Abstract. 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 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 π₁, π₂ of G, when π₁ and the contragredient of π₂ differ only at the "tame level" (more precisely, π₁ and π₂^∨ 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 π₁ by tamely ramified quasi-characters. Our results generalise the special case π₁=π₂^∨ totally wildly ramified, due to Bushnell and Henniart.