Roots of a compact connected Lie group

Assume the Axiom of Choice (The Axiom of Choice). Let $G$ be a compact connected Lie group with maximal torus $T$, and put $\mathfrak{g}=\mathrm{Lie}G$, $\mathfrak{t}=\mathrm{Lie}T$. Complexification is over $\mathbb{R}$.

Circle convention. The lattice definition Character and cocharacter lattices uses $\mathbb{R}\mathrm{/}\mathbb{Z}$. Here its character values are transported to the multiplicative complex circle $\mathbb{U}=\{z\in \mathbb{C}:\mathrm{\mid }z\mathrm{\mid }=1\}$ by $E:\mathbb{R}\mathrm{/}\mathbb{Z}⟶\mathbb{U},[r]⟼e^{2\pi ir}\mathrm{.}$ The exponential addition law makes $E$ a homomorphism ($\exp (z+w)=\exp z\exp w$, and the complex exponential extends the real exponential), and its kernel is trivial by $\ker (\exp )=2\pi i\mathbb{Z}$, and $\exp z=\exp w$ exactly when $z-w\in 2\pi i\mathbb{Z}$. The modulus formula $\exp (x+iy)=e^x(\cos y+i\sin y)$, $\mathrm{\mid }\exp (x+iy)\mathrm{\mid }=e^x$, and $e^{i\pi }+1=0$ and surjectivity of the complex exponential The complex exponential maps $\mathbb{C}$ onto $\mathbb{C}\setminus \{0\}$ show that its image is all of $\mathbb{U}$: a logarithm of a unit-modulus number has real part zero. Thus $E$ is a continuous bijection from a compact circle to a Hausdorff circle, with continuous inverse. It identifies addition in $\mathbb{R}\mathrm{/}\mathbb{Z}$ with multiplication in $\mathbb{U}$. All scalar character values below use this identification; they are not elements of the additive group $\mathbb{C}$.

The smooth adjoint representation of $G$ (Adjoint is a smooth Lie-group representation) extends complex-linearly to ${\mathfrak{g}}_{\mathbb{C}}$. A root of $(G,T)$ is a nontrivial continuous character $\alpha :T\to \mathbb{U}$ whose weight space ${\mathfrak{g}}_{\alpha }=\{v\in {\mathfrak{g}}_{\mathbb{C}}:\mathrm{Ad}(t)v=\alpha (t)v\text{ for every }t\in T\}$ is nonzero. This is a character in $X^{\ast }(T)$ via $E^{-1}$. Write $\mathrm{\Phi }(G,T)$ for the set of roots. The trivial character is denoted $1$, and its weight space is denoted ${\mathfrak{g}}_0$ to agree with additive weight notation: ${\mathfrak{g}}_0=\{v:\mathrm{Ad}(t)v=v\text{ for every }t\in T\}\mathrm{.}$ The trivial character is not a root.

There is a finite direct sum decomposition ${\mathfrak{g}}_{\mathbb{C}}={\mathfrak{g}}_0\oplus {⨁}_{\alpha \in \mathrm{\Phi }(G,T)}{\mathfrak{g}}_{\alpha }\mathrm{.}$ Indeed, unitarizability under AC (Finite-dimensional compact-group representations are unitarizable) makes the commuting operators $\mathrm{Ad}(t)$ normal. The complex spectral theorem and simultaneous diagonalization (Complex spectral theorem: a normal endomorphism of a finite-dimensional complex inner product space has an orthonormal eigenbasis, and conversely, A family of diagonalisable endomorphisms of a finite-dimensional space is simultaneously diagonalisable if and only if its members commute pairwise) give a common eigenbasis. For a nonzero common eigenvector $v$, its eigenvalue function is $\alpha (t)=⟨\mathrm{Ad}(t)v,v⟩\mathrm{/}⟨v,v⟩$, with inner product linear in the first argument, so is continuous. The representation law makes it multiplicative and unitarity gives $\mathrm{\mid }\alpha (t)\mathrm{\mid }=1$. Only finitely many joint eigenvalue functions occur. Separating the trivial one gives exactly the displayed sum.

Differential convention. Continuous Lie-group homomorphisms are smooth under countable choice (Continuous homomorphisms between Lie groups are smooth), which follows from the assumed AC. For $H\in \mathfrak{t}$ define $d\alpha (H)={\frac{d}{ds}\mid }_{s=0}\alpha ({\exp }_G(sH))\in i\mathbb{R},$ and extend this real-linear differential complex-linearly to ${\mathfrak{t}}_{\mathbb{C}}$. The displayed derivative is taken only for real $H\in \mathfrak{t}$; no exponential in the real group is applied to a general complex tangent vector. If $a=E^{-1}\circ \alpha$ is the additive circle-valued character and $d{a}_e:\mathfrak{t}\to \mathbb{R}$, then $d\alpha =2\pi id{a}_e$ on $\mathfrak{t}$. We often also denote $d\alpha$ by $\alpha$, explicitly switching from the group character to its differential.

Differentiating the weight equation and using The differential of Ad is ad gives $[H,v]=d\alpha (H)v$, first for real $H$, then by complex linearity for $H\in {\mathfrak{t}}_{\mathbb{C}}$. Restricted to the real form ${\mathfrak{t}}_{\mathbb{R}}=i\mathfrak{t}$, this functional is real-valued since $d\alpha (iH)=id\alpha (H)$. It is nonzero for a root: naturality of the exponential (Exponential map is natural for Lie-group homomorphisms) shows that a character with zero differential is trivial on an exponential identity neighborhood, using The exponential map is a local diffeomorphism at zero. Its kernel is then an open subgroup of connected $T$, hence all of $T$. Applying this to the quotient of two characters also shows that their differentials determine them uniquely. No global torus lattice classification is needed here.

• The zero weight space is the centralizer of ${\mathfrak{t}}_{\mathbb{C}}$: one direction follows by differentiation; in the other direction Adjoint exponential identity makes all ${\exp }_G(H)$, $H\in \mathfrak{t}$, act trivially on the vector, and these generate $T$ by local invertibility and connectedness. Every root character is trivial on $T\cap Z(G)$, since conjugation by such an element is the identity, and its differential vanishes on $\mathfrak{t}\cap \mathfrak{z}(\mathfrak{g})$ by the bracket formula.
• If $G$ is a torus, its adjoint action is trivial and $\mathrm{\Phi }(G,T)$ is empty. A trivial maximal torus also has no nontrivial characters, so the root set is empty. These cases do not introduce a zero root.
• For $T^{\mathrm{\prime }}=gT{g}^{-1}$, the map ${\mathrm{Ad}}_g$ sends ${\mathfrak{g}}_{\alpha }$ to the weight space for $\alpha \circ C_{g^{-1}}{\mathrm{\mid }}_{T^{\mathrm{\prime }}}$, by the representation law. Conjugacy of maximal tori (Conjugacy of maximal tori) therefore identifies these root sets and weight spaces for different maximal tori.