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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  generalized Steinberg map  and an  exotic moment map . Although the framework is general, in this paper we focus on the pair   $(G,K) = (\textrm{GL}_{2n}({\mathbb{C}}), \textrm{GL}_n({\mathbb{C}}) \times \textrm{GL}_n({\mathbb{C}}))$  . Then the generalized Steinberg map is a map from  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, that is, the set of nilpotent   $ K$  -orbits in the Cartan space   $(\textrm{Lie}(G)/\textrm{Lie}(K))^* $  . We explain the geometric background of the theory and combinatorial algorithms, which produce the above-mentioned maps.

A Generalization of Steinberg Theory and an Exotic Moment Map