Let \(G/P\) be the Grassmannians of \(k\)-planes in \(\mathbb{C}^ n\). Let \(\theta\) be an involution of \(G\) \((=\text{GL}(n))\), and \(K\) the connected component of the fixed point set of \(\theta\). In this paper, the authors prove that the closures of \(K\)-orbits in \(G/P\) are normal, have rational singularities (and hence are Cohen-Macaulay). For other generalized flag varieties, these orbit closures are not always normal. By studying those orbit closures which are normal, the authors prove a PRV-conjecture type results for real groups. This paper makes an important contribution to representation theory.