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

Using recent results of the second author which explicitly identify the “$(1,2,1,2)$-avoiding” ${\rm GL}(p,{{ \BBC}}) \times {\rm GL}(q,{{ \BBC}})$-orbit closures on the flag manifold ${\rm GL}(p+q,{{ \BBC}})/B$ 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 $P_{\tau ,\gamma }(q)$ in the case where $\gamma$ 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 $\bar {Q}$. This combines the aforementioned result of the second author, results of Knutson et al., and a formula of Lascoux and Schützenberger which computes the ordinary (type $A$) Kazhdan–Lusztig polynomial $P_{x,w}(q)$ whenever $w \in S_n$ is cograssmannian.