2022/10/13 by Sonja Žunar, Žunar, Sonja
Mathematics · #11F03 #11F46 #Advanced Algebra and Geometry #Advanced Mathematical Identities #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2210.07192
openalex publication_date 2022/10/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let Γ be a congruence subgroup of Sp2n(\mathbb Z) . Using Poincaré series of K -finite matrix coefficients of integrable discrete series representations of Sp2n(\mathbb R) , we construct a spanning set for the space Sm(Γ) of Siegel cusp forms of weight m∈\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 Sp2n(\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 Sm(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.