Exercise 9, § 3

Let $\mathfrak{g}$ be a compact Lie algebra, $\mathfrak{t}$ a Cartan subalgebra of $\mathfrak{g}$; let $(X_{\alpha})_{\alpha\in R}$ be a Chevalley system in the split reductive algebra $(\mathfrak{g}_{\mathbf{C}},\mathfrak{t}_{\mathbf{C}})$ such that $X_{\alpha}$ and $X_{-\alpha}$ are conjugate (with respect to $\mathfrak{g}$) for all $\alpha \in R$.

a) Let $\mathscr{T}$ be a $\mathbf{Z}$-submodule of $\mathfrak{t}_{\mathbf{C}}$ containing the $iH_{\alpha}(\alpha \in R)$ and such that $\alpha (\mathscr{T})\subset \mathbf{Z}i$ for all $\alpha \in R$. Show that the $\mathbf{Z}$-submodule $\mathscr{G}$ of $\mathfrak{g}_{\mathbf{C}}$ generated by $\mathscr{T}$ and the elements $u_{\alpha}$ and $v_{\alpha}(\alpha \in R)$ is a $\mathbf{Z}$-Lie subalgebra of $\mathfrak{g}$.

b) Assume that $\mathfrak{g}$ (resp. $\mathfrak{t}$) is the Lie algebra of a compact group G (resp. of a maximal torus T of G). Let $\Gamma (T)$ be the kernel of the homomorphism exp$_T:\mathfrak{t}\rightarrow T$. Show that the $\mathbf{Z}$-module $(2\pi )^{-1}\Gamma (T)$ satisfies the hypotheses of a).

c) Let $\langle ,\rangle$ be an invariant scalar product on $\mathfrak{g}$; let $\mu$ (resp. $\tau$ ) be the Haar measure on $\mathfrak{g}$ (resp. $\mathfrak{t}$) corresponding to the Lebesgue measure when $\mathfrak{g}$ (resp. $\mathfrak{t}$) is identified with a space $\mathbf{R}^n$ by means of an orthonormal basis. Denote also by $\mu$ (resp. $\tau$ ) the measure on $\mathfrak{g}/\mathscr{G}$ (resp. $\mathfrak{t}/\mathscr{T}$) that is the quotient of $\mu$ (resp. $\tau$ ) by the normalized Haar measure on $\mathscr{G}$ (resp. $\mathscr{T}$). Prove the formula $\mu(\mathfrak{g}/\mathscr{G}) =\tau (\mathfrak{t}/\mathscr{T}).\prod_{\alpha\in R_+}\langle iH_{\alpha},iH_{\alpha}\rangle$.