Type AIII orbits in the affine flag variety of type A

TL;DR

This work extends the Matsuki–Oshima clan framework to the affine setting of type AIII by constructing explicit bijections between $K$-orbits on the affine flag variety $\widetilde{Fl}_n$ and affine $(p,q)$-clans, where $K=\textsf{GL}_p(\mathbb{K}((t))) \times \textsf{GL}_q(\mathbb{K}((t)))$ and $n=p+q$. It provides two complementary viewpoints: (i) a direct matrix-multiplication approach yielding affine $(p,q)$-clan matrices as distinct double coset representatives of $K\backslash G/B$, and (ii) an invariant-based combinatorial description using the coordinates $(i;+)$, $(i;-)$, $(i;\mathbb{N})$, and $(i;j)$ that classify affine flags up to $K$-equivalence. The main results include explicit bijections between $K$-orbits and affine $(p,q)$-clans (and a lattice-flag viewpoint) and a detailed algorithm to pass between flags and affine clans, with examples and lemmas ensuring injectivity. The findings generalize Yamamoto’s classical finite-type correspondence to the affine world, enabling potential extensions to other Lie types and new combinatorial weak-order descriptions of orbit closures. Overall, the paper provides concrete, workable tools for indexing and distinguishing affine $K$-orbits via affine clan data, bridging linear-algebra constructions with combinatorial clan theory in the affine setting.

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.

Theorems & Definitions (65)

• Theorem 1.1: Yamamoto
• Example 1.2
• Theorem 1.3
• Example 1.4
• Theorem 1.5
• Remark 1.6
• Proposition 2.1: Magyar
• Remark 2.2
• Theorem 2.3: Mann
• Lemma 3.1