On generalized Steinberg theory for type AIII

TL;DR

This paper extends Steinberg theory for type AIII symmetric pairs, providing a combinatorial description of orbit parametrization and Steinberg maps using partial permutations, generalizing classical algorithms.

Contribution

It offers a comprehensive generalization of Steinberg maps for type AIII, describing orbits and nilpotent correspondences via combinatorial algorithms on partial permutations.

Findings

Parametrization of orbits by pairs of partial permutations

Explicit combinatorial algorithms for Steinberg maps

Extension of classical Robinson--Schensted procedure

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.