A Generalization of Steinberg Theory and an Exotic Moment Map

Abstract

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 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.