A note on branching of V(ρ)

Abstract

Let g be a complex simple Lie algebra and let g0 be the sub-algebra fixed by a diagram automorphism of g. Let G be the complex, simply-connected, simple algebraic group with Lie algebra g, and let G0 be the connected subgroup of G with Lie algebra g0. Let ρ be the half sum of positive roots of g. In this article, we give a necessary and sufficient condition for a highest weight g0-representation V0(dμ) to occur in the representation resg0V(dρ), for any saturation factor d of the pair (G0,G).