Root and weight lattice sandwich

Each root $\alpha$ is an honest character of $T$ by [L1], so $\mathbb{Z}\alpha \subseteq X^{\ast }(T)$ for each root and hence $Q\subseteq X^{\ast }(T)$.

If $\chi \in X^{\ast }(T)$, then for every root $\alpha$ with cocharacter ${\alpha }^{\vee }$ the composite $\chi \circ {\alpha }^{\vee }:S^1\to S^1$ is $z\mapsto z^n$ for some $n\in \mathbb{Z}$, and $n=⟨\chi ,{\alpha }^{\vee }⟩$; and by [L6] and the normalization in [L1] this integer is $d\chi (i{h}_{\alpha })\mathrm{/}i=d\chi (h_{\alpha })=(\chi ,{\alpha }^{\vee })$; hence $(\chi ,{\alpha }^{\vee })\in \mathbb{Z}$ for every root and $\chi \in P$. Thus $X^{\ast }(T)\subseteq P$.

Since $Q$ and $P$ have full rank by [L2], a basis of $Q$ is a nonsingular integer matrix in a basis of $P$, so $[P:Q]<\mathrm{\infty }$ by [L8]. In rank zero both are zero and the index is one.

Let $p:\tilde{G}\to G$ be the simply connected covering with discrete central kernel $\mathrm{\Gamma }$ from [L3]. Choose a symmetric relatively compact identity neighborhood $U$. The subgroup it generates is open, so equals connected $\tilde{G}$. Finitely many sets $p(a_{i}U)$ cover compact $G$. Include $a_i=e$ and set $C={\bigcup }_i{a}_i\bar{U}$, so $\tilde{G}=C\mathrm{\Gamma }$ and $e\in C$. For $\gamma =u_1\cdots u_n\in \mathrm{\Gamma }$, write the partial products $u_1\cdots u_j=c_j{\gamma }_j$ with $c_j\in C$, ${\gamma }_j\in \mathrm{\Gamma }$, choosing endpoints $(c_0,{\gamma }_0)=(e,e)$ and $(c_n,{\gamma }_n)=(e,\gamma )$. Centrality gives ${\gamma }_j{\gamma }_{j-1}^{-1}=c_j^{-1}{c}_{j-1}{u}_j$. The intersection $\mathrm{\Gamma }\cap C^{-1}C\bar{U}$ is finite: it is a closed discrete subset of a compact set. It generates $\mathrm{\Gamma }$ by multiplying these increments. Thus $\mathrm{\Gamma }$ is finitely generated abelian.

Suppose $\mathrm{\Gamma }$ were infinite. Its decomposition in [L5] then has a nonzero free summand, so it has subgroups $K$ with finite indices $\mathrm{\mid }\mathrm{\Gamma }\mathrm{/}K\mathrm{\mid }$ arbitrarily large (reduce one free coordinate modulo any positive integer). Each $K$ is closed and central in $\tilde{G}$, so $G_K=\tilde{G}\mathrm{/}K$ is a connected Lie group by [L7]. The induced map $p_K:G_K\to G$ is a finite covering with kernel $D=\mathrm{\Gamma }\mathrm{/}K$: an evenly covered neighborhood for p descends to disjoint sheets indexed by $\mathrm{\Gamma }\mathrm{/}K$. Hence $G_K$ is compact by [L5], with the same semisimple Lie algebra. Let $T_K$ be maximal. Every central element of $D$ lies in $T_K$, since [L5] puts it in some maximal torus and conjugacy moves that torus to $T_K$ while fixing the element. The image $T=p_K(T_K)$ is a closed connected abelian Lie subgroup, hence a torus. It is maximal: if $T^{\mathrm{\prime }}$ contains it, the inverse image of $\mathrm{Lie}(T^{\mathrm{\prime }})$ under the Lie-algebra isomorphism $d{p}_K$ is abelian and contains ${\mathfrak{t}}_K$. The latter is maximal abelian, since exponentiating a larger abelian algebra and taking its compact connected closure would give a larger torus. Thus $\mathrm{Lie}(T^{\mathrm{\prime }})=\mathrm{Lie}(T)$; exponential charts make $T$ open in connected $T^{\mathrm{\prime }}$, so $T^{\mathrm{\prime }}=T$.

