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

TL;DR

This paper characterizes the invariants that classify $Sp(n)$-orbits in the Grassmannian of certain complex and $ extSigma$-complex subspaces within a Hermitian quaternionic vector space, revealing their structural decomposition and geometric properties.

Contribution

It provides a complete set of invariants for $Sp(n)$-orbits in the Grassmannian of complex and $ extSigma$-complex subspaces, based on their decomposition and isoclinic properties.

Findings

Subspaces decompose into orthogonal sums of 4D complex addends and 2D totally complex parts.

4D complex addends are isoclinic subspaces with equal principal angles.

The invariants fully characterize the $Sp(n)$-orbit of these subspaces.

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 $Gr^\R(2k,4n)$.