Finite Symplectic Automorphisms of Hyperkähler Manifolds of $K3^{[n]}$-Type Act Trivially on Discriminant Groups

We give an alternative proof of Mongardi's result that every finite group of symplectic automorphisms of a hyperkähler manifold of $K3^{[n]}$-type acts trivially on the discriminant groups of its second cohomology lattice and its coinvariant lattice. The proof combines sphere-packing bounds, Vinberg