Harmonic analysis is concentrated mostly on studying locally compact groups. I would like to ask people about examples of non-locally compact topological groups that are interesting in connection with some applications in mathematics (or physics). I must confess that I know only two examples:
topological vector spaces are studied as examples of topological groups (with the additive group operation) to which Pontryagin duality is sometimes transferred from the class of locally compact abelian groups,
the groups of diffeomorphisms of smooth manifolds are studied in the theory of infinite dimensional manifolds.
Can people enlighten me about other similar subjects? (If possible, with motivations.)