$Sp(n)$-orbits in the Grassmannians of complex and $Σ$-complex subspaces of an Hermitian quaternionic vector space

Abstract: We determine the invariants characterizing the $Sp(n)$-orbits in the real Grassmannian $Gr^\R(2k,4n)$ of the $2k$-dimensional complex and $\Sigma$-complex subspaces of a $4n$-dimensional Hermitian quaternionic vector space. A $\Sigma$-complex subspace is the orthogonal sum of complex subspaces by different, up to sign, compatible complex structure. The result is obtained by considering two main features of such subspaces. The first is that any such subspace admits a decomposition into an Hermitian orthogonal sum of 4-dimensional complex addends plus a 2-dimensional totally complex subspace if $k$ is odd, meaning that the quaternionification of the addends are orthogonal in pairs. The second is that any 4-dimensional complex addend $U$ is an isoclinic subspace i.e. the principal angles of the pair $(U,AU)$ are all the same for any compatible complex structure $A$. Using these properties we determine the full set of the invariants characterizing the $Sp(n)$-orbit of such subspaces in $Gr^\R(2k,4n)$.