• Corpus ID: 17195173

Representation Theory ∗ Its Rise and Its Role in Number Theory

@inproceedings{Langlands1990RepresentationT,
  title={Representation Theory ∗ Its Rise and Its Role in Number Theory},
  author={Robert P. Langlands},
  year={1990}
}

• R. Langlands
• Published 1990
• Mathematics

By representation theory we understand the representation of a group by linear transformations of a vector space. Initially, the group is finite, as in the researches of Dedekind and Frobenius, two of the founders of the subject, or a compact Lie group, as in the theory of invariants and the researches of Hurwitz and Schur, and the vector space finite-dimensional, so that the group is being represented as a group of matrices. Under the combined influences of relativity theory and quantum mechanics, in the context of relativistic field theories in the nineteen-thirties, the Lie group ceased to be compact and the space to be finite-dimensional, and the study of infinite-dimensional representations of Lie groups began. It never became, so far as I can judge, of any importance to physics, but it was continued by mathematicians, with strictly mathematical, even somewhat narrow, goals, and led, largely in the hands of Harish-Chandra, to a profound and unexpectedly elegant theory that, we now appreciate, suggests solutions to classical problems in domains, especially the theory of numbers, far removed from the concerns of Dirac and Wigner, in whose papers the notion first appears. Physicists continue to offer mathematicians new notions, even within representation theory, the Virasoro algebra and its representations being one of the most recent, that may be as fecund. Predictions are out of place, but it is well to remind ourselves that the representation theory of noncompact Lie groups revealed its force and its true lines only after an enormous effort, over two decades and by one of the very best mathematical minds of our time, to establish rigorously and in general the elements of what appeared to be a somewhat peripheral subject. It is not that mathematicians, like cobblers, should stick to their lasts; but that humble spot may nevertheless be where the challenges and the rewards lie. J. Willard Gibbs, too, offered much to the mathematician. An observation about the convergence of Fourier series is classical and now known as the Gibbs phenomenon. Of far greater importance, in the last quarter-century the ideas of statistical mechanics have been put in a form that is readily accessible to mathematicians, who are, as a community, very slowly becoming aware of the wealth of difficult problems it poses. His connection with representation theory is more tenuous. In the period between his great researches on thermodynamics and statistical mechanics, he occupied himself with linear algebra, even in a somewhat polemical fashion. In particular, he introduced notation that was more than familiar to mathematics students of my generation, but that perhaps survives today only in electromagnetic theory and hydrodynamics. In these two subjects, the dot product α · β and the cross product α × β introduced by Gibbs are ubiquitous and extremely convenient. The usual orthogonal group comes to us provided with a natural representation because it is a group of 3 × 3 matrices, and all objects occurring in elementary physical theories must transform simply under it. The