  Let $G$ be the group of rational points of a general linear group over a
non-archimedean local field $F$. We show that certain representations of open,
compact-mod-centre subgroups of $G$, (the maximal simple types of Bushnell and
Kutzko) can be realized as concrete spaces. In the level zero case our result
is essentially due to Gelfand. This allows us, for a supercuspidal
representation $\pi$ of $G$, to compute a distinguished matrix coefficient of
$\pi$. By integrating, we obtain an explicit Whittaker function for $\pi$. We
use this to compute the epsilon factor of pairs, for supercuspidal
representations $\pi_1$, $\pi_2$ of $G$, when $\pi_1$ and the contragredient of
$\pi_2$ differ only at the `tame level' (more precisely, $\pi_1$ and
$\check{\pi}_2$ contain the same simple character). We do this by computing
both sides of the functional equation defining the epsilon factor, using the
definition of Jacquet, Piatetskii-Shapiro, Shalika. We also investigate the
behaviour of the epsilon factor under twisting of $\pi_1$ by tamely ramified
quasi-characters. Our results generalise the special case $\pi_1=\check{\pi}_2$
totally wildly ramified, due to Bushnell and Henniart.
