Orbit Duality in Ind-Varieties of Maximal Generalized Flags

TL;DR

This paper extends Matsuki duality to ind-varieties of maximal generalized flags, providing explicit combinatorial parametrizations in finite dimensions and establishing conditions for orbit existence in infinite dimensions.

Contribution

It generalizes Matsuki duality to infinite-dimensional ind-varieties and characterizes orbit structures for classical ind-groups.

Findings

Matsuki duality is extended to ind-varieties of maximal generalized flags.

Explicit combinatorial parametrization of orbits in finite-dimensional case.

Necessary and sufficient conditions for open and closed orbits in infinite-dimensional case.

Abstract

We extend Matsuki duality to arbitrary ind-varieties of maximal generalized flags, in other words, to any homogeneous ind-variety $\mathbf{G}/\mathbf{B}$ for a classical ind-group $\mathbf{G}$ and a splitting Borel ind-subgroup $\mathbf{B}\subset\mathbf{G}$. As a first step, we present an explicit combinatorial version of Matsuki duality in the finite-dimensional case, involving an explicit parametrization of $K$- and $G^0$-orbits on $G/B$. After proving Matsuki duality in the infinite-dimensional case, we give necessary and sufficient conditions on a Borel ind-subgroup $\mathbf{B}\subset\mathbf{G}$ for the existence of open and closed $\mathbf{K}$- and $\mathbf{G}^0$-orbits on $\mathbf{G}/\mathbf{B}$, where $\left(\mathbf{K},\mathbf{G}^0\right)$ is an aligned pair of a symmetric ind-subgroup $\mathbf{K}$ and a real form $\mathbf{G}^0$ of $\mathbf{G}$.