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

In this paper we present an algorithm to compute the orbits of a minimal parabolic k -subgroup acting on a symmetric k -variety and most of the combinatorial structure of the orbit decomposition. This algorithm can be implemented in LiE, GAP4, Magma, Maple or in a separate program. These orbits are essential in the study of symmetric k -varieties and their representations. In a similar way to the special case of a Borel subgroup acting on the symmetric variety, (see A. G. Helminck. Computing B -orbits on G / H. J. Symb. Comput.,21 , 169–209, 1996.) one can use the associated twisted involutions in the restricted Weyl group to describe these orbits (see A. G. Helminck and S. P. Wang. On rationality properties of involutions of reductive groups. Adv. Math., 99, 26–96, 1993). However, the orbit structure in this case is much more complicated than the special case of orbits of a Borel subgroup. We will first modify the characterization of the orbits of minimal parabolic k -subgroups acting on the symmetric k -varieties given in Helminck and Wang (1993), to illuminate the similarity to the one for orbits of a Borel subgroup acting on a symmetric variety in Helminck (1996). Using this characterization we show how the algorithm in Helminck (1996) can be adjusted and extended to compute these twisted involutions as well.