Plancherel and Poisson summation formulae for a family of affine $\Psi$-bundles

Technical Summary: Plancherel and Poisson Summation Formulae for a Family of Affine $\Psi$-Bundles

Problem and Setting
This paper addresses the Poisson summation conjecture for a specific family of singular spaces arising in the context of triple product $L$-functions. The conjecture, originally posed by Braverman and Kazhdan and refined by Ngô and Sakellaridis, posits the existence of a Schwartz space, a Fourier transform satisfying twisted equivariance, and a Poisson summation formula for spherical varieties.

The authors focus on a family of affine $\Psi$-bundles over a specific spherical variety. Let $F$ be a number field and $r \geq 3$. The underlying geometric object is the space $X^\circ_r = N \backslash GL_r$, where $N$ is the unipotent radical of a standard parabolic subgroup. This is paired with the space $M^\circ$ of rank 1 matrices in $M_{r-2,2}$. The authors consider the product $X^\circ = X^\circ_r \times M^\circ$ and its fibers $X^\circ_\ell$ over points $\ell \in \mathbb{P}^1(F)$. Over these fibers, they study Hermitian line bundles $H$ (affine $\Psi$-bundles) whose sections satisfy a specific transformation law involving a character $\psi$ and a pairing on the matrix components.

The primary motivation for this specific choice of space is its application to the integral representation of triple product $L$--functions explored in a companion paper by the authors and Leslie [GGHL25]. In that context, proving the analytic continuation of these $L$-functions requires a pair of Poisson summation formulae: one for a vector space (known) and one for the singular space $X^\circ_\ell$ studied here.

Methodology
The authors employ a spectral method, utilizing known properties of automorphic representations and $L$-functions to deduce geometric results. The core technical strategy involves the following steps:

1. Local Harmonic Analysis: The authors first establish a Plancherel formula for the Harish-Chandra space $C(X^\circ_\ell(F_v), H)$ at a local place $v$. This involves defining the space of functions, constructing local zeta integrals, and relating them to Whittaker models of representations of $GL_{r,r-2}$.
2. Zeta Integrals and Bounds: A significant portion of the work is dedicated to proving the absolute convergence and establishing precise bounds for local zeta integrals $Z(f, W, \chi)$. These integrals are defined using the $\Psi$-bundle structure and Whittaker functions. The authors prove that these integrals converge for rapidly decreasing functions and establish their meromorphic continuation.
3. Matrical Plancherel Theorem: By combining the local Plancherel formula for $GL_{r,r-2}$ with the structure of the zeta integrals, the authors derive a "matrical" Plancherel theorem for the space $X^\circ_\ell(F_v)$. This theorem provides a topological isomorphism between the Harish-Chandra space of the bundle and a space of sections over the tempered spectrum of $GL_{r,r-2}$.
4. Fourier Transform Construction: Using the Plancherel isomorphism, the authors define a local Fourier transform. This transform is constructed by describing its image under the Plancherel map in terms of $\gamma$-factors associated with the relevant $L$-functions.
5. Asymptotic Schwartz Spaces: The authors define an asymptotic Schwartz space $Sas(X_\ell(F_v), H)$ adapted to the geometry of the space. They prove that the basic function (a specific test function defined via characteristic functions of compact subgroups) lies in this space and is $\Xi$-rapidly decreasing.
6. Global Unfolding: To prove the global Poisson summation formula, the authors unfold a global zeta integral involving a cusp form and the Schwartz function. They utilize the spectral decomposition of the automorphic kernel and the functional equations of the local zeta integrals (derived from the local Fourier transform) to relate the sum over rational points to the sum over the Fourier transform.

Key Contributions and Results

• Plancherel Formula for Affine $\Psi$-Bundles: The paper proves a Plancherel formula for the Harish-Chandra space of a family of affine $\Psi$-bundles over singular spherical varieties (Theorem 1.1, Theorem 8.6). This is notable because the Plancherel decomposition involves admissible representations of arbitrarily high rank ($GL_{r,r-2}$), distinguishing it from previous results on spherical varieties which often involve lower-rank groups or specific monoids.
• Construction of Fourier Transforms: The authors construct local and global Fourier transforms for these spaces (Theorem 9.2, Theorem 10.2). These transforms are shown to be topological isomorphisms on the asymptotic Schwartz spaces.
• Poisson Summation Formula: Under suitable assumptions (A1 and A2) regarding the cuspidal image and rapid decay of the test functions, the authors prove the Poisson summation formula:
  $$\sum_{x \in X^\circ_\ell(F)} f(x) = \sum_{x \in X^\circ_\ell(F)} \mathcal{F}(f)(x)$$
  (Theorem 1.2, Theorem 12.7).
• Technical Estimates: The paper provides rigorous bounds for Whittaker functions and zeta integrals in both Archimedean and non-Archimedean settings, including estimates for non-tempered representations and the behavior of the basic function.

Significance and Claims
The authors claim that this work represents the first case of the Poisson summation conjecture proved for singular spaces whose Plancherel decomposition involves admissible representations of arbitrarily high rank.

The paper explicitly states that the technique of passing back and forth between spectral and geometric data to deduce results about spectral data is ubiquitous in automorphic representation theory. In this specific setting, the authors use known properties of $L$-functions (specifically their analytic continuation and functional equations) to deduce the Poisson summation formula. They note that this approach is somewhat circular in the sense that Poisson summation is often used to prove properties of $L$-functions, but they argue that the technique is valid and necessary for their intended application in [GGHL25].

The authors acknowledge that they do not know how to construct the Fourier transform or prove the Poisson summation formula without using the Plancherel formula and the spectral method. They also note that while they assume the test functions satisfy strong conditions (A1 and A2) to simplify the proof, they believe many functions in the Schwartz space satisfy the rapid decay condition (A2), though a full Paley-Wiener theorem for the open orbit remains a future goal.

In summary, the paper provides a rigorous harmonic analytic framework for a specific class of singular spaces, establishing the necessary Plancherel and Poisson summation tools required to advance the study of triple product $L$-functions via the fiber bundle method.