| Overview | Symmetric Spaces | Invariant Theory | Representation Theory | Symbolic Computation | Flag Varieties | Graph Theory |
Structure of p-adic symmetric spaces
(1) Classification of p-adic symmetric spaces:
In order to study p-adic symmetric spaces and their representation theory it is important to have a classification of the k-involutions together with their fine structure of restricted root systems with multiplicities and Weyl groups. A characterization of the isomorphy classes of k-involutions was given in [4]. This characterization of the isomorphy classes of k-involutions required 3 invariants. One was a Satake type diagram which determines the fine structure. For k the real numbers, p-adic numbers, number fields and finite fields these diagrams were classified in [4] as well. The other two invariants were isomorphy classes of involutions of anisotropic groups and isomorphy classes of quadratic elements. For k the p-adic numbers the latter two still need to be classified and this is one of the projects I am currently working on. These results will be essential for a study of the representations associated with these p-adic symmetric spaces.
(2) Classification of Cartan subspaces:
For real symmetric spaces the natural representation of Gk on the Hilbert space L2(Gk /Hk) of square integrable functions on Gk/Hk decomposes into several series, one for each Hk-conjugacy class of Cartan subspaces of Lie(Gk/Hk). The most extreme of these are, respectively, the most discrete series (in the group case called fundamental series) corresponding to the conjugacy class of Cartan subspaces with maximal compact part and the most continuous series, corresponding to the conjugacy class with maximal non-compact part. For the p-adic symmetric spaces one gets a similar decomposition of L2(Gk /Hk). The equivalent of the "Cartan subspaces of Lie(Gk/Hk)'' for general symmetric spaces are the maximal s-split k-tori, where s is the involution related to the symmetric space.
(3) Buildings for p-adic symmetric spaces:
The work of Bruhat and Tits [1] has been of fundamental importance in the study of p-adic groups and their representations. Most of the advances in this area would not have been accomplished without this monumental work. Recently I have started with studying a generalization of this work of Bruhat and Tits to the setting of p-adic symmetric spaces. Some results for the p-adic were already obtained in [3] and recently I have obtained a few additional results. This includes the construction of a building type structure related to these p-adic symmetric spaces and a number of results for the corresponding BN-pair. Most of these generalizations of Bruhat and Tits [1] to this setting of p-adic symmetric spaces have turned out to be far from straightforward due to an additional twisting by the involution of the symmetric space. A lot more work remains to be done. Given the size of this work of Bruhat and Tits I expect this project to take a while.
Motivation: That a study of p-adic symmetric spaces is a natural extension of the study of real semisimple symmetric spaces and also of p-adic groups is not easily seen. In the remainder of this subsection I will try to explain this a little further. The basic philosophy behind the work of Bruhat and Tits can be described by making an analogy with the real case. In the study of real reductive Lie groups and their representations the Riemannian symmetric spaces play an essential role. One of the main reasons behind this is the fact that in a complete simply connected Riemannian space of negative curvature (i.e. non-compact) every compact group of isometries has a fixed point. This result (called the fixed point theorem) was originally due to Cartan. He used this to show that all maximal compact subgroups of a real simple Lie group are conjugate. On the other hand a maximal compact subgroup determines a Cartan involution of the real Lie group and this involution determines the real Lie group as a real form of its complexification. Most of the structure of this real Lie group and the associated Riemannian space can be derived from the structure of the complex group with the involution (for more details see [2]).
The analog of the Riemannian symmetric spaces for p-adic groups are the Euclidean buildings. Bruhat and Tits showed in [1] that most properties of the Riemannian symmetric spaces carry over to these Euclidean buildings. For example they showed that a compact group of isometries of a Euclidean building has a fixed point. They used this to show that in a simply connected simple p-adic group the maximal compact subgroups are parahoric subgroups. There are many other similarities between the Riemannian symmetric spaces and the Euclidean buildings. For example for both a geodesic joining two points is unique. There is also a p-adic curvature on the building. The Euclidean building also replaces the role of the Riemannian symmetric spaces in the cohomology of discrete subgroups. Let me finally remark that in a similar manner there is a relation between spherical buildings and Riemannian symmetric spaces. This has to do with compactifications.
Semisimple symmetric spaces are a natural generalization of the Riemannian symmetric spaces. Recall that the Riemannian symmetric spaces can be defined as those Riemannian manifolds M such that any point x in M has a normal neighborhood on which the geodesic symmetry with respect to x is an isometry. By weakening the last condition from ``isometry'' to ``affine transformation'' one gets a larger class of varieties called affine symmetric spaces or also semisimple symmetric spaces. Another way to define these symmetric spaces is as follows. Let R denote the field of the real numbers, let G be a real reductive connected R-group and s an R-involution of G. The reductive symmetric spaces are precisely the homogeneous spaces GR/HR, where H is a R-open subgroup of the fixed point group of s. This is exactly the definition of a symmetric R-variety.
Underlying a reductive symmetric space there is also a Riemannian symmetric space which basically describes the structure of the real group GR. This Riemannian symmetric space comes from a Cartan involution t of GR which commutes with s. All the structure of the semisimple symmetric space follows now by combining the structure of this Riemannian symmetric space with the structure of the involutorial automorphism s of G. For more details see [2].
As in the real case, the p-adic symmetric spaces are a natural extension of the p-adic groups. To get a detailed description of their structure one needs to combine the structure of the Euclidean building with the structure of the involutorial automorphism s. Note that the resulting structure is in general no longer an Euclidean building, but it is very similar. There is an extra twisting coming from the action of the involution. This building type structure and the corresponding structure of the p-adic symmetric spaces is what I am studying at this moment.
(4) Other problems:
There is, of course, a vast assortment of open problems about the structure of these p-adic symmetric spaces. Among those which I hope to pursue are the following: the study of various orbits on p-adic symmetric spaces, for example Hk-orbits on Gk/Hk and orbits of maximal compact subgroups on Gk/Hk; a classification of s-stable maximal k-split tori; and closures of orbits of (minimal) parabolic k-subgroups acting on the p-adic symmetric spaces. Each of these problems is of importance for a study of the representations associated with these p-adic symmetric spaces.
References: