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How and when did the dedicated study of locally compact groups begin?
Specific instances from literature, recorded stories, etc., may help supplement the answers. There seems to be no reason why I would wake up on a Sunday morning and think of locally compact topological groups.
Thanks!
The symbolic starting date is 1933, when Haar introduced left invariant measures and proved their existence on second-countable locally compact groups, see his Der Massbegriff in der Theorie der Kontinuierlichen Gruppen. However, invariant integral on Lie groups was used by Hurwitz already in 1897, as Haar himself noted. In 1934 Pontryagin undertook a systematic study of locally compact Abelian groups and proved Pontryagin duality for second countable ones, compact or discrete. The general Abelian case was considered by van Kampen in 1935. Weil proved existence of invariant measure and its uniqueness in the general case in 1938, it is covered in his 1940 book L'intégration dans les groupes topologiques et ses applications.
A detailed theory for some non-compact non-Abelian locally compact groups (real semi-simple Lie groups) was developed by Harish-Chandra in 1952, who relied on earlier 1943 work of Gelfand and Raikov concerning their irreducible representations. Further details and references can be found in Remarks on History of Abstract Harmonic Analysis by Stankovic et al.
