  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.
