2006/03/02 by Vytautas Paškūnas, Paskunas, Vytautas, Shaun Stevens +1 · 1 citation
Mathematics · #22E50 #Advanced Algebra and Geometry #Advanced Topics in Algebra #FOS: Mathematics #Finite Group Theory Research #Number Theory (math.NT) #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.math/0603051
openalex publication_date 2006/03/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 π1, π2 of G, when π1 and the contragredient of π2 differ only at the `tame level' (more precisely, π1 and \checkπ2 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=\checkπ2 totally wildly ramified, due to Bushnell and Henniart.