Collisions of Nilpotent Higgs Fields

We prove the non-characteristicness of the completed self-sum of the universal nilpotent cone in $T^{\ast }{\mathrm{B}\mathrm{u}\mathrm{n}}_G$ relative to the moduli of curves, for reductive groups $G$ of semi-simple rank $1$. In particular, this implies the non-characteristicness of the $\mathcal{H}om$-sheaf between Betti automorphic sheaves for groups of semi-simple rank $1$. We also construct an example of a collision of universal nilpotent Higgs fields, proving that the completed self-sum is not non-characteristic for ${\mathrm{G}\mathrm{L}}_n$, $n\ge 3$, relative to the moduli of curves of genus $g\ge 4$.