Let $\mathbb{H}_g$ denote Siegel space, and $M$ denote an order 4 element of the unitary subgroup $U(n)(\mathbb{R})$with $p$ eigenvalues equal to $i$, and $q$ eigenvalues equal to $-i$, $p+q=g$. Consider $M$ as an element of the symplectic group $SP_{2g}(\mathbb{R})$ by identifying $U(n)(\mathbb{R})$ with a stabilizer of a point.If we consider the fixed point $S$ set of $M$ acting on $\mathbb{H}_g$, we get a complex, totally geodesic subvariety, which must therefore be a hermitian symmetric subspace. By computing at the tangent space, we can check that $S$ has complex dimension $pq$. Going through the classification table, it is natural enough to guess that $S$ is in the AIII family, isomorphic to the non-compact dual of the Grassmanian $G(p+q,q)$.

**Question:** Is $S$ indeed this hermitian symmetric space?

I would be interested in how one goes about determining what $S$ is through group theoretic methods.

Thanks!