On the realization of maximal simple types and epsilon factors of pairs

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 Gel$'$fand. 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 $\varepsilon$-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 behavior of the $\varepsilon$-factor under twisting of $\pi_1$ by tamely ramified quasi-characters. Our results generalize the special case $\pi_1=\check{\pi}_2$ totally wildly ramified, due to Bushnell and Henniart.