Weight spaces of Lie algebra modules

Abstract

Let V V be a finite-dimensional module for the finite-dimensional Lie algebra L L over a field of characteristic zero. If V λ = { v ∈ V | all x ∈ L , [ x − λ ( x ) ] i v = 0 for some } {V^\lambda } = \{ v \in V|\;{\text {all }}x \in L,{[x - \lambda (x)]^i}v = 0\;{\text {for some }}\} is nonzero, then λ ∈ L ∗ \lambda \in {L^*} and is a character of L L . Moreover, the corresponding eigenspace { v ∈ V | all x ∈ L , x v = λ ( x ) v } \{ v \in V|{\text {all }}x \in L,xv = \lambda (x)v\} is nonzero and V λ {V^\lambda } is an L L submodule of V V .