Orthogonal and symplectic orbits in the affine flag variety of type A

TL;DR

This paper provides the first combinatorial description of orthogonal and symplectic orbits in the affine flag variety of type A, extending classical finite results to the affine setting with explicit bijections.

Contribution

It introduces a novel combinatorial framework for describing $K$-orbits in the affine flag variety of type A, linking double cosets to twisted affine involutions.

Findings

Constructed explicit bijections between double cosets and twisted affine involutions.

First combinatorial description of $K$-orbits in the affine flag variety of type A.

Extended classical finite orbit results to the affine case.

Abstract

It is a classical result that the set $K\backslash G /B$ is finite, where $G$ is a reductive algebraic group over an algebraically closed field with characteristic not equal to two, $B$ is a Borel subgroup of $G$, and $K = G^{\theta}$ is the fixed point subgroup of an involution of $G$. In this paper, we investigate the affine counterpart of the aforementioned set, where $G$ is the general linear group over formal Laurent series, $B$ is an Iwahori subgroup of $G$, and $K$ is either the orthogonal group or the symplectic group over formal Laurent series. We construct explicit bijections between the double cosets $K \backslash G/B$ and certain twisted affine involutions. This is the first combinatorial description of $K$-orbits in the affine flag variety of type A.