Theorem 2 (H. Weyl)

For all $\lambda \in X_{++}$, we have $J(\rho ).\chi_{\lambda}|T = J(\lambda \rho )$.

The function $J(\rho ).\chi_{\lambda}|T$ is anti-invariant under W, and is a linear combination with integer coefficients of elements of X(T). Thus, by Chap. VI, §3, no. 3, Prop. 1, it can be written as $\sum_\mu a_\mu J(\mu\rho )$, where $\mu$ belongs to $X_{++}$, and the $a_\mu$ are integers all but finitely-many of which are zero; since $\int_G|\chi_{\lambda}(g)|^2dg= 1$

(Spectral Theories), it follows from Prop. 3 and Lemma 4 that $\sum_\mu(a_\mu)^2= 1$; thus, the $a_\mu$ are all zero, except one which must be equal to 1 or $-1$. But the coefficient of $\lambda$ in $\chi_{\alpha}|T$ is equal to 1 (Th. 1), hence the coefficient of $\lambda \rho$ in $J(\rho ).\chi_{\lambda}|T$ is equal to 1 (Chap. VI, §3, no. 3, Remark 2), which implies that $a_{\lambda \rho}= 1$, hence the theorem.