On a family of Siegel Poincar\'e series
Sonja \v{Z}unar

TL;DR
This paper constructs a spanning set for Siegel cusp forms using Poincaré series derived from discrete series representations, proving non-vanishing and analyzing inner products through representation theory, extending Muić's methods to higher degree forms.
Contribution
It generalizes Muić's representation-theoretic approach from holomorphic modular forms to Siegel cusp forms of higher degree, providing new constructions and insights.
Findings
Constructed a spanning set for Siegel cusp forms using Poincaré series.
Proved non-vanishing of certain Poincaré series elements.
Reproduced known results on the kernel function via representation theory.
Abstract
Let be a congruence subgroup of . Using Poincar\'e series of -finite matrix coefficients of integrable discrete series representations of , we construct a spanning set for the space of Siegel cusp forms of weight . We prove the non-vanishing of certain elements of this spanning set using Mui\'c's integral non-vanishing criterion for Poincar\'e series on locally compact Hausdorff groups. Moreover, using the representation theory of , 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 . Our results are obtained by generalizing representation-theoretic methods developed by Mui\'c in…
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Taxonomy
TopicsAdvanced Algebra and Geometry · Advanced Mathematical Identities · Algebraic Geometry and Number Theory
