Let
$G$
be an affine Kac-Moody group,
${{\pi }_{0}},...,{{\pi }_{r}},{{\pi }_{\delta }}$
its fundamental irreducible representations and
${{\chi }_{0}},...,{{\chi }_{r}},{{\chi }_{\delta }}$
their characters. We determine the set of all group elements
$x$
such that all
${{\pi }_{i}}(x)$
act as trace class operators, i.e., such that
${{\chi }_{i}}(x)$
exists, then prove that the
${{\chi }_{i}}$
are class functions. Thus, (
$\chi \,:=\,({{\chi }_{0}},...,{{\chi }_{r}},\,{{\chi }_{\delta }})$
)factors to an adjoint quotient
$\bar{\chi }$
for
$G$
. In a second part, following Steinberg, we define a cross-section
$C$
for the potential regular classes in
$G$
. We prove that the restriction
$\chi \text{ }\!\!|\!\!\text{ }c$
behaves well algebraically. Moreover, we obtain an action of
${{\mathbb{C}}^{\times }}$
on
$C$
, which leads to a functional identity for
$\text{ }\chi |\text{ c}$
which shows that
$\text{ }\chi |\text{ c}$
is quasi-homogeneous.