Properties of linear representations with a highest weight for the semisimple Lie algebras

Abstract

A theory for the representations with a highest and/or lowest weight is given for the semisimple complex Lie algebras (and their real forms). These representations are either irreducible finite-dimensional, irreducible infinite-dimensional or reducible, but not completely reducible, infinite-dimensional (called elementary representations), depending upon the property of the associated highest (or lowest) weight Λ. No restriction is made to those representations of the semisimple Lie algebras which can be integrated to form representations of the corresponding Lie group. The algebra A1 is chosen (Sec. III) as a simple and familiar example upon which, however, much of the proof for the results obtained for the theory of representations with a highest (and/or lowest) weight for the general case of a semisimple Lie algebra rests (Sec. IV). It is demonstrated that the irreducible representations D (Λ) with a highest (and/or lowest) weight Λ of the semisimple Lie algebras decompose with respect to any (regularly) embedded subalgebra of the type A1 in the manner that either (a) the subrepresentations subduced on A1 are all irreducible finite-dimensional, or (b) all infinite-dimensional. If for case (b) the complex number Mα ≡ 2(M,α)/(α,α), α the (simple) root of A1 and M a weight of D (Λ) extremal with respect to A1, is not a nonnegative integer, then the representation subduced on A1 is irreducible. If, however, Mα is a nonnegative integer, then a reducible but not completely reducible, representation is subduced on A1. Based upon the results of Sec. IV a generalization of Freudenthal’s formula is obtained in Sec. V, valid for irreducible infinite-dimensional representations with highest (or lowest) weight. In Sec. VI generalizations are given of Racah’s recurrence relation for the multiplicity of weights, Weyl’s character formula and Kostant’s formula for the multiplicity of weights for infinite-dimensional irreducible representations with a highest (or lowest) weight of the semisimple Lie algebras. These formulas are derived utilizing theorems and lemmas obtained by Verma, I. M. Gel’fand, S. I. Gel’fand, Bernstein, Harish-Chandra and the results of Sec. IV. In Sec. VII some of the infinite-dimensional representations of the algebra A2 are discussed as examples, employing the geometrical methods developed by Antoine and Speiser and by Biedenharn and others.