  For a reductive group $G$, Steinberg established a map from the Weyl group to
the set of nilpotent $G$-orbits by using moment maps on double flag varieties.
In particular, in the case of the general linear group, it provides a geometric
interpretation of the Robinson-Schensted correspondence between permutations
and pairs of standard tableaux of the same shape.
  We extend Steinberg's approach to the case of a symmetric pair $(G,K)$ to
obtain two different maps, namely a \emph{generalized Steinberg map} and an
\emph{exotic moment map}.
  Although the framework is general, in this paper we focus on the pair $(G,K)
= (\mathrm{GL}_{2n}(\mathbb{C}), \mathrm{GL}_n(\mathbb{C}) \times
\mathrm{GL}_n(\mathbb{C}))$. Then the generalized Steinberg map is a map from
\emph{partial} permutations to the pairs of nilpotent orbits in $
\mathfrak{gl}_n(\mathbb{C}) $. It involves a generalization of the classical
Robinson--Schensted correspondence to the case of partial permutations.
  The other map, the exotic moment map, establishes a combinatorial map from
the set of partial permutations to that of signed Young diagrams, i.e., the set
of nilpotent $K$-orbits in the Cartan space
$(\mathrm{Lie}(G)/\mathrm{Lie}(K))^* $.
  We explain the geometric background of the theory and combinatorial
algorithms which produce the above mentioned maps.
