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arXiv · 2501.16269

Type AIII orbits in the affine flag variety of type A

Kam Hung Tong
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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.

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Kam Hung Tong. 2025-01-27. Type AIII orbits in the affine flag variety of type A. https://arxiv.org/abs/2501.16269

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