PRODUCTS OF SYMPLECTIC GROUPS ACTING ON ISOTROPIC SUBSPACES

Let A be a finite dimensional commutative semisimple algebra over a field k, and let (V, B) be a finitely generated symplective space over A. We examine the action of the symplectic group SPA (V, B) on the wt of B'-isotropic k-subspaces of V, where B' = phi . B is the k-symplectic form induced by a 'trace' map phi : A --> k. The case of A being a field was studied earlier and here we consider the case where A has several simple components. The orbits are completely classified when A = k x k and for maximal B'-isotropic subspaces when dim (k)A = 3; the number of orbits of maximal B'-isotropic subspaces is infinite if dim (k)A greater-than-or-equal-to 4 and k is infinite.