Type AIII orbits in the affine flag variety of type A

TL;DR

This paper extends the concept of clans, which parametrize classical group orbits, to the affine setting, establishing a bijection between affine orbits and affine clans characterized by involutions with signed fixed points.

Contribution

It introduces affine $(p,q)$-clans and constructs an explicit bijection with affine $K$-orbits, generalizing classical orbit parametrization to the affine flag variety.

Findings

Established a bijection between affine orbits and affine clans.

Affine clans are characterized as involutions with signed fixed points.

Provides a concrete combinatorial model for affine $K$-orbits.

Abstract

Matsuki and Oshima introduced the notion of clans, which are incomplete matchings with positive or negative signs on isolated vertices. They discovered that clans parametrise $K$-orbits in the flag varieties for classical linear groups, where $K$ is a fixed point subgroup of an involution in the same classical linear group. In this work we investigate the affine version of these orbits. For a field $\Bbbk$ with characteristic not equal to two, we construct an explicit bijection between the $\textsf{GL}_p(\Bbbk(\hspace{-0.5mm}(t)\hspace{-0.5mm})) \times \textsf{GL}_q(\Bbbk(\hspace{-0.5mm}(t)\hspace{-0.5mm}))$-orbits in the affine flag variety and certain objects called affine $(p,q)$-clans. These affine $(p,q)$-clans can be concretely interpreted as involutions in the affine permutation group with positive or negative signs on fixed points.