[Paper Review] Wave packets in the Schwartz space of a reductive p-adic symmetric space

Patrick Delorme, Pascale Harinck|arXiv (Cornell University)|Mar 5, 2012

Advanced Algebra and Geometry16 references3 citations

TL;DR

This paper constructs wave packets in the Schwartz space of a reductive p-adic symmetric space H\G using families of tempered functions defined via Eisenstein integrals. By introducing types I, I', and II' families based on exponent and weak constant term conditions, the authors generalize real-differential methods of Harish-Chandra and Baaj–Clement to the p-adic setting, proving that such wave packets exist and are well-behaved in the Schwartz space framework.

ABSTRACT

We form wave packets in the Schwartz space of a reductive p-adic symmetric space for certain famillies of tempered functions. We show how to construct such families from Eisenstein integrals.

Motivation & Objective