| Symmetric varieties occur in many areas of mathematics. They are defined as the homogeneous spaces G/H with G a reductive algebraic group and H the fixed point group of an involution sigma. Similarly, symmetric k-varieties are the homogeneous spaces Gk/Hk where Gk and Hk are the k-points of G and H and k is not necessarily algebraically closed. They occur in many problems in representation theory, geometry, singularity theory, and number theory. Perhaps the best known application is in the representation theory of Lie groups.;To study the representation theory of the symmetric k-varieties over real and local fields much structure of these symmetric k-varieties is needed. For example the orbits of parabolic k-subgroups acting on a symmetric k-variety are of fundamental importance in the study of induced representations. The characterization of these orbits involves conjugacy classes of sigma-stable maximal k-split tori and for each of these sigma-stable maximal k-split tori a quotient of Weyl groups. This thesis focuses on refining the characterization found in Helminck and Wang [18] and use this to classify the conjugacy classes of sigma-stable maximal k-split tori over the real numbers, building on partial results obtained in [9] and [11]. This problem is not only of importance for the representation theory of symmetric spaces but also for several other of fields mathematics and physics. |
