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Fourier-Jacobi expansion of automorphic forms generating quaternionic discrete series

Hiro-aki Narita

TL;DR

This work develops a comprehensive Fourier-Jacobi theory for automorphic forms on adjoint simple groups admitting quaternionic discrete series, organizing the analysis around Heisenberg parabolic subgroups and cubic norm structures. It provides explicit archimedean realizations of generalized Whittaker functions for Schrödinger representations, derives reduction to tractable differential equations, and establishes multiplicity formulas for Fourier-Jacobi models. A key achievement is proving the Koecher principle for automorphic forms generating quaternionic discrete series, showing that nonzero central-character Fourier-Jacobi coefficients do not contribute to the discrete spectrum in the Jacobi group, and that cusp forms lie in the continuous spectrum for ξ≠0; the adelic theory yields detailed expansions with Theta-series data and a robust framework for Pollack’s cusp forms. Collectively, these results give structural insight into the Fourier-Jacobi expansion, enabling explicit realizations and applications to explicit constructions via Pollack’s cusp forms, while clarifying the spectral composition of Jacobi-type coefficients.

Abstract

We provide a theory of the Fourier-Jacobi expansion for automorphic forms on simple adjoint groups of some general class. This theory respects the Heisenberg parabolic subgroups, whose unipotent radicals are the Heisenberg groups uniformly explained in terms of the notion of cubic norm structures. Based on this theory of the Fourier expansion, we prove that automorphic forms generating quaternionic discrete series representations automatically satisfy the moderate growth condition except for the cases of the group of $G_2$-type and special orthogonal groups of signature $(4,N)$. This should be called ``Köcher principle'' verified already for the case of the quaternion unitary group $Sp(1,q)$ for $q>1$ by the author. We also prove that every term of the Fourier expansion with a non-trivial central character for cusp forms generating quaternionic discrete series has no contribution by the discrete spectrum of the Jacobi group, which is a non-reductive subgroup of the Heisenberg parabolic subgroup. This is obtained by showing that generalized Whittaker functions of moderate growth for the Schrödinger representations are zero under some assumption of the separation of variables, which suffices for our purpose to establish such consequence.

Fourier-Jacobi expansion of automorphic forms generating quaternionic discrete series

TL;DR

This work develops a comprehensive Fourier-Jacobi theory for automorphic forms on adjoint simple groups admitting quaternionic discrete series, organizing the analysis around Heisenberg parabolic subgroups and cubic norm structures. It provides explicit archimedean realizations of generalized Whittaker functions for Schrödinger representations, derives reduction to tractable differential equations, and establishes multiplicity formulas for Fourier-Jacobi models. A key achievement is proving the Koecher principle for automorphic forms generating quaternionic discrete series, showing that nonzero central-character Fourier-Jacobi coefficients do not contribute to the discrete spectrum in the Jacobi group, and that cusp forms lie in the continuous spectrum for ξ≠0; the adelic theory yields detailed expansions with Theta-series data and a robust framework for Pollack’s cusp forms. Collectively, these results give structural insight into the Fourier-Jacobi expansion, enabling explicit realizations and applications to explicit constructions via Pollack’s cusp forms, while clarifying the spectral composition of Jacobi-type coefficients.

Abstract

We provide a theory of the Fourier-Jacobi expansion for automorphic forms on simple adjoint groups of some general class. This theory respects the Heisenberg parabolic subgroups, whose unipotent radicals are the Heisenberg groups uniformly explained in terms of the notion of cubic norm structures. Based on this theory of the Fourier expansion, we prove that automorphic forms generating quaternionic discrete series representations automatically satisfy the moderate growth condition except for the cases of the group of -type and special orthogonal groups of signature . This should be called ``Köcher principle'' verified already for the case of the quaternion unitary group for by the author. We also prove that every term of the Fourier expansion with a non-trivial central character for cusp forms generating quaternionic discrete series has no contribution by the discrete spectrum of the Jacobi group, which is a non-reductive subgroup of the Heisenberg parabolic subgroup. This is obtained by showing that generalized Whittaker functions of moderate growth for the Schrödinger representations are zero under some assumption of the separation of variables, which suffices for our purpose to establish such consequence.
Paper Structure (13 sections, 35 theorems, 142 equations)

This paper contains 13 sections, 35 theorems, 142 equations.

Table of Contents

  1. Introduction
  2. Generalized Whittaker functions and Fourier-Jacobi models for quaternionic discrete series representations
  3. Cubic norm structures and associated groups
  4. Jacobi groups and their irreducible unitary representations
  5. Generalized Whittaker functions
  6. Generalized Whittaker functions for Schrödinger representations
  7. Reduction of the differential equations (Proof of Proposition \ref{['Diff-eq_general-2']})
  8. Results on the generalized Whittaker functions and Fourier-Jacobi models
  9. Fourier-Jacobi expansion (in the adelic formulation)
  10. A general theory
  11. Automorphic forms generating quaternionic discrete series and Köcher principle
  12. Application: cusp forms constructed by Pollack
  13. The non-adelic Fourier-Jacobi expansion of cusp forms generating quaternionic discrete series

Key Result

Theorem 1.1

For $\xi\in{\mathbb Q}\setminus\{0\}$ let $F_{\xi}$ be as above for a general automorphic form $F$ on ${\cal G}({\mathbb A})$. For a character $\chi$ of ${\cal N}({\mathbb Q})\backslash{\cal N}({\mathbb A})$, $F_{\chi}$ denotes the Fourier transformation of $F$ by $\chi$. Using the coordinate $n(w,t with where $\chi'_{\xi}$ (respectively $\chi"_{\xi}$) runs over characters of ${\cal N}({\mathbb Q

Theorems & Definitions (62)

  • Theorem 1.1: Theorem \ref{['adelic-FJ-exp']}
  • Corollary 1.2: Corollary \ref{['Determination-by-zeroth-coeff']}
  • Theorem 1.3: Köcher principle
  • Theorem 1.4: Theorem \ref{['FJ-model']}
  • Theorem 1.5: Theorem \ref{['adelicFJ-exp-QDS']}
  • Proposition 2.1
  • proof
  • Definition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • ...and 52 more