On uniqueness of branching to fixed point Lie subalgebras

Abstract

Abstract Let 𝔤 {\mathfrak{g}} be a complex semisimple Lie algebra and let θ be a finite-order automorphism of 𝔤 {\mathfrak{g}} . Let 𝔤 0 {\mathfrak{g}_{0}} be the subalgebra { X ∈ 𝔤 : θ ⁢ ( X ) = X } {\{X\in\mathfrak{g}:\theta(X)=X\}} . In this article, we study for which pairs ( V 1 , V 2 ) {(V_{1},V_{2})} , consisting of two irreducible finite-dimensional representations of 𝔤 {\mathfrak{g}} , we have res 𝔤 0 ⁡ V 1 ≃ res 𝔤 0 ⁡ V 2 . \operatorname{res}_{\mathfrak{g}_{0}}V_{1}\simeq\operatorname{res}_{\mathfrak{% g}_{0}}V_{2}. In many cases, we show that V 1 {V_{1}} and V 2 {V_{2}} have isomorphic restrictions to 𝔤 0 {\mathfrak{g}_{0}} if and only if V 1 {V_{1}} is isomorphic to V 2 σ {V_{2}^{\sigma}} for some outer automorphism σ of 𝔤 {\mathfrak{g}} .