The early development of the representation theory of semisimple Lie groups: A. Hurwitz, I. Schur, H. Weyl (Q1305402)

This article starts from the celebrated ``character formula'' by H. Weyl, and then looks back to the work by \textit{A. Hurwitz}, The invariant integral on Lie groups in 1897. Meanwhile, \textit{I. Schur} determined the irreducible polynomial representations of \(\text{GL}_n\), developed a character formula in his dissertation (1901). Weyl, in a letter to I. Schur in 1924, sketched the four sections: 1) Introduction; 2) The derivation of the integral formula for the unitary groups; 3) The derivation of the character and dimension formula for unitary groups; 4) A sketch of 2), 3) for semisimple groups. The definite breakthrough had to wait until his joint work with \textit{F. Peter} in 1927 [Math. Ann. 97, 737-755 (1927; JFM 53.0387.02)].