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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:
 2.2e+51 Representations of Lie and linear algebraic groups over local fields
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##### References:
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