Combinatorial curve neighborhoods for the affine flag manifold of type $A_1^1$

Abstract

Let $X$ be the affine flag manifold of Lie type $A_1^1$. Its moment graph encodes the torus fixed points (which are elements of the infinite dihedral group $D_{\infty }$) and the torus stable curves in $X$. Given a fixed point $u\in D_{\infty }$ and a degree $d=\left(d_0,d_1\right)\in {\mathbb{Z}}_{\ge 0}^2$, the combinatorial curve neighborhood is the set of maximal elements in the moment graph of $X$ which can be reached from $u$ using a chain of curves of total degree $\le d$. In this paper we give a formula for these elements, using combinatorics of the affine root system of type $A_1^1$.