Computing Orbits of Minimal Parabolic k-subgroups Acting on Symmetric k-varieties

The orbits of a minimal parabolic k-subgroup acting on a symmetric k-variety are essential to the study of symmetric k-varieties and their representations. This paper gives an algorithm to compute these orbits and most of the combinatorial structure of the orbit decomposition. There are several ways to describe these orbits, see for example [22, 28, 35]. Fundamental in all these descriptions are the associated twisted involutions in the restricted Weyl group. These describe the combinatorial structure of the orbit decomposition in a similar manner to the special case of orbits of a Borel subgroup acting on a symmetric variety, see [19]. However, the orbit structure in the general case is much more complicated than the special case of orbits of a Borel subgroup. In this paper we first modify the characterization of the orbits of minimal parabolic k-subgroups acting on a symmetric k-varieties given in [22], to illuminate the similarity to the one for orbits of a Borel subgroup acting on a symmetric variety in [19]. Using this characterization we show how the algorithm in [19] can be adjusted and extended to compute these twisted involutions as well.