In Knapp's book Representation Theory of Semisimple Groups: An Overview Based on Examples, he proves the following theorem of Weyl: If $G$ is a compact connected semisimple Lie group, then $\pi_1(G)$ is finite. I am confused about a detail in the proof, which I now explain.
First, Knapp shows (by exploiting compactness of $G$) that $\pi_1(G)$ is a finitely generated abelian group.
Let $\widetilde{G}$ be the universal cover of $G$, so that $G = \widetilde{G}/Z$ for a discrete subgroup $Z \subset Z(\widetilde{G})$ isomorphic to $\pi_1(G)$. Knapp assumes for the sake of contradiction that $Z$ is infinite. Then (since it is a finitely generated abelian group, so by the structure theory…) there exist subgroups $Z_1 \subset Z$ of arbitrarily large finite index.
Now for some notation and recall of facts. Let $\mathfrak{g}$ be the Lie algebra of $G$ (and fix a Cartan $\mathfrak{h}$ in $\mathfrak{g} \otimes \mathbf{C}$), and let $d$ be the index of the root lattice in the weight lattice for $\mathfrak{g} \otimes \mathbf{C}$ (in particular $d$ is a finite number). To make my notation (which comes from Fulton–Harris, which I studied much more carefully) consistent with Knapp, my "weight lattice" is what Knapp calls the "algebraically integral forms" — it is the set of $\lambda \in \mathfrak{h}^*$ such that $\langle \lambda, \alpha^\vee \rangle \in \mathbf{Z}$ for all roots $\alpha$.
Given a compact connected Lie group $G' = \widetilde{G}/Z'$ with Lie algebra $\mathfrak{g}$, we can consider the lattice $\Lambda_{G'}$ of “analytically integral weights for $G'$'', namely the $\lambda \in \mathfrak{h}^*$ such that $\lambda(X) = 1$ for all $X \in \mathfrak{h} \cap \mathfrak{g}$ with the property that $\exp_{G'}(X) = 1$. It is a fact that $\Lambda_{G'}$ contains the root lattice $\Lambda_R$ and is contained in the weight lattice $\Lambda_W$(this is Knapp (4.15)). Moreover, if $Z(\widetilde{G}) \supset Z'' \supset Z'$ and $G'' = \widetilde{G}/Z''$, then the number of sheets in the covering map $G' \to G''$ is $[\Lambda_{G'} : \Lambda_{G''}]$ (this is Knapp 4.25).
It seems to me that the rest of the proof of Weyl's theorem is a direct consequence of the above paragraph: choose $Z_1$ so that $[Z : Z_1] > d$. Since $[Z : Z_1]$ is the number of sheets in the covering of compact connected groups $\widetilde{G}/Z_1 \to \widetilde{G}/Z = G$, by the above paragraph we have $[\Lambda_{\widetilde{G}/Z_1} : \Lambda_{\widetilde{G}/Z}] > d$. This contradicts the inclusions $\Lambda_R \subset \Lambda_{\widetilde{G}/Z} \subset \Lambda_{\widetilde{G}/Z_1} \subset \Lambda_W$ and the fact that $[\Lambda_W : \Lambda_R] = d$.
However, the proof in Knapp doesn't quite finish here: he concludes (somehow) from this that $\widetilde{G}/Z_1$ is not a linear group, which contradicts the fact that it is compact (which is because $[Z : Z_1]$ is finite and $G$ is compact). Why is this extra step necessary, and why is it clear from this that $\widetilde{G}/Z_1$ is not linear ?
Best Answer
It seems to me that you are right and that the proof does not need the detour to linear Lie groups. I think the reason why Knapp writes the proof of his Theorem 4.26 in this way is that throughout paragraph IV.5 he assumes that G is a compact linear Lie group. Strictly speaking, Knapp shows the necessary results about lattices only for such groups.