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On the non-vanishing of Poincaré series on irreducible bounded symmetric domains

Sonja Žunar

TL;DR

The paper develops a vector-valued non-vanishing criterion for Poincaré series on irreducible bounded symmetric domains, connecting holomorphic automorphic forms of polynomial type to K-finite matrix coefficients of integrable discrete series via a precise lift. Using Harish-Chandra realization and explicit holomorphic/discrete series, it establishes an isomorphism between spaces of holomorphic automorphic forms and weight components in L^2 of the quotient, then applies Muić’s integral non-vanishing criterion to obtain concrete conditions ensuring non-vanishing of Poincaré series. The main result provides a verifiable inequality on the F_f lift over a suitable S subset of the positive Weyl chamber, guaranteeing nontrivial automorphic Poincaré series and their lifts for finite-covolume subgroups. The AIII (SU(p,q)) specialization yields explicit non-vanishing thresholds N_0 for arithmetic subgroups, with determinant-based Poincaré series and practical computations illustrated for small ranks, highlighting the method’s applicability to explicit families of automorphic forms.

Abstract

Let $ \mathcal D\equiv G/K $ be an irreducible bounded symmetric domain. Using a vector-valued version of Muić's integral non-vanishing criterion for Poincaré series on locally compact Hausdorff groups, we study the non-vanishing of holomorphic automorphic forms on $ \mathcal D $ that are given by Poincaré series of polynomial type and correspond via the classical lift to the Poincaré series of certain $ K $-finite matrix coefficients of integrable discrete series representations of $ G $. We provide an example application of our results in the case when $ G=\mathrm{SU}(p,q) $ and $ K=\mathrm S(\mathrm U(p)\times\mathrm U(q)) $ with $ p\geq q\geq1 $.

On the non-vanishing of Poincaré series on irreducible bounded symmetric domains

TL;DR

The paper develops a vector-valued non-vanishing criterion for Poincaré series on irreducible bounded symmetric domains, connecting holomorphic automorphic forms of polynomial type to K-finite matrix coefficients of integrable discrete series via a precise lift. Using Harish-Chandra realization and explicit holomorphic/discrete series, it establishes an isomorphism between spaces of holomorphic automorphic forms and weight components in L^2 of the quotient, then applies Muić’s integral non-vanishing criterion to obtain concrete conditions ensuring non-vanishing of Poincaré series. The main result provides a verifiable inequality on the F_f lift over a suitable S subset of the positive Weyl chamber, guaranteeing nontrivial automorphic Poincaré series and their lifts for finite-covolume subgroups. The AIII (SU(p,q)) specialization yields explicit non-vanishing thresholds N_0 for arithmetic subgroups, with determinant-based Poincaré series and practical computations illustrated for small ranks, highlighting the method’s applicability to explicit families of automorphic forms.

Abstract

Let be an irreducible bounded symmetric domain. Using a vector-valued version of Muić's integral non-vanishing criterion for Poincaré series on locally compact Hausdorff groups, we study the non-vanishing of holomorphic automorphic forms on that are given by Poincaré series of polynomial type and correspond via the classical lift to the Poincaré series of certain -finite matrix coefficients of integrable discrete series representations of . We provide an example application of our results in the case when and with .
Paper Structure (9 sections, 21 theorems, 98 equations, 2 tables)

This paper contains 9 sections, 21 theorems, 98 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Basic notation
  3. Representation-theoretic preliminaries
  4. Harish-Chandra's realization of irreducible bounded symmetric domains
  5. Holomorphic and anti-holomorphic discrete series of $G$
  6. Determination of matrix coefficients
  7. A space of holomorphic automorphic forms on $\mathcal{D}$
  8. A non-vanishing result
  9. The case of bounded symmetric domains of type $\textbf{A\,III}$

Key Result

Theorem 1.1

The space ${\mathcal{H}}_\rho^\infty(\Gamma)$ is the space of (absolutely and locally uniformly convergent) Poincaré series where $f$ goes over the space $\mathcal{P}({\mathcal{D}},V)$ of polynomial functions $\mathcal{D}\to V$.

Theorems & Definitions (37)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 3.1
  • Lemma 3.2
  • Remark 3.3
  • Proposition 3.4
  • proof
  • Lemma 5.1
  • proof
  • Lemma 5.2
  • ...and 27 more