Type AIII orbits in the affine flag variety of type A (2501.16269v1)

Abstract: Matsuki and Oshima introduced the notion of clans, which are incomplete matchings between $[1,n]$ with positive or negative signs on isolated vertices. They discovered that clans can 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. Yamamoto gave a full proof for the type AIII $\textsf{GL}_p(\mathbb{C}) \times \textsf{GL}_q(\mathbb{C})$-orbits in type A flag variety. In this work we investigate the affine version of type AIII orbits. For a field $\mathbb{K}$ with characteristic not equal to two, we construct explicit bijections between the $\textsf{GL}_p(\mathbb{K}(\hspace{-0.5mm}(t)\hspace{-0.5mm})) \times \textsf{GL}_q(\mathbb{K}(\hspace{-0.5mm}(t)\hspace{-0.5mm}))$-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 in fixed points.