Orbits of a Fixed-point Subgroup of the Symplectic Group on Partial Flag Varieties of Type A

In this paper we compute the orbits of the symplectic group Sp2n on partial flag varieties GL2n/P and on partial flag varieties enhanced by a vector space, C ×GL2n/P . This extends analogous results proved by Matsuki on full flags. The general technique used in this paper is to take the orbits in the full flag case and determine which orbits remain distinct when the full flag variety GL2n/B is projected down to the partial flag variety GL2n/P . The recent discovery of a connection between abstract algebra and the classical combinatorial Robinson-Schensted (RS) correspondence has sparked research on related algebraic structures and relationships to new combinatorial bijections, such as the RobinsonSchensted-Knuth (RSK) correspondence, the “mirabolic” RSK correspondence, and the “exotic” RS correspondence. We conjecture an exotic RSK correspondence between the orbits described in this paper and semistandard bi-tableaux, which would yield an extension to the exotic RS correspondence found in a paper of Henderson and Trapa.