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

Abstract

In the framework of the study of the $Sp(n)$-orbits in the real Grassmannian $G^\R(k,4n)$ 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 $θ^A$. 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 $(ξ,χ,η)$ and a pair $(Γ, Δ)$ where $Γ$ itself is a function of $(ξ,χ,η)$. We prove that the angles of isoclinicity $(θ^I,θ^J,θ^K)$ together with $(ξ,χ,η, Δ)$ 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 $(ξ= \pm 1,χ= \pm 1)$.