On the realisation 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 Gelfand. This allows us, for a supercuspidal representation $π$ of $G$, to compute a distinguished matrix coefficient of $π$. By integrating, we obtain an explicit Whittaker function for $π$. We use this to compute the epsilon factor of pairs, for supercuspidal representations $π_1$, $π_2$ of $G$, when $π_1$ and the contragredient of $π_2$ differ only at the `tame level' (more precisely, $π_1$ and $\checkπ_2$ 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 $π_1$ by tamely ramified quasi-characters. Our results generalise the special case $π_1=\checkπ_2$ totally wildly ramified, due to Bushnell and Henniart.