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A generalization of Steinberg theory and an exotic moment map

Lucas Fresse, Kyo Nishiyama

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.

A generalization of Steinberg theory and an exotic moment map

Abstract

For a reductive group , Steinberg established a map from the Weyl group to the set of nilpotent -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 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 . Then the generalized Steinberg map is a map from \emph{partial} permutations to the pairs of nilpotent orbits in . 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 -orbits in the Cartan space . We explain the geometric background of the theory and combinatorial algorithms which produce the above mentioned maps.

Paper Structure

This paper contains 35 sections, 31 theorems, 207 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Image of the conormal variety
  3. Organization of the paper
  4. Acknowledgements
  5. Preliminaries on miscellanea and a review of Steinberg theory
  6. Robinson-Schensted correspondence
  7. Partitions, Young diagrams, tableaux
  8. Row-insertion and column-insertion algorithms
  9. The Robinson-Schensted correspondence
  10. Jeu de taquin
  11. Moment maps and conormal varieties
  12. Conormal variety
  13. Moment map for double flag variety
  14. Moment maps for rational representations
  15. The Steinberg map
  16. ...and 20 more sections

Key Result

Proposition 1.1

Assume that Then, the conormal variety $\mathcal{Y}$ is equidimensional and its irreducible components are in one-to-one correspondence with the $K$-orbits of $\mathfrak{X}$. More precisely, whenever $\mathbb{O}\subset\mathfrak{X}$ is a $K$-orbit, the closure of the conormal bundle $T^*_\mathbb{O}\mathfrak{X}$

Figures (3)

  • Figure 1: The correspondence $\tau\mapsto(T_1,T_2,\nu)$ for $\mathfrak{T}_3$ ($\mathrm{rank}\,\tau=3$); the maps $\Phi$, $\Xi_{\mathfrak{k}}$, and $\Xi_{\mathfrak{s}}$
  • Figure 2: The correspondence $\tau\mapsto(T_1,T_2,\nu)$ for $\mathfrak{T}_3$ ($\mathrm{rank}\,\tau=2$); the maps $\Phi$, $\Xi_{\mathfrak{k}}$, and $\Xi_{\mathfrak{s}}$
  • Figure 3: The correspondence $\tau\mapsto(T_1,T_2,\nu)$ for $\mathfrak{T}_3$ ($\mathrm{rank}\,\tau\leq 1$); the maps $\Phi$, $\Xi_{\mathfrak{k}}$, and $\Xi_{\mathfrak{s}}$

Theorems & Definitions (67)

  • Proposition 1.1
  • Remark 1.3
  • Theorem 1.4: see Theorem \ref{['T3']}
  • Theorem 1.5: Theorems \ref{['T4']} and \ref{['T5']}
  • Theorem 1.6
  • Theorem 1.7: Theorems \ref{['T2']} and \ref{['thm:3.9']}
  • Proposition 3.1
  • proof
  • Theorem 4.1: Steinberg2
  • Definition 5.1
  • ...and 57 more