Abstract
We find explicit multiplicity-free branching rules of some series of irreducible finite dimensional representations of simple Lie algebras $¥mathfrak g$ to the fixed point subalgebras $¥mathfrak g$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 ($¥mathfrak g$, $¥mathfrak g$σ) includes an exceptional symmetric pair (E6, F4) and also a non-symmetric pair (D4, G2) 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 $¥mathfrak g$ in terms of the subalgebras $¥mathfrak g$σ on one hand, and the length function on the other hand.
