Highest weights for compact connected groups

We spell out the connectedness and scalar arguments. By [L6] any continuous finite-dimensional representation is smooth and satisfies $\pi (\exp X)=\exp (d\pi (X))$. An exponential neighborhood generates a connected group: the generated subgroup is open, and all its cosets are open, so its complement is open and connectedness forces it to be the whole group. Thus a subspace invariant under the differentiated representation is group invariant, and a linear map intertwining differentials intertwines the group representations, by the power series for the matrix exponential and generation. Conversely group invariance and intertwining differentiate. Finally any endomorphism commuting with an irreducible complex representation is scalar: it has an eigenvalue, and the nonzero eigenspace is invariant, hence the whole space. In particular every central group element acts by a scalar. These scalars form a continuous character; for a compact subgroup its values have modulus one, since a compact subgroup of the positive real numbers is trivial (the powers of any other element are unbounded in one direction).

Use the cover $p:Z(G{)}^0\times S\to G$ of [L4], where $S=G_{\mathrm{d}\mathrm{e}\mathrm{r}}^{\mathrm{s}\mathrm{c}}$ is compact semisimple and simply connected, and put $C=\ker p$. The central torus $Z(G{)}^0$ lies in the fixed maximal torus $T$: the product $Z(G{)}^{0}T$ is compact, connected, and abelian, hence a torus by [L7], so maximality of $T$ forces equality with $T$. The closed subgroup $p^{-1}(T)$ is embedded by [L7]. Its identity component $K$ has abelian Lie algebra because $dp$ identifies it with $\mathfrak{t}$; exponential neighborhoods generate $K$, so $K$ is abelian and hence a torus by [L7]. Since $Z(G{)}^0\times \{e\}\subseteq K$, one has $K=Z(G{)}^0\times T_S,T_S:=K\cap (\{e\}\times S)\mathrm{.}$ Indeed, multiplication by the inverse of the first component puts every element of $K$ in the second factor. The image $p(K)$ is a connected subgroup of $T$ with Lie algebra $\mathfrak{t}$, so exponential charts make it open in connected $T$ and hence equal to $T$. The torus $T_S$ is maximal in $S$, for a larger torus would make the image of $Z(G{)}^0$ times that torus a torus properly containing $T$. Finally every $c\in C$ is central in the compact connected product. By [L7] it lies in some maximal torus, and conjugacy carries that torus to $K$; centrality fixes $c$ under the conjugation, so $c\in K=Z(G{)}^0\times T_S$. Thus characters of this product torus split into their two restrictions, and every pullback of a character on $T$ is trivial on $C$.

Let V be an irreducible finite-dimensional continuous G-module. The central torus acts by a character ${\mu }_0$ by step 1.1. Its differentiated central algebra is scalar. Therefore restriction to ${\mathfrak{s}}_{\mathbb{C}}$ is irreducible: an invariant subspace would be invariant under the entire differentiated algebra, and then under G by step 1.1. By [L1] it has a dominant highest weight ${\lambda }^{\mathrm{\prime }}$ and by [L3] a one-dimensional highest-weight space. The entire ${\mathfrak{t}}_{\mathbb{C}}$ preserves this line, since the central part acts by scalars; step 1.1 for connected T makes it T-invariant. T acts on it by a continuous character $\lambda \in X^{\ast }(T)$ (its image has modulus one by compactness). Its central restriction is ${\mu }_0$ and its semisimple differential is ${\lambda }^{\mathrm{\prime }}$, so it is dominant.

Conversely take a dominant $\lambda \in X^{\ast }(T)$ and write its pullback under step 1.2 as ${\mu }_0\otimes {\lambda }_S$. By [L2], ${\lambda }_S$ corresponds to a dominant weight in P. By [L1] it gives a finite-dimensional irreducible complex ${\mathfrak{s}}_{\mathbb{C}}$-module V. Restrict its action to the real algebra $\mathfrak{s}$ and apply [L5] with target $\mathrm{GL}(V)$ regarded as a real Lie group. This produces a smooth representation $\rho$ of S with the specified differential. It is irreducible by step 1.1, since a real-algebra invariant complex subspace is invariant under its complexification. The highest-weight line is $T_S$-invariant by the same exponential argument; its character has the prescribed differential and thus is exactly ${\lambda }_S$ by [L2].

Define $\tilde{\pi }(z,s)={\mu }_0(z)\rho (s)$. This product representation is irreducible because its S-restriction is irreducible. For $c\in C$, centrality and step 1.1 show that $\tilde{\pi }(c)$ is scalar. Since c lies in the product torus, evaluate that scalar on the nonzero highest line: it equals $({\mu }_0\otimes {\lambda }_S)(c)=(p^{\ast }\lambda )(c)=1$. Thus the representation factors through G. Covering charts make the descended homomorphism smooth and hence continuous; its invariant subspaces are exactly those of its surjective pullback, so it remains irreducible. Its highest character is $\lambda$, since the pullback on the product torus is the one just constructed and that torus maps onto T.

Two irreducible G-modules with the same highest character have the same central character and the same semisimple highest weight by step 2.1. The classification [L1] supplies a semisimple-algebra intertwining isomorphism. The central algebra acts by the same scalars, so this is an intertwiner for the full differentiated representations; step 1.1 makes it a G-intertwiner. Conversely an isomorphism preserves highest characters. When the semisimple algebra is zero the scalar argument gives a one-dimensional module, and the construction simply returns each torus character; when G is trivial it returns only the trivial character and its one-dimensional representation. The zero highest character gives the trivial representation, which is unique by this argument.

Step 2.1 assigns a dominant character to every irreducible module, step 3.1 realizes each dominant character, and step 3.2 proves uniqueness and invariance under equivalence. These are the claimed inverse bijections. The Axiom of Choice supplies the assumptions of [L1]–[L6].