  Given a symmetric pair
$(G,K)=(\mathrm{GL}_{p+q}(\mathbb{C}),\mathrm{GL}_{p}(\mathbb{C})\times
\mathrm{GL}_{q}(\mathbb{C}))$ of type AIII, we consider the diagonal action of
$K$ on the double flag variety
$\mathfrak{X}=\mathrm{Grass}(\mathbb{C}^{p+q},r)\times K/B_K$ whose first
factor is a Grassmann variety for $G$ and whose second factor is a full flag
variety of $K$. There is a finite number of orbits for this action, and our
first result is a description of these orbits: parametrization, dimensions,
closure relations, and cover relations. Specifically, the orbits are
parametrized by certain pairs of partial permutations.
  Each orbit in $\mathfrak{X}$ gives rise to a conormal bundle. As in the
references [5] and [6], by using the moment map associated to the action, we
define a so-called symmetrized Steinberg map, respectively an exotic Steinberg
map, which assigns to each such conormal bundle (thus to each orbit) a
nilpotent orbit in the Lie algebra of $K$, respectively in the Cartan
complement of that Lie algebra. Our main result is an explicit description of
these Steinberg maps in terms of combinatorial algorithms on partial
permutations, extending the classical Robinson--Schensted procedure on
permutations. This is a thorough generalization of the results in [5], where we
supposed $p=q=r$ and considered orbits of special forms.
