For compact Lie groups one considers a maximal torus to define the weight space decomposition of a representation. For a complex semisimple Lie algebra one considers a Cartan subalgebra. How does one define weights for a general semisimple Lie group?
According to wikipedia the irreducible representations of semisimple Lie groups are parametrized by highest weights. Can anyone confirm that this is indeed true and shed any light on this theory?
"Representations" here means "complex representations." A complex representation of a Lie group $G$ gives rise to a complex representation of its Lie algebra $\mathfrak{g}$, and hence to a complex representation of the complexification $\mathfrak{g} \otimes \mathbb{C}$ of its Lie algebra. If $G$ is semisimple then so is this complex Lie algebra. There is no need to bring in either the complexification of $G$ or any of its other real forms.