Combinatorial Results on (1,2,1,2)-Avoiding GL(p, C) x GL(q, C)-Orbit Closures on GL(p plus q, C) / B

Using recent results of the second author which explicitly identify the "(1, 2, 1, 2)-avoiding" GL(p,C) x GL(q,C)-orbit closures on the flag manifold GL(p+ q, C)/B as certain Richardson varieties, we give combinatorial criteria for determining smoothness, lci-ness, and Gorensteinness of such orbit closures. (In the case of smoothness, this gives a new proof of a theorem of W.M. McGovern.) Going a step further, we also describe a straightforward way to compute the singular locus, the non-lci locus, and the non-Gorenstein locus of any such orbit closure. We then describe a manifestly positive combinatorial formula for the Kazhdan-Lusztig-Vogan polynomial Pt-tau,Pt-gamma(q) in the case where. corresponds to the trivial local system on a (1, 2, 1, 2)-avoiding orbit closure Q and tau corresponds to the trivial local system on any orbit Q' contained in (Q) over bar. This combines the aforementioned result of the second author, results of Knutson et al., and a formula of Lascoux and Schutzenberger which computes the ordinary (type A) Kazhdan-Lusztig polynomial P-x,P-w(q) whenever w is an element of S-n is cograssmannian.