Exercise 3, § 6

Let $\mu$ be a Haar measure on the additive group $\mathfrak{g}$.

a) There exists a unique Haar measure $\mu_G$ on G with the following property: if $\omega_G$ and $\omega_{\mathfrak{g}}$ are invariant differential forms of degree $n$ on G and $\mathfrak{g}$, respectively, such that $\omega_G(e) =\omega_{\mathfrak{g}}(0)$ and $|\omega_{\mathfrak{g}}|=\mu$, then $|\omega_G|=\mu_G$. The map $\mu \rightarrow \mu_G$ is a bijection from the set of Haar measures on $\mathfrak{g}$ to the analogous set for G. b) Choose a scalar product $(|)$ invariant under $\mathfrak{g}$, and a Haar measure $\tau$ on $\mathfrak{t}$ such that $\mu$ (resp. $\tau$ ) corresponds to the Lebesgue measure when $\mathfrak{g}$ is identified with $\mathbf{R}^n$ (resp. $\mathfrak{t}$ with $\mathbf{R}^r$) by means of an orthonormal basis. Prove the formula

c) Identify X(T) with a subset of the Hilbert space $\mathfrak{t}^*$ by means of the map $(2\pi i)^{-1}\delta ($§4, no. 2), and put $P(x) =\prod_{\alpha\in R_+}\langle \alpha , x\rangle$ for $x\in \mathfrak{t}$. With the notations

of Exerc. 1 and 2, show that

$l$

$\frac{\tau_T(T)}{\mu_G(G)}= (2\pi )^Nw(G)^{-1}(P|P) =\pi^N\prod(\alpha |\alpha )\prod m_i$!

$\alpha \in R_+i=1$ d) With the notations of Exerc. 9 of §3, let $\mathfrak{g}_{\mathbf{Z}}$ be the $\mathbf{Z}$-Lie subalgebra of $\mathfrak{g}$ generated by $(2\pi )^{-1}\Gamma (T)$ and the elements $u_{\alpha}, v_{\alpha}$ for $\alpha \in R$. Prove the equality

$\mu_G(G) =\mu(\mathfrak{g}/\mathfrak{g}_{\mathbf{Z}})2\prod^r_i\pi m^{N+}_i$!$^r$.

e) Assume that G is simply-connected. Show that $l=r$ and that

$\mu_G(G) = 2^{l/2}\pi^{-N}f^{1/2}\prod_{\alpha\in R_+}(\alpha |\alpha )^{-1}\prod_{i=1}^l(\alpha_i|\alpha_i)^{-1/2}\prod_i(m_i$!)$^{-1}$,

where $\{\alpha_1, . . . , \alpha_l\}$ is a basis of R and $f$ is the connection index of R.

f) Assume further that R is irreducible and that all its roots are of the same length; take the scalar product on $\mathfrak{g}$ to be the opposite of the Killing form. Prove that $\mu_G(G) = (2\pi )^{N+r}(2h)^{n/2}f^{1/2}\prod_i(m_i$!)$^{-1}$.