Let F be a non-Archimedean local field of characteristic zero. Let G = GL(2, F) and $3\widetildeG = \widetilde{GL}(2,F)$ be the metaplectic group. Let τ 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 τ is a dualizing involution. In this paper, we make some modifications and adapt a topological argument of Tupan to the metaplectic group $\widetildeG$ and give an elementary proof that any lift of the standard involution to $\widetildeG$ ; is also a dualizing involution.

中文翻译：

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METAPLECTIC GL(2) à la TUPAN 的双重化

让F是特征为零的非阿基米德局部场。让G= GL(2,F） 和$3\widetildeG = \widetilde{GL}(2,F)$是元理组。让τ成为标准对合G. Gelfand 和 Kazhdan 的一个著名定理说，标准对合取G与其相反。在这种情况下，我们说τ是对偶对合。在本文中，我们进行了一些修改，并将 Tupan 的拓扑论证适用于 metaplectic 群$\宽波浪号G$并给出一个基本证明，证明标准对合的任何提升$\宽波浪号G$; 也是一个二元对合。