Double Covers of Symplectic Dual Polar Graphs

Let $\Gamma=\Gamma(2n,q)$ be the dual polar graph of type $Sp(2n,q)$. Underlying this graph is a $2n$-dimensional vector space $V$ over a field ${\mathbb F}_q$ of odd order $q$, together with a symplectic (i.e. nondegenerate alternating bilinear) form $B:V\times V\to{\mathbb F}_q$. The vertex set of $\Gamma$ is the set ${\mathcal V}$ of all $n$-dimensional totally isotropic subspaces of $V$. If $q\equiv1$ mod 4, we obtain from $\Gamma$ a nontrivial two-graph $\Delta=\Delta(2n,q)$ on ${\mathcal V}$ invariant under $PSp(2n,q)$. This two-graph corresponds to a double cover $\widehat{\Gamma}\to\Gamma$ on which is naturally defined a $Q$-polynomial $(2n+1)$-class association scheme on $2|{\mathcal V}|$ vertices.