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 Gel'fand. This allows us, for a supercuspidal representation p of G, to compute a distinguished matrix coefficient of p. By integrating, we obtain an explicit Whittaker function for p. We use this to compute the epsilon factor of pairs, for supercuspidal representations p1, p2 of G, when p1 and the contragredient of p2 differ only at the "tame level" (more precisely, p1 and p2? 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 p1 by tamely ramified quasi-characters. Our results generalise the special case p1=p2? totally wildly ramified, due to Bushnell and Henniart.