  We study the nullcone in the multi-vector representation of the symplectic
group with respect to a joint action of the general linear group and the
symplectic group. By extracting an algebra over a distributive lattice
structure from the coordinate ring of the nullcone, we describe a toric
degeneration and standard monomial theory of the nullcone in terms of double
tableaux and integral points in a convex polyhedral cone.
