Combinatorial results on (1,2,1,2)-avoiding $GL(p,\mathbb{C}) \times GL(q,\mathbb{C})$-orbit closures on $GL(p+q, \mathbb{C})/B$

Using recent results of the second author which explicitly identify the "$(1,2,1,2)$-avoiding" $GL(p,\mathbb{C}) \times GL(q,\mathbb{C})$-orbit closures on the flag manifold $GL(p+q,\mathbb{C})/B$ 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 $P_{\tau,\gamma}(q)$ 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 $\overline{Q}$. 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 Sch\"{u}tzenberger which computes the ordinary (type $A$) Kazhdan-Lusztig polynomial $P_{x,w}(q)$ whenever $w \in S_n$ is cograssmannian.