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 of 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 the Ngô action to obtain universal symmetries of the corresponding quantum systems. Our construction of the Ngô action and its quantization comes as the Langlands dual form of a simple feature of groupoid algebras (the presence of multiplication alongside convolution).