Computing orbits of minimal parabolic k-subgroups acting on symmetric k-varieties

In this paper we present an algorithm to compute the orbits uf a minimal pa rabolic k-subgroup acting on a symmetric k-variety and most of the combinat orial structure of the orbit decomposition. This algorithm can Le implement ed in LiE, GAP4, Magma, Maple or in a separate program. These orbits are es sential 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 symmet ric variety, (see A. G. Helminck. Computing B-orbits on G/H. J. Symb. Compu t., 21, 169-209, 1996.) one can use the associated twisted involutions in t he restricted Weyl group to describe these orbits (sce A. G. Helminck and S . P. Wang, On rationality properties of involutions of reductive groups. Ad v. Math., 99, 26-96, 1993). However, the orbit structure in this case is mu ch 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 subg roup acting on a symmetric variety in Helminck (1996). Using this character ization we show how the algorithm in Helminck (1996) can be adjusted and ex tended to compute these twisted involutions as well. (C) 2000 Academic Pres s.