For the adjoint endpoint put $G_{ad}=G\mathrm{/}Z(G)$. The center is closed, so this quotient is a compact connected Lie group by [L7]. Its Lie algebra is the original semisimple algebra: the center has zero Lie algebra, because a vector whose exponentials are central has zero adjoint bracket and semisimplicity is centerless. The adjoint action of $G_{ad}$ is faithful, since an element of connected G acting trivially on its Lie algebra commutes with every exponential and hence with the group they generate. Write a character $\chi$ of the maximal torus $T_{ad}$ of the adjoint group in the basis of simple roots: $\chi ={\sum }_j{c}_j{\alpha }_j$ with $c_j\in \mathbb{R}$. Let $H_j\in \mathfrak{t}$ be defined by ${\alpha }_i(H_j)=2\pi i{\delta }_{ij}$. Every root has integral simple-root coordinates, so $\mathrm{Ad}(\exp H_j)$ acts trivially on the root spaces and on the torus, and since its adjoint action is faithful we get $\exp H_j=1$; analytic integrality of $\chi$ then forces $\chi (H_j)=2\pi i{c}_j\in 2\pi i\mathbb{Z}$ for every $j$, so all $c_j$ are integers and $\chi \in Q$. Together with step 1.1 this gives $X^{\ast }(T_{ad})=Q$.

Identify the two torus Lie algebras by $d{p}_K$. Naturality gives exponential lattices ${\mathrm{\Lambda }}_K\subseteq \mathrm{\Lambda }$, and surjectivity of ${\exp }_{T_K}$ identifies $D$ with $\mathrm{\Lambda }\mathrm{/}{\mathrm{\Lambda }}_K$. If a basis of ${\mathrm{\Lambda }}_K$ is the matrix A in a basis of $\mathrm{\Lambda }$, [L8] gives $\mathrm{\mid }D\mathrm{\mid }=\mathrm{\mid }\det A\mathrm{\mid }$. With character coordinates $d\chi (Y_j)\mathrm{/}(2\pi i)$, restriction of characters from T to $T_K$ is the integer matrix $A^{\mathsf{T}}$, so [L1, L8] give $[X^{\ast }(T_K):X^{\ast }(T)]=\mathrm{\mid }D\mathrm{\mid }$. Rank zero has trivial lattices and D and needs no determinant theorem. The roots agree under the differential by [L6], so steps 1.1–1.2 give $Q\subseteq X^{\ast }(T)\subseteq X^{\ast }(T_K)\subseteq P$. Thus $\mathrm{\mid }D\mathrm{\mid }\le [P:Q]$, contrary to step 2.1 and step 1.3. Consequently $\mathrm{\Gamma }$ is finite and $G_{sc}=\tilde{G}$ is compact by [L5].

Every dominant integral element of $P$, in particular every fundamental weight, is the highest weight of a finite-dimensional representation of ${\mathfrak{g}}_{\mathbb{C}}$ by [L4]; restricting the infinitesimal representation to the compact real form $\mathrm{Lie}(G_{sc})$ and integrating it by [L3] gives a finite-dimensional representation of the simply connected compact group $G_{sc}$, whose highest weight line is invariant under $T_{sc}$: for $H\in {\mathfrak{t}}_{sc}$ its exponential acts on that line by $e^{\lambda (H)}$ by [L7], and those exponentials generate $T_{sc}$. The line therefore defines a continuous homomorphism $T_{sc}\to {\mathbb{C}}^{\times }$. Its image is compact, so its modulus is identically one (a nontrivial positive modulus has unbounded positive or negative powers). It is a character with differential that fundamental weight; hence $P\subseteq X^{\ast }(T_{sc})$ and, with step 1.2, $X^{\ast }(T_{sc})=P$.

Steps 1.1 and 1.2 give the sandwich $Q\subseteq X^{\ast }(T)\subseteq P$ for every compact connected semisimple $G$, and steps 2.2 and 4.1 realise the two endpoints; the Axiom of Choice entered only through the integration and covering theory used.