A decomposition of the natural embedding spaces for the symplectic dual polar spaces

Let e be the Grassmann-embedding of the symplectic dual polar space DW(2n-1,K) into PG(W), where W is a -dimensional vector space over K. For every point z of DW(2n-1,K) and every i∈N, Δi(z) denotes the set of points at distance i from z. We show that for every pair {x,y} of mutually opposite points of DW(2n-1,K),W can be written as a direct sum W0⊕W1⊕⋯⊕Wn such that the following four properties hold for every i∈{0,…,n}: (1) 〈e(Δi(x)∩Δn-i(y))〉=PG(Wi); (2) 〈e(⋃j⩽iΔj(x))〉=PG(W0⊕W1⊕⋯⊕Wi); (3) 〈e(⋃j⩽iΔj(y))〉=PG(Wn-i⊕Wn-i+1⊕⋯⊕Wn); (4) .