Wobbly moduli of chains, equivariant multiplicities and $\mathrm{U}(n_0,n_1)$-Higgs bundles

We give a birational description of the reduced schemes underlying the irreducible components of the nilpotent cone and the ${\text{CC}}^{\times }$-fixed point locus of length two in the moduli space of Higgs bundles. Using these results, we prove Drinfeld’s conjecture for the sublocus of type $(n_0,n_1)$ fixed points. We introduce the notion of $\text{U}(n_0,n_1)$-wobbliness (stronger than the one of wobbliness) and show that fixed point components of type $(n_0,n_1)$ are wobbly in rank higher than three, if and only if they are also $\text{U}(n_0,n_1)$-wobbly. This yields a computable criterion to check wobbliness of fixed point components, that simplifies the existing ones. We analyse the virtual equivariant multiplicities of fixed points of type $(n_0,n_1)$ and their Euler pairings with downward flows for type $(1,\dots ,1)$ fixed points. We find that both invariants fail to fully detect all wobbly components for ordered partitions other than $(2,1)$.