论文标题
在西格尔庞加莱系列的家族中
On a family of Siegel Poincaré series
论文作者
论文摘要
令$γ$为$ \ mathrm {sp} _ {2n}（\ mathbb z）$的一致子组。使用$ k $ -finite矩阵系列的POINCARé系列，包括$ \ Mathrm {sp} _ {2n}（\ Mathbb r）$的可集成离散系列表示的系列，我们为Space $ s_m（γ）$ siegel cusp $ m m \ mathbb z $ s的space $ s_m（γ）$构建一个跨度集。我们证明了使用Muić在本地紧凑的Hausdorff组上使用Muić的庞加莱系列中不可或缺的非逐渐划分标准来证明该跨度集的某些元素。此外，使用$ \ mathrm {sp} _ {2n}（\ mathbb r）$的表示理论，我们研究了彼得森内部产品的相应cuspidal自动形态形式，从而恢复了代表性的证据，以理论证明了在重现$ s_m（γ）kernel kernel功能的一些众所周知的结果。我们的结果是通过Muić在上半平面上的霍明型尖尖的工作中开发的，从而获得了较高程度的siegel cusp形式的设置，从而获得了我们的结果。
Let $ Γ$ be a congruence subgroup of $ \mathrm{Sp}_{2n}(\mathbb Z) $. Using Poincaré series of $ K $-finite matrix coefficients of integrable discrete series representations of $ \mathrm{Sp}_{2n}(\mathbb R) $, we construct a spanning set for the space $ S_m(Γ) $ of Siegel cusp forms of weight $ m\in\mathbb Z_{>2n} $. We prove the non-vanishing of certain elements of this spanning set using Muić's integral non-vanishing criterion for Poincaré series on locally compact Hausdorff groups. Moreover, using the representation theory of $ \mathrm{Sp}_{2n}(\mathbb R) $, we study the Petersson inner products of corresponding cuspidal automorphic forms, thereby recovering a representation-theoretic proof of some well-known results on the reproducing kernel function of $ S_m(Γ) $. Our results are obtained by generalizing representation-theoretic methods developed by Muić in his work on holomorphic cusp forms on the upper half-plane to the setting of Siegel cusp forms of a higher degree.
