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

It is a classical result that the set $K\mathrm{\backslash }G\mathrm{/}B$ 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 $K\mathrm{\backslash }G\mathrm{/}B$ and certain twisted affine involutions. This is the first combinatorial description of $K$-orbits in the affine flag variety of type A.