Orbit embedding for double flag varieties and Steinberg map

TL;DR

This paper reviews Steinberg theory for double flag varieties in symmetric pairs, classifies orbits for a specific case, and develops a general embedding theory to relate different double flag varieties.

Contribution

It introduces a general embedding theorem for double flag varieties, enabling analysis of complex varieties via known simpler cases, with explicit orbit classifications for type AIII.

Findings

Classification of $K$-orbits in a specific double flag variety.

Explicit combinatorial description of Steinberg maps for type AIII.

Proved a general embedding theorem relating different double flag varieties.

Abstract

In the first half of this article, we review the Steinberg theory for double flag varieties for symmetric pairs. For a special case of the symmetric space of type AIII, we will consider $ X = GL_{2n}/P_{(n,n)} \times GL_n / B_n^+ \times GL_n / B_n^- $ on which $ K = GL_n \times GL_n $ acts diagonally. We give a classification of $ K $-orbits in $ X $, and explicit combinatorial description of the Steinberg maps. In the latter half, we develop the theory of embedding of a double flag variety into a larger one. This embedding is a powerful tool to study different types of double flag varieties in terms of the known ones. We prove an embedding theorem of orbits in full generality and give an example of type CI which is embedded into type AIII.