Hermann Weyl and representation theory

Abstract

Weyl was a universal mathematician whose fundamental contributions to mathematics encompassed all areas, and provided a unification seldom seen. His work on the theory of Lie groups was motivated by his life-long interest in quantum mechanics and relativity. WhenWeyl entered Lie theory, it mostly focussed on the infinitesimal, and he strove to bring in a global perspective. Time and again, Weyl’s ideas arising in one context have been adapted and applied to wholly new contexts. In 1925–26, Weyl wrote four epochal papers in representation theory of Lie groups which solved fundamental problems, and also gave birth to the subject of harmonic analysis of semisimple Lie groups. In these papers, Weyl proved complete reducibility theorems and introduced many techniques which have become the standard way to study representations of Lie groups and their various generalizations in the last seven decades. Weyl’s work covers several parts of mathematics, as well as parts of physics. In this article, we discuss mainly his contributions to the representation theory of Lie groups via the four papers mentioned above.