Realizability of Cohomology Classes in Orthogonal Grassmannians

Building on the work of Coskun and Ross, we investigate which cohomology classes of the orthogonal Grassmannian $OG(k,n)$ can be realized as irreducible subvarieties of $OG(k,n)$. We give a criterion for when the realizable classes in the orthogonal Grassmannian agree with those in the Grassmannian. We give a characterization for realizability in low dimension and codimension and give necessary conditions for realizability in certain orthogonal Grassmannians.