Classical and quantum Hamiltonian actions of reductive groups, respectively, give rise to ubiquitous families of commuting flows and of commutative rings of operators. I will explain how a construction independently due to Knop and Ngô (from the proof of the Fundamental Lemma) provides a universal integration of these flows for classical systems. I will then explain, following joint work with Sam Gunningham, how to quantize this action to obtain universal symmetries of the corresponding quantum systems. The action can be understood as a symmetry in supersymmetric gauge theories which is a nonabelian generalization of the invariance of Maxwell theory under shifts of the electromagnetic potential.