Discrete Series

Abstract

One of the greatest achievements of mathematics in the 20th century is Harish-Chandra’s classification of discrete series representations of semisimple Lie groups. Let G be a noncompact semisimple Lie group with a maximal compact subgroup K. Discrete series representations are those irreducible unitary representations of G which occur as subrepresentations in the Plancherel decomposition of L²(G). Harish-Chandra proved that a necessary and sufficient condition for G to have a discrete series is to have a compact Cartan subgroup. He constructed the characters of all discrete series representations. Speaking of Harish-Chandra’s work on discrete series, we quote Varadarajan in his article “Harish-Chandra, His Work, and its Legacy” [Va]: “In my opinion the character problem and the problem of constructing the discrete series were the ones that defined him, by stretching his formidable powers to their limit. The Harish-Chandra formula for the characters of discrete series is the single most beautiful formula in the theory of infinite-dimensional unitary representations.” Harish-Chandra “actually wrote down all the proofs in an extraordinary sequence of 8 papers [1964a]–[1966b], totaling 461 journal pages constituting one of the most remarkable series of papers in the annals of scientific research in our times—remarkable because of how long it took him to reach his goal, remarkable for how difficult the journey was and how it was punctuated by illness, remarkable for how unaided his achievement was, and finally, remarkable for the beauty and inevitability of his theorems.”