Classification of irreducible representations via Casimirs

Physicists almost always label irreducible representations via Casimirs (e.g., characterizing the irreducible representations of $SO(3)$ by spin). I've been looking far and wide to see the general statement that justifies this labeling made precise, so that, in particular, I would have some idea of why this works and in what cases it won't work. I have already checked several books on representation theory (e.g. Fulton and Harris, Humphreys, Knapp), but have found nothing. Could someone point me in the right direction?

Thanks in advance.

Solution 1:

I am using Humphreys as my source.

I don't think that the Casimir element alone is sufficient. At least I don't see a reason why it should (at this time - I'm a bit rusty on this topic). The relevant concept in Humphreys is the action of the center $\frak{Z}$ of the universal enveloping algebra. By Schur's lemma any element $c$ of $\frak{Z}$ acts as a scalar $f_V(c)$ on an irreducible representation $V$. Thus an irreducible module gives us a character $f_V:\frak{Z}\to \mathbb{C}$. The results there state that the characters related to standard cyclic modules of two highest weights are equal, if and only if the two weights are in the same orbit of the Weyl group under the so called dot-action (=the usual action conjugated with a translation by the sum of fundamental dominant weights). This is the essence of Harish-Chandra theorem. Note that within such orbits there is ever only a single dominant weight. Therefore if we restrict ourselves to finite dimensional representations (that are parametrized by the dominant highest weights), the character by which $\frak{Z}$ acts on the module is sufficient information to distinguish one irreducible representation from another.

We know that the universal Casimir element $c_L$ belongs $\frak{Z}$. If two characters of $\frak{Z}$ agree on $c_L$, then they necessarily agree on the subalgebra of $\frak{Z}$ generated by $c_L$. So the question is whether the restriction of the character to this subalgebra gives enough information to distinguish one finite dimensional rep from another. I don't think that it does in general. IIRC the number of independent generators that you need to get all of $\frak{Z}$ equals the rank of the root system. That suggests that the subalgebra generated by $c_L$ cannot be large enough unless that rank is equal to one. In other words, $c_L$ is enough in the case of $SO(3)$ and its universal cover $SU(2)$, but is not enough, if we have a rank $\ge2$ root system.

This is a bit vague, sorry about that. I do think that this is about what's happening.

Solution 2:

To point you in the right direction:

Try checking Casimir's thesis (1931) and, in particular, Chapter III.

(Don't worry about the beginning; the dissertation is in English!)