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Recall the following paracomplex analogues of hypercomplex and Hermitian hypercomplex structures on a finite-dimensional, real vector space $\Bbb V$, necessarily of dimension $4 m$ for some positive integer $m$:

A hyper-paracomplex structure on $\Bbb V$ is a triple $(I, J, K)$ of endomorphisms $\Bbb V \to \Bbb V$, where $I$ is a complex structure and $J, K$ are paracomplex structures satisfying the usual paraquaternion identity $I J = - J I = -K$.

A Hermitian hyper-paracomplex structure on $\Bbb V$ is a quadruple $(g; I, J, K)$ where $g$ is an inner product on $\Bbb V$ (necessarily of neutral signature $(2 m, 2 m)$, $(I, J, K)$ is a hyper-paracomplex structure on $\Bbb V$, $(g, I)$ is a Hermitian structure on $\Bbb V$, and $(g, J)$ and $(g, K)$ are para-Hermitian structures on $\Bbb V$. (This latter condition is just that $I, J, K$ are all $g$-skew.)

The subgroup $$G := \operatorname{Stab}_{GL(\Bbb V)}(g; I, J, K)$$ of $GL(\Bbb V)$ fixing $(g; I, J, K)$ is isomorphic to the real symplectic group $Sp(2 m, \Bbb R)$. (The complex analogue of this statement is that the stabilizer of a Hermitian hypercomplex structure, with inner product not necessarily definite, is the symplectic group $Sp(p, q)$.)

Now, the defining action of $GL(\Bbb V)$ maps rays (based at $0$) to rays, so it descends to an action on the space $\Bbb P_+(\Bbb V) \cong S^{4 m - 1}$ of rays, and we can restrict the action to $G \subset GL(\Bbb V)$.

What is the orbit structure of the action of $G$ on $S^{4 m - 1}$?

There are at least three orbits: Forgetting about the triple $(I, J, K)$ (and so remembering only the inner product $g$) defines an inclusion $$Sp(2 m, \Bbb R) \hookrightarrow SO(2 m, 2 m),$$ and the restriction to $SO(2 m, 2 m)$ of the $GL(\Bbb V)$-action on $S^{4 m - 1}$ has precisely three orbits, parameterized by the causality type of each ray (i.e., whether it is spacelike, lightlike, or timelike). The orbits of spacelike and timelike rays are both homeomorphic to the definite sphere $S^{2 m - 1, 2 m}$, and the (hypersurface) orbit of lightlike rays is homeomorphic to $S^{2 m - 1} \times S^{2 m - 1}$.

One can check that in the analogous Hermitian hypercomplex setting, where the inclusions are $Sp(p, q) \subset SO(4 p, 4 q) \subset GL(\Bbb V)$, that the $Sp(p, q)$-orbit decomposition is no finer than the $SO(4 p, 4 q)$-decomposition.

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  • $\begingroup$ NB the comparison with the hypercomplex setting in the last paragraph might be misleading. In the analogous Hermitian situation, where one has a $2\ell$-dimensional vector space $\Bbb W$ equipped with a Hermitian structure $(g, J)$ of signature $(2 p, 2 q)$, (if $g$ is indefinite, i.e., if $p q > 0$) the action of the stabilizer $\operatorname{Stab}_{GL(\Bbb W)}(g, J) \cong U(p, q)$ on $S^{2 \ell - 1}$ has exactly three orbits, again corresponding to the causality type of the rays... $\endgroup$ – Travis Willse Dec 5 '19 at 9:54
  • $\begingroup$ ...whereas if instead $(g, J)$ has a para-Hermitian structure, the action of the stabilizer $\operatorname{Stab}_{GL(\Bbb W)}(g, J) \cong GL(\ell, \Bbb R)$ on $S^{2 \ell - 1}$ has five orbits; the two "extra" orbits are the sets of rays in the $(\pm 1)$-eigenspaces of $J$. $\endgroup$ – Travis Willse Dec 5 '19 at 9:57

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