Fundamental Involutory Root Systems and a Branching Theorem for Symmetric Pairs

Abstract

Let 𝔤 be a complex semisimple Lie algebra with an involutory automorphism θ. Let 𝔨 be the corresponding symmetric subalgebra and 𝔥 be a fundamental Cartan subalgebra of 𝔤 containing a Cartan subalgebra 𝔱 of 𝔨. Let Δ(𝔤, 𝔥) and Δ(𝔤, 𝔱) be the respective root systems of 𝔤 with respect to 𝔥 and 𝔱. The pair (Δ(𝔤, 𝔥), θ) is a fundamental involutory root system. We prove that there exists natural 1-1 correspondences between θ-stable Weyl chambers of Δ(𝔤, 𝔥) and Weyl chambers of Δ(𝔤, 𝔱) and that the Weyl group of Δ(𝔤, 𝔱) acts simply transitively on the above two sets of Weyl chambers. Regard a finite dimensional complex irreducible 𝔤-module V as a 𝔨-module, based on the above results we show that some certain irreducible modules of 𝔨 will occur in V.