$Sp(n)$-orbits of isoclinic subspaces in the real Grassmannians

In the framework of the study of the $Sp(n)$-orbits in the real Grassmannian $G^\R(k,4n)$ of $k$-dimensional non oriented subspaces of a real $4n$-dimensional vector space $V$, here we consider the case of the isoclinic subspaces whose set we indicate with $\mathcal{IC}$. Endowed $V$ with an Hermitian quaternionic structure $(\mathcal{Q},<,>)$, a subspace $U$ is isoclinic if for any compatible complex structure $A \in \mathcal{Q}$ the principal angles of the pair $(U,AU)$ are all the same, say $\theta^A$. We will show that, fixed an admissible hypercomplex basis $(I,J,K)$, to any such subspace $U$ we can associate two set of invariants, namely a triple $(\xi,\chi,\eta)$ and a pair $(\Gamma, \Delta)$ where $\Gamma$ itself is a function of $(\xi,\chi,\eta)$. We prove that the angles of isoclinicity $(\theta^I,\theta^J,\theta^K)$ together with $(\xi,\chi,\eta, \Delta)$ determine its $Sp(n)$-orbit. In particular if $\dim U= 8k+2$ or $\dim U= 8k+6$ with $k \geq 0$ the last set reduce to the pair $(\xi= \pm 1,\chi= \pm 1)$.