arXiv: Representation Theory | 2019
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. \nWe 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}. \nAlthough 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. \nThe 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))^* $. \nWe explain the geometric background of the theory and combinatorial algorithms which produce the above mentioned maps.