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What is a good reference to learn about real representations of Lie groups ? I've parsed through the very enlightening book of Fulton and Harris, but it is extremely (if not exclusively) example-oriented, and I need the more general statements. More precisely, I'm looking for statements about the dimension of real irreducible representations of real Lie groups.

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    $\begingroup$ Have a look at the book Representations of compact Lie groups by Brocker and tom Dieck, Grad. Texts in Math, vol. 98, Springer. $\endgroup$
    – Liviu Nicolaescu
    Commented May 17, 2013 at 15:38
  • $\begingroup$ For the dimensions look for Weyl's dimension formula for compact Lie groups. $\endgroup$
    – Dietrich Burde
    Commented May 17, 2013 at 18:44
  • $\begingroup$ @Samuel: From the context I assume you are interested in finite dimensional representations. These are well-studied, but usually indirectly via their Lie algebras and complexifications. As Dietrich points out, the (relatively easy) dimensions depends on Weyl's formula, for compact or complex Lie groups (or Lie algebras). But working with real forms sometimes doubles dimensions, since irreducible over $\mathbb{R}$ may not mean irreducible over $\mathbb{C}$. There are lots of textbooks, but what works best depends on what you know. $\endgroup$
    – Jim Humphreys
    Commented May 17, 2013 at 21:21
  • $\begingroup$ @Jim Humphrey: I am indeed interested in finite dimentional representations. I am getting more or less comfortable with the theory of complex representations, but when it come to real ones, I have trouble finding good sources. @Livu Nicolaescu: Thanks for this reference, I'll check it out. $\endgroup$
    – Samuel Tinguely
    Commented May 22, 2013 at 14:14

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MR2041548 Onishchik, Arkady L. Lectures on real semisimple Lie algebras and their representations. ESI Lectures in Mathematics and Physics. European Mathematical Society (EMS), Zürich, 2004. x+86 pp.

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