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Are the irreducible finite-dimensional modules of the lie algebra $\frak{so}(n)$ uniquely determined by dimension? If so, can one then conclude that all finite-dimensional modules of $\frak{o}(n)$ are self-dual?

Abo Kutis-Felan
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Alesandro Levi
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  • Your question should be edited one more time: "Are the irreducible finite-dimensional modules..." – Moishe Kohan Feb 03 '18 at 10:41
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    For the second question: If you work over ${\mathbb C}$: If $n$ is even then $w_0\ne −1$, hence, the opposition involution $\iota$ is a nontrivial automorphism of the Dynkin diagram, hence, for each even $n$ there are non-self-dual irreducible representations. They are given by highest weights λ such that ι(λ)≠λ. But for all odd $n$, the opposition involution is trivial, hence, each irreducible representation is self-dual. Over real numbers, every representation of a compact Lie algebra is self-dual. – Moishe Kohan Feb 03 '18 at 10:55
  • Over the reals there are many examples for the first question as well, see https://mathoverflow.net/questions/53955/which-compact-groups-have-nonisomorphic-irreducible-representations-of-the-same – Moishe Kohan Feb 03 '18 at 15:35

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