Multiplicity-free branching rules for outer automorphisms of simple Lie algebras

Abstract

We find explicit multiplicity-free branching rules of some series of irreducible finite dimensional representations of simple Lie algebras 𝔤 to the fixed point subalgebras 𝔤 σ of outer automorphisms σ . The representations have highest weights which are scalar multiples of fundamental weights or linear combinations of two scalar ones. Our list of pairs of Lie algebras ( 𝔤 , 𝔤 σ ) includes an exceptional symmetric pair ( E 6 , F 4 ) and also a non-symmetric pair ( D 4 , G 2 ) as well as a number of classical symmetric pairs. Some of the branching rules were known and others are new, but all the rules in this paper are proved by a unified method. Our key lemma is a characterization of the ``middle'' cosets of the Weyl group of 𝔤 in terms of the subalgebras 𝔤 σ on one hand, and the length function on the other hand.