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That characters and group representations might have something to do with Fourier analysis seems to have first been recognized by Hermann Weyl (1885-1955) in 1927. But an essential first step was taken by Schur in 1924.
Because of connections with the branch of algebraic geometry known as "invariant theory," Schur became interested in studying representations of the rotation group in n dimensions and discovered that he could carry over the main features of the character and representation theory of finite groups if he replaced summation over the elements of a finite group by a suitable integration over the compact manifold constituted by the elements of the rotation group. Hurwitz had made use of such an integration in 1897 in a method he discovered for constructing invariants. Schur adapted Hurwitz's integral to his needs. From a modern point of view, Schur and Hurwitz made use of the fact (proved by A. Haar in 1933) that every separable locally-compact group admits a measure (unique up to a multiplicative constant) that is defined on all Borel sets, is finite on compact sets, is invariant under right translation, and is not identically zero. When the group is a Lie group, the existence of this measure can be established easily using concepts from differential geometry. Using integration with respect to "Haar measure" to replace sums over the group elements, Schur was able to carry over Maschke's argument and prove the decomposability into irreducibles of an arbitrary representation of the rotation group. He was able to show also that an irreducible representation is determined to within equivalence by its character and to find all irreducible representations together with their characters for the groups with which he concerned himself.