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On generalized Steinberg theory for type AIII

Lucas Fresse, Kyo Nishiyama

Abstract

Given a symmetric pair $(G,K)=(\mathrm{GL}_{p+q}(\mathbb{C}),\mathrm{GL}_{p}(\mathbb{C})\times \mathrm{GL}_{q}(\mathbb{C}))$ of type AIII, we consider the diagonal action of $K$ on the double flag variety $\mathfrak{X}=\mathrm{Grass}(\mathbb{C}^{p+q},r)\times K/B_K$ whose first factor is a Grassmann variety for $G$ and whose second factor is a full flag variety of $K$. There is a finite number of orbits for this action, and our first result is a description of these orbits: parametrization, dimensions, closure relations, and cover relations. Specifically, the orbits are parametrized by certain pairs of partial permutations. Each orbit in $\mathfrak{X}$ gives rise to a conormal bundle. As in the references [5] and [6], by using the moment map associated to the action, we define a so-called symmetrized Steinberg map, respectively an exotic Steinberg map, which assigns to each such conormal bundle (thus to each orbit) a nilpotent orbit in the Lie algebra of $K$, respectively in the Cartan complement of that Lie algebra. Our main result is an explicit description of these Steinberg maps in terms of combinatorial algorithms on partial permutations, extending the classical Robinson--Schensted procedure on permutations. This is a thorough generalization of the results in [5], where we supposed $p=q=r$ and considered orbits of special forms.

On generalized Steinberg theory for type AIII

Abstract

Given a symmetric pair of type AIII, we consider the diagonal action of on the double flag variety whose first factor is a Grassmann variety for and whose second factor is a full flag variety of . There is a finite number of orbits for this action, and our first result is a description of these orbits: parametrization, dimensions, closure relations, and cover relations. Specifically, the orbits are parametrized by certain pairs of partial permutations. Each orbit in gives rise to a conormal bundle. As in the references [5] and [6], by using the moment map associated to the action, we define a so-called symmetrized Steinberg map, respectively an exotic Steinberg map, which assigns to each such conormal bundle (thus to each orbit) a nilpotent orbit in the Lie algebra of , respectively in the Cartan complement of that Lie algebra. Our main result is an explicit description of these Steinberg maps in terms of combinatorial algorithms on partial permutations, extending the classical Robinson--Schensted procedure on permutations. This is a thorough generalization of the results in [5], where we supposed and considered orbits of special forms.

Paper Structure

This paper contains 20 sections, 22 theorems, 128 equations, 6 figures.

Table of Contents

  1. Introduction
  2. A multiple flag variety and its orbital decomposition
  3. Conormal variety and Steinberg maps
  4. Main results
  5. Combinatorial notation on pairs of partial permutations
  6. Graphic representation of a pair of partial permutations
  7. Numerical invariants for the graph
  8. Orbit decomposition of the multiple flag variety $\mathfrak{X}$
  9. Description of symmetrized and exotic Steinberg maps
  10. An extension of the Robinson--Schensted correspondence
  11. On the decomposition of $\mathfrak{X}$ into $K$-orbits
  12. A preliminary lemma
  13. Parametrization and dimension formula -- proof of Theorem \ref{['T1']} (1)--(3)
  14. Closure relations -- proof of Theorems \ref{['T1']} (4) and \ref{['C1']}
  15. Further remarks
  16. ...and 5 more sections

Key Result

Theorem 2.2

Figures (6)

  • Figure 1: Elementary moves yielding cover relations in the poset $(\{\overline{\mathbb{O}_\omega}\},\subset)$.
  • Figure 2: The parameters of the $K$-orbits of $\mathfrak{X}$ and the cover relations for $p=q=r=2$.
  • Figure 3: Calculation of $\Phi_\mathfrak{k}(\mathbb{O}_\omega)=\mathfrak{O}_{\lambda,\mu}$ and $\Phi_\mathfrak{s}(\mathbb{O}_\omega)=\mathfrak{O}_\Lambda$ for $p=q=r=2$.
  • Figure 4: The correspondence $\omega\mapsto(T_1,T_2;\lambda',\mu';\nu)$ in the case $(p,q,r)=(3,2,2)$.
  • Figure 5: The parameters of the $K$-orbits of $\mathrm{Gr}(V,r)$ and the cover relations for $p=q=r=2$.
  • ...and 1 more figures

Theorems & Definitions (49)

  • Example 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Example 2.4
  • Theorem 2.5
  • Example 2.6
  • Remark 2.7
  • Remark 2.8
  • Theorem 2.9
  • proof
  • ...and 39 more