Let $\mathfrak g$ be a real simple noncompact Lie algebra. Are there any correspondence between irreducible real representations of $\mathfrak g$ and the highest weight correspond to some positive system of restricted roots.

There is a complete classification of the complex representation $\mathfrak g \otimes \mathbb C$. There is a classification of real irreducible representations using the irreducible representations of $\mathfrak g\otimes \mathbb C$ in "Onishchik, Lectures on real semi simple lie algebras and their representations". But in a paper I read, it is claimed that the real representations can also be classified using highest weight like complex one. But I can not find a reference for that.

  • 2
    $\begingroup$ It might help to specify which paper you read. Comparison of real and complex forms of the Lie algebras, along with their finite dimensional representations, goes back a century or so. Most approaches to classification rely on working out the easier complex case first, then studying restriction to real forms and extension of scalars. See for example Helgason's books. The 100 page EMS lecture notes by Onishchik probably provide a good overview, as Dietrich indicates. $\endgroup$ – Jim Humphreys Dec 10 '13 at 14:50
  • $\begingroup$ Probably I did not make the question very clear. I explained it in the comment to the answer given by Dietrich. $\endgroup$ – ronggang Dec 10 '13 at 21:20

I'm still unsure what you are looking for (or where you saw the material you recall), but the work of Borel-Tits and also Satake on reductive groups over arbitrary fields including $\mathbb{R}$ might be relevant here. (In characteristic 0 studying the groups is almost equivalent to studying the Lie algebras and their representations.) This line of research doesn't necessarily reveal much that is new about Lie groups, but it does place those groups in a framework which extends to other fields, etc. And it's possible to work over a given field of definition, within the framework of a minimal $k$-parabolic and a maximal $k$-split torus.

In the foundational work of Borel-Tits here, see especially $\S 12$. There is also a later paper by Tits which deals systematically with representations over arbitrary fields: Representations lineaires irreductibles d’un groupe reductif sur un corps quelconque. J. Reine Angew. Math. 247 (1971) 196–220. Still, for most practical purposes it's enough to follow the classical outline of E. Cartan and others, relating real groups or Lie algebras and their representations to the complexifications.


A reference for this is also "Lectures on Real Semisimple Lie Algebras and Their Representations" by A. L. Onishchik. A first result here is: any irreducible real representation $\rho\colon \mathbb{g}\rightarrow \mathbb{gl}(V)$ of a real Lie algebra $\mathbb{g}$ satisfies precisely one of the following two conditions.
$(1)$ $\rho^{\mathbb{C}}$ is an irreducible complex representation.
$(2)$ $\rho=\rho'_{\mathbb{R}}$, where $\rho'$ is an irreducible complex representation admitting no invariant real structure.
Conversely, any real representation satisfying $(1)$ or $(2)$ is irreducible.

For the results concerning highest weigts etc. see Onishchik's book, section $8$, about "Real representations of real semisimple Lie algebras", including the classification of them.

  • $\begingroup$ I have read section 8 of Onishchik's book before I am posing the question. So it is clear to me that there is some classification. What I am asking is whether there is a classification using highest weight of the representation with respect to the restricted root system of the real Lie algebra. $\endgroup$ – ronggang Dec 10 '13 at 21:16

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