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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?

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  • $\begingroup$ it is true for finite-dimensional representations (they can be extended to holomorphic representations of the corresponding complex Lie group, and again restricted to representations of the compact real form of the group) $\endgroup$
    – user8268
    Nov 17, 2014 at 23:17
  • $\begingroup$ @user8268 Do you know any books that give the details about what you are talking about? You are saying one defines weights by passing to the compact real form of the complexification? (Where does semisimple come into play here?) $\endgroup$
    – Seth
    Nov 17, 2014 at 23:24

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"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.

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  • $\begingroup$ So basically you relate reps of a real group $G$ to reps of it's complexified Lie algebra? Why would it be necessary to start with a complex rep then, since we can just complexify the vector space too? The only concern is, it doesn't seem clear which representations of the complexified algebra lift to representations of $G$. $\endgroup$
    – Seth
    Nov 18, 2014 at 1:10
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    $\begingroup$ @Seth: of course you can complexify the vector space too, but the original real representation contains more information than its complexification, and not every complex representation is a complexification of a real representation. As for your other question, every complex representation of $\mathfrak{g}$ lifts to a complex representation of the simply connected Lie group $\widetilde{G}$ with Lie algebra $\mathfrak{g}$, which is some cover of $G$ (here I assume that $G$ is connected). $G$ is a quotient of $\widetilde{G}$ by a discrete normal subgroup of the center of $\widetilde{G}$, and... $\endgroup$
    – Qiaochu Yuan
    Nov 18, 2014 at 1:22
  • $\begingroup$ ... a representation of $\widetilde{G}$ descends to $G$ iff this subgroup acts trivially. It's a good exercise to work out what all this looks like for $G = \text{SO}(3), \widetilde{G} = \text{SU}(2)$. $\endgroup$
    – Qiaochu Yuan
    Nov 18, 2014 at 1:23
  • $\begingroup$ @QiaochuYuan: What precisely do you mean when you say "the original real representation contains more information than its complexification"? I would think that at least for semisimple $\mathfrak{g}$, one can recover a rep. with real coefficients from its extension to complexified coefficients, at least up to isomorphism (maybe that's what you mean). The other point, that not every complex rep. comes from a real one, seems more important to me. In other words, complexifying the coefficients is injective, but far from surjective on isoclasses of representations, correct? $\endgroup$
    – Torsten Schoeneberg
    Jul 26, 2019 at 19:03

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