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On the realization of maximal simple types and epsilon factors of pairs. (English) Zbl 1159.22009

Authors’ 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 \(\pi\) 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.

MSC:

22E50 Representations of Lie and linear algebraic groups over local fields
20G05 Representation theory for linear algebraic groups
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