A generalization of Steinberg theory and an exotic moment map

Abstract

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.