Very stable parabolic $G$-Higgs bundles and affine flag varieties

Abstract

Let $G$ be a semisimple complex algebraic group. We study the $\mathbb C^\times$-action on the moduli space of strongly parabolic $G$-Higgs bundles over a punctured curve with weights in the interior of the standard Weyl alcove. We describe its fixed points and, for the components with generically regular Higgs field, we classify the fixed points that have closed Bialynicki-Birula upward flows, called very stable. The classification is stated in terms of a naturally associated divisor valued in the extended affine Weyl group of $G$ which encodes the information of the fixed point. Under this point of view, very stable fixed points naturally coincide with divisors where the coefficients are minimal under the Bruhat order. Furthermore, when the divisor is supported at the parabolic punctures, the corresponding very stable upward flow is a section of the Hitchin map. For these upward flows, we propose a conjectural mirror line bundle over the dual moduli space, and verify its validity for $G=\mathrm{PGL}_n(\mathbb{C})$.