Some subspaces of the kth exterior power of a symplectic vector space

Let K be an arbitrary field, let n,k,l be nonnegative integers satisfying n⩾1, 1⩽k⩽2n, 0⩽l⩽min(n,k), and let V be a 2n-dimensional vector space over K equipped with a nondegenerate alternating bilinear form f. Let Wk,l denote the subspace of ⋀kV generated by all vectors , where are k linearly independent vectors of V such that is totally isotropic with respect to f. We prove that . We give a recursive method for constructing a basis of Wk,l and give a decomposition of Wk,l relative to a given hyperbolic basis of V. We also study two linear mappings, one between the spaces Wk,l and Wk-2,l-1 and another one between Wk,l and W2n-k,n+l-k.