Let $\mathfrak{g}$ be a semisimple finite dimensional Lie algebra and $V_\lambda$, resp. $V_\mu$
its finite dimensional highest weight modules with highest weights $\lambda$, resp. $\mu$. Let
$\chi_\lambda, \chi_\mu : C(\mathcal{U}(\mathfrak{g})) \rightarrow \mathbb{C}$
be the corresponding central characters. Harish-Chandra theorem asserts that $\chi_\lambda = \chi_\mu$ if and only if
$w(\lambda+\delta)-\delta = \lambda$ for some $w$ in the Weyl group, where
$\delta$ is the Weyl vector, i.e. the sum of all fundamental weights.
Is it also true in this setting, that $V_\lambda \simeq V_\mu$ if and only if $\chi_\mu = \chi_\nu$ ?
How is this version of the theorem related to the fact that Harish-Chandra homomorphism is an isomorphism ?
Thank you very much for your answers. (I am studying a program in mathematical physics and trying to figure out
how general is the procedure of labeling irreducible representations by values of Casimir operators, as physicists do so often)
Best Answer
If I interpret your question as asking: are irred. finite-dimensional rep's determined by their infinitesimal character (i.e. by the eigenvalues of the centre of the enveloping algebra on them) the answer is yes. As you essentially observe, if one has such an irrep. $V$, and you want to write it as $V_{\lambda}$, you can realize $\lambda$ as the unique dominant weight that $\chi_V$ (the central character of $V$) is equal to $\chi_{\lambda+\delta}$. (I am not sure what normalization you are using, but if you are using the normalized HC isomorphism, the one that identifies the centre of the enveloping algebra with $W$-invariants in the enveloping algebra of the Cartan, then $\chi_V$ will be the homomorphism corresponding to the character $\chi+\delta$ of the Cartan.)
The proof of this statement is closely tied up with the proof of the HC isomorphism (at least, with the proofs that I know). I learned the HC isomorphism from Knapp's overview by examples book, and I think I learned this fact, and its relationship to the HC isomorphism, from that book.
By the way, if you have a physics background, then you probably know one case of this: for the spherical harmonics, the Casimir (= spherical Laplacian) eigenvalue determines the irrep. of $SO(3)$ to which a given spherical harmonic belongs.