  We show that closed, immersed, minimal hypersurfaces in a compact symmetric
space satisfy a lower bound on the index plus nullity, which depends linearly
on their first Betti number. Moreover, if either the minimal hypersurface
satisfies a certain genericity condition, or if the ambient space is a product
of two CROSSes, we improve this to a lower bound on the index alone, which is
affine in the first Betti number. To prove these, we introduce a generalization
of isometric immersions in Euclidean space. Compact symmetric spaces admit (and
in fact are characterized by) such a structure with skew-symmetric second
fundamental form.
