Dimension of maximal tori

Question

Let $G$ be a compact Lie group. $T$ $-$ its maximal torus. Is there a simple reasoning to show that dimensions of $T$ and $G$ have the same parity? I am sorry if this quesion is for children, but nevertheless.

Answer 1

Yes. To show Weyl Jacobian is positive definite we show that Ad$_{G/T}(t^{-1})-1$ acting on $\mathfrak{g}/\mathfrak{t}$ with $\langle t \rangle$ dense in $T$ has only conjugate pairs as its eigenvalues. Therefore $\dim(G/T) = \dim(\mathfrak{g}/\mathfrak{t})$ is even.

The proof consists of three steps:

(1) $\mbox{Ad}_{G/T}(t^{-1}) - 1$ acting on $\mathfrak{g}/\mathfrak{t}$ with $\langle t \rangle$ dense in $T$ has only conjugate pairs as its eigenvalues is equivalent to Ad$_{G/T}(t^{-1})$ does not have $\pm 1$ eigenvalue since it is an orthogonal transformation thus all of its eigenvalues are in $S^1$, or alternatively treat the eigenvalues as functions on $T$ which is compact

(2) this can be translated into $\mbox{Ad}_{G/T}(t^2)$ having no nonzero fixed vector in $\mathfrak{g}/\mathfrak{t}$, which is equivalent to $(\mathfrak{g}/\mathfrak{t})^{t} = 0$ as $t \mapsto t^2$ is a surjection on $T$

(3) since $\langle t \rangle$ is dense in $T$ this implies $(\mathfrak{g}/\mathfrak{t})^T = 0$, to prove it show $((\mathfrak{g}/\mathfrak{t})_{\mathbb{C}})^T = (\mathfrak{g}_{\mathbb{C}}^T)/\mathfrak{t}_{\mathbb{C}}$ using complexification (because we need complete reducibility to isotope pieces), then we have $((\mathfrak{g}/\mathfrak{t})^T)_{\mathbb{C}} = (\mathfrak{g}^T/\mathfrak{t})_{\mathbb{C}}$ hence $(\mathfrak{g}/\mathfrak{t})^T = 0$ is equivalent to $\mathfrak{g}^T = \mathfrak{t}$ which follows from $\mathfrak{g}^T = \text{Lie}(Z_G(T)) = \text{Lie}(Z_G(T)^0) = \text{Lie}(N_G(T)^0)$ and $N_G(T)^0 = T$

Reference: Stanford University MATH 210C Brian Conrad's handout on Weyl Jacobian

Answer 2

A conceptual reason is that $G/T$ has a complex structure. This is easiest to see in examples: for example, if $G = U(n), T = U(1)^n$, then $G/T$ is the flag variety of complete flags in $\mathbb{C}^n$. Another conceptual reason is that $G/T$ has a symplectic structure, which arises due to the fact that it is a coadjoint orbit.