

Real semisimple symmetric spaces are often characterized by a pair of commuting involutions of a reductive group and a lot of their properties are studied in this setting. A problem about these groups with a pair of commuting involutions which had not been studied so far is a characterization of the orbits and invariants associated with such a group with a pair of commuting involutions. This problem is not only of importance for the representation theory of real semisimple symmetric spaces (see for example[1] or [9]) but also for a study of character sheaves (see for example [3] and [8]). Recently Gerry Schwarz (Brandeis University) and I have started a study of these orbits and their invariants which has already led to a number of results. Results include a characterization of the closed orbits, slice representations, polar decompositions and a variant of the "Chevalley restriction theorem'' (see [4]). Some of the results we have obtained were surprising since the finite Weyl group involved contains translations besides the usual reflections. This phenomena often leads to considerable complications in a description of the orbits. Another problem is that the closed orbits can contain non semisimple elements. Our results generalize similar results of Kostant and Rallis for the case of Riemannian symmetric spaces (see [7]) and results of Richardson [10] in the case of one involution. Although we have made considerable progress in understanding the case of the geometry and invariant theory of these double cosets in the algebraically closed case, there are many interesting questions left to study. An example of this is the question when the quotient is a polynomial ring. Currently we are also looking at extensions of our results for non algebraically closed fields, like the real numbers.
We have already shown that our results in the algebraically closed case also lead to a description of the two extreme real cases, namely the case that the group is compact and the case that one of the involutions is a Cartan involution. For all the other real cases we expect to get a mixture of these two cases. We intend to expand our results to p-adic groups as well, starting with the case that the two involutions are the same. In this case these double cosets are of importance for the representation theory of p-adic symmetric k-varieties.

(2) Normality and desingularization problems for symmetric varieties:

Many problems in the representation theory of a complex reductive group G involve questions about the geometry of the underlying Schubert varieties. These varieties are the closures of the orbits of a Borel subgroup on the generalized flag variety G/P where P is a parabolic subgroup of G (see for example[2]). Consequently these varieties have been studied extensively over the last few decades by many mathematicians, including Seshadri, Lakshmibai, Ramanathan, Littleman and many others. Another class of varieties, similar to these Schubert varieties, that is of importance in the representation theory of real reductive groups, are the closures of K-orbits on the generalized flag variety G/P. Here K is the connected component of the fixed point group G^s of an involutorial automorphism s of G. Some of the questions about the geometry of these closures of K-orbits on G/P which are of importance for representation theory, are:
1) Describe the singularities of closures of double cosets or, equivalently, of K-orbit closures in G/P. Are these closures normal and which of them have rational singularities?
2) For such an orbit closure X and a homogeneous line bundle L on G/P having non-zero global sections, describe the K-module H⁰(X,L) and the image of the restriction map res_X :H⁰(G/P,L) to H⁰(X,L). These questions are of importance for the study of Harish-Chandra modules. If the closure of a K-orbit is normal, then one can expect to deduce a number of results for the representations. In particular a study of sections of line bundles over closures of K-orbits will pick out the K-types in a finite dimensional representation of G. Not all closures of K-orbits on G/P are normal, but many of them are. Even when the orbit closures are not normal their singularities seem to be relatively nice. Some partial results for these problems were obtained in [5] and [6]. We hope to further develop this in the future.

References:

1. E. van den Ban and H. Schlichtkrull, The most continuous part of the Plancherel decomposition for a reductive symmetric space I, Ann. Math. 145 (1997), 267-364.
2. M. Demazure, Desingularisation des varietes de Schubert generalisees, Ann. Scient. Ec. Norm. Sup. 4-em serie t 7 (1974), 53-88.
3. I. Grojnowski, Character sheaves on symmetric spaces, Ph.D. thesis, Massachusetts Institute of Technology, June 1992.
4. A. G. Helminck and G. Schwarz, Orbits and invariants associated with commuting involutions. Duke Math. Journal. Vol. 106, No.2, 237-280.
5. A. G. Helminck and Michel Brion, On orbit closures of symmetric subgroups in flag varieties. Can. Journal Math., Vol 52 (2), (2000), pp. 265-292.
6. A. G. Helminck and Michel Brion, On orbit closures of symmetric subgroups in flag varieties II. In preparation.
7. B. Kostant and S. Rallis, Orbits and representations associated with symmetric spaces, Amer. J. Math. 93 (1971), 753-809.
8. G. Lusztig, Character sheaves, Adv. in Math. 56 (1985), 195-237.
9. T. Oshima and T. Matsuki, A description of discrete series for semisimple symmetric spaces, Adv. Stud. in Pure Math., vol. 4, Academic Press, Orlando, FL, 1984, pp.331-390.
10. R.W. Richardson, Orbits, invariants and representations associated to involutions of reductive groups, Invent. Math. 66 (1982), 287-312.