ComputingB-orbits onG/H

The orbits of a Borel subgroup acting on a symmetric varietyG/Hoccur in several areas of mathematics. For example, these orbits and their closures are essential in the study of HarishChandra modules (see Vogan, 1983). There are several descriptions of these orbits, but in practice it is actually very difficult and cumbersome to compute the orbits and their closures. Since the characterizations of theseorbits are very combinatorial in nature, this work could conceivably be done by a computer. In this paper we prove a number of additional properties of these orbits and combine these with properties of the various descriptions of these orbits to obtain an efficient algorithm. This algorithm can be implemented on a computer by using existing symbolic manipulation programs or by writing an independent program.