Alexander Woo and Benjamin

Using recent results of the second author which explicitly identify the “(1, 2, 1, 2)avoiding” GL(p,C)×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-LusztigVogan polynomial Pτ,γ(q) in the case where γ corresponds to the trivial local system on a (1, 2, 1, 2)-avoiding orbit closure Q and τ corresponds to the trivial local system on any orbit Q′ contained in Q. This combines the aforementioned result of the second author, results of A. Knutson, the first author, and A. Yong, and a formula of Lascoux and Schützenberger which computes the ordinary (type A) Kazhdan-Lusztig polynomial Px,w(q) whenever w ∈ Sn is cograssmannian.