Classification of Finite-Dimensional Irreducible Representations of Connected Complex Semisimple Lie Groups

Abstract

Using the results of Malcev and Dynkin, the irreducible representations of connected components of complex semisimple Lie groups are classified. The simple Lie algebras A1, Bn, Cn, G2, F4, E7, and E8 have no outer automorphism and, for the Lie algebra D2k, the automorphism corresponding to the contragredient transformation is inner. Hence, all the irreducible representations of all simple Lie algebras except An (n > 1), D2k+1, and E6 are self-contragredient. For An (n > 1), D2k+1, and E6, an irreducible representation is self-contragredient, provided its highest weight possesses the symmetry to reduce the outer automorphism of the algebra into an inner one for the representation. All the self-contragredient irreducible representations are classified into orthogonal and symplectic types by reducing the problem to the case of the angular-momentum algebra.