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Dualizing involutions on the metaplectic \(\operatorname{GL}(2)\). (English) Zbl 1441.22027
Summary: Let \(F\) be a non-Archimedean local field of characteristic zero. Let \(G=\operatorname{GL}(2, F)\) and \(\widetilde{G}=\widetilde{\operatorname{GL}}(2, F)\) be the metaplectic group. Let \(\tau\) be the standard involution on \(G\). A well known theorem of Gelfand and Kazhdan says that the standard involution takes any irreducible admissible representation of \(G\) to its contragredient. In such a case, we say that \(\tau\) is a dualizing involution. In this paper, we show that any lift of the standard involution to \(\widetilde{G}\) is also a dualizing involution.
MSC:
22E50 Representations of Lie and linear algebraic groups over local fields
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