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

Abstract

In the framework of the study of the $Sp(n)$-orbits in the real Grassmannian $G^{\mathbb{R}}(k,4n)$ 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 $\mathcal{I}\mathcal{C}$. 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 $k\ge 0$ the last set reduce to the pair $(\xi =\pm 1,\chi =\pm 1)$.