Abstract:Let $\mathbf {G}$ be an algebraic group over a local field $\mathbf {k}$ of characteristic zero. We show that the locally compact group $\mathbf {G}(\mathbf {k})$ consisting of the $\mathbf {k}$-rational points of $\mathbf {G}$ is of type I. Moreover, we complete Lipsman’s characterization of the groups $\mathbf {G}$ for which every irreducible unitary representation of $\mathbf {G}(\mathbf {k})$ is a CCR representation and show at the same time that such groups belong to the family of trace class groups, recently studied by Deitmar and van Dijk.

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中文翻译：

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局部域上代数群的酉表示

摘要：令 $\mathbf {G}$ 是特征为零的局部域 $\mathbf {k}$ 上的代数群。我们证明由 $\mathbf {G}$ 的 $\mathbf {k}$-有理点组成的局部紧群 $\mathbf {G}(\mathbf {k})$ 是类型 I。此外，我们完整的 Lipsman 对群 $\mathbf {G}$ 的表征，其中 $\mathbf {G}(\mathbf {k})$ 的每个不可约幺正表示都是 CCR 表示，同时表明这些群属于跟踪类群家族，最近由 Deitmar 和 van Dijk 研究。

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