Exercise 4, § 8

Choose a separating positive quadratic form Q on $\mathfrak{g}$ invariant under Ad(G). For $x\in \mathfrak{t}$, put $\vartheta_0(x) =\sum_{u\in\Gamma(T)}e^{-Q(x+u)}$.

a) Show that $\vartheta_0$ is a function of class $C^{\infty}$ on $\mathfrak{t}$, and that there exists a function $\vartheta_1$ of class $C^{\infty}$ on T such that $\vartheta_1\circ$ exp$_T=\vartheta_0$. b) Show that there exists a unique central function $\vartheta$ on G, of class $C^{\infty}$, whose restriction to T is equal to $\vartheta_1$. For every maximal torus S of G and all $x\in L(S)$, we have

$\vartheta$(exp $x$) $=\sum_{u\in\Gamma(S)}e^{-Q(x+u)}$.

c) Let A be an alcove of $\mathfrak{t},dx$ a Haar measure on $\mathfrak{t}$, and $h$ a locally integrable function on $\mathfrak{t}$ invariant under the affine Weyl group $W_a'($§5, no. 2). Prove the equality

d) For $x\in \mathfrak{g}$, put $\xi (x) =\lambda_{\mathfrak{g}}(x)e^{-Q(x)}(\vartheta$(exp $x$))$^{-1}$, with $\lambda_{\mathfrak{g}}(x) =$ det$\frac{e^{\mathrm{ad}\ x}-1}{\mathrm{ad}\ x}$ (§6, no. 3). Show that $\xi$ is a function of class $C^{\infty}$ on $\mathfrak{g}$ and that if $\mu$ is a Haar measure on $\mathfrak{g}$, the image under exp$_G$ of the measure $\xi \mu$ is a Haar measure on G (use $c$) as well as Cor. 2 of Th. 1 and Prop. 4 (§6, no. 2 and 3)).

e) Prove the formula $\vartheta_0(x) =m\sum_{\lambda\in X(T)}$ exp($\delta (\lambda )(x)-^1_4Q'(\delta (\lambda ))$) for $x\in \mathfrak{t}$, where $Q'$ is the quadratic form on $\mathfrak{t}^*_{\mathbf{C}}$ inverse to the quadratic form on $\mathfrak{t}_{\mathbf{C}}$ induced by Q and where $m$ is a constant that should be calculated (use Poisson’s formula, cf. Spectral Theory, Chap. II, §1, no. 8). Deduce the equality $\vartheta (t) =m\sum_{\lambda\in X(T)}e^{-Q'(\delta(\lambda))/4}t^{\lambda}$ for $t\in T$.