Abstract Let F be a non-Archimedean local field of characteristic zero. Let G = GL ( 2 , F ) and G ˜ = 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 show that any lift of the standard involution to G ˜ is also a dualizing involution.

中文翻译：

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对元对合对合 GL(2)

摘要 令 F 为特征为零的非阿基米德局部场。设 G = GL ( 2 , F ) 和 G ～ = GL ～ ( 2 , F ) 为元组。令 τ 是 G 的标准对合。 Gelfand 和 Kazhdan 的一个众所周知的定理说标准对合将 G 的任何不可约的可容许表示取到它的逆参。在这种情况下，我们说 τ 是一个二元对合。在本文中，我们表明标准对合对 G 的任何提升也是对偶对合。