Corollary

Let $f$ be a continuous central function on G. Then $f$ is infinitely-differentiable if and only if the restriction of $f$ to T is infinitely-differentiable.

Indeed, by Cor. 4 of §7, no. 4,

$\langle \chi_u|f\rangle =\int_G\overline{\lambda(u)(t)}\varphi (t)dt$, where $\varphi (t) =\prod_{\alpha >0}(1-\alpha (t)^{-1})f(t)$.

If $f|T$ is infinitely-differentiable, so is $\varphi$; by Prop. 5, applied to the group T, the function $\mu \rightarrow \int_T\overline{\mu(t)}\varphi (t)dt$ on $\widehat{T} = X(T)$ is then rapidly decreasing, and so is the function $u \rightarrow \langle \chi_u|f\rangle$; hence the function $f$ is infinitely-differentiable (Prop. 5). The converse is clear.