We find the number sk⁢(p,Ω)s_{k}(p,\Omega) of cuspidal automorphic representations of GSp⁢(4,AQ)\mathrm{GSp}(4,\mathbb{A}_{\mathbb{Q}}) with trivial central character such that the archimedean component is a holomorphic discrete series representation of weight k≥3k\geq 3, and the non-archimedean component at 𝑝 is an Iwahori-spherical representation of type Ω and unramified otherwise. Using the automorphic Plancherel density theorem, we show how a limit version of our formula for sk⁢(p,Ω)s_{k}(p,\Omega) generalizes to the vector-valued case and a finite number of ramified places.

中文翻译：

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关于计数GSp（4）的尖峰自构表示

我们发现GSp⁢（4，AQ）\ mathrm {GSp}（4，\ mathbb {A} _ {\ mathbb {Q}}），其中心特征是琐碎的，使得阿基米德分量是权重k≥3k\ geq 3的全纯离散序列表示，而at处的非阿基米德分量是Ω类型的Iwahori球面表示，否则不加分叉。使用自构的Plancherel密度定理，我们显示sk⁢（p，Ω）s_ {k}（p，\ Omega）的公式的极限形式如何推广到向量值的情况和有限数量的分支位置。