Representations of complex semi-simple Lie groups and Lie algebras

TL;DR

This paper reviews the 1967 foundational work on irreducible admissible Harish-Chandra modules over complex semisimple Lie groups and algebras, highlighting its impact, new insights, and subsequent research developments.

Contribution

It provides an accessible exposition of a seminal paper, emphasizing its contributions to classification, tensor products, and determinants in representation theory.

Findings

Classification of irreducible Harish-Chandra modules

Results on the PRV conjecture and tensor multiplicities

Development of PRV determinants for affine and semisimple Lie algebras

Abstract

This article is an exposition of the 1967 Annals paper by Parthasarathy, Ranga Rao, and Varadarajan, on irreducible admissible Harish-Chandra modules over complex semisimple Lie groups and Lie algebras. It was written in Winter 2012 to be part of a special collection organized to mark 10 years and 25 volumes of the series "Texts and Readings in Mathematics (TRIM)". Each article in this collection is intended to give nonspecialists in its field, an appreciation for the impact and contributions of the paper being surveyed. Thus, the author has kept the prerequisites for this article down to a basic course on complex semisimple Lie algebras. While it arose out of the grand program of Harish-Chandra on admissible representations of semisimple Lie groups, the work by Parthasarathy et al also provided several new insights on highest weight modules and related areas, and these results have been the subject of extensive research over the last four decades. Thus, we also discuss its results and follow-up works on the classification of irreducible Harish-Chandra modules; on the PRV conjecture and tensor product multiplicities; and on PRV determinants for (quantized) affine and semisimple Lie algebras.