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 ${𝔤}^{\sigma }$ of outer automorphisms $\sigma$. 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 $(𝔤,{𝔤}^{\sigma })$ 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 ${𝔤}^{\sigma }$ on one hand, and the length function on the other hand.