Proposition 5

Let $f$ be a continuous central function on G. Then $f$ is infinitely-differentiable if and only if the function $u \rightarrow |\langle \chi_u|f\rangle |$ is rapidly decreasing on $\widehat{G}$; in that case,

for all $g\in G$.

By Th. $1b)$, the function $\overline{f}$ is infinitely-differentiable if and only if the function $u \rightarrow \|u(\overline{f})\|_{\infty}$ is rapidly decreasing; but, by (20),

hence the first assertion, since the functions $d(u)$ and $\frac{1}{d(u)}$ are moderately increasing.

Assume that $f$ is infinitely-differentiable; by Th. $1a),f(g) =\sum_{u\in\widehat{G}}f_u(g)$ for all $g\in G$, so

$f_u(g) =\langle u(g)|u(f)\rangle =d(u$)Tr($u(g)^{-1}.u(f)$) $=d(u$)Tr$(u(g)^{-1}\langle \overline{\chi}_u|f\rangle \frac{\varepsilon_u}{d(u)})$

$=\langle \overline{\chi}_u|f\rangle$Tr($u(g)^{-1}$) $=\langle \overline{\chi}_u|f\rangle \overline{\chi}_u(g)$.

Hence, $f(g) =\sum_{u\in\widehat{G}}\langle \overline{\chi}_u|f\rangle \overline{\chi}_u(g)$; but, for all $u\in \widehat{G}$, the contragredient representation $u'$ of $u$ satisfies $\overline{\chi}_u=\chi_{u'}$ and the map $u \rightarrow u'$ is a permutation of $\widehat{G}$; so we also have $f(g) =\sum_{u\in\widehat{G}}\langle \chi_u|f\rangle \chi_u(g)$, hence the proposition.

$^8$ We embed End(E$_u$) in the product $F( \widehat{G}) =\prod_{v\in\widehat{G}}$ End(E$_v$) by associating to any

$A\in$ End(E$_u$) the family $(A_v)_{v\in\widehat{G}}$ such that $A_u= A$ and $A_v= 0$ for $v\not=u$.