Type AIII orbits in the affine flag variety of type A

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 $\mathbb{k}$ with characteristic not equal to two, we construct an explicit bijection between the ${\mathsf{\text{GL}}}_p(\mathbb{k}((t)))\times {\mathsf{\text{GL}}}_q(\mathbb{k}((t)))$-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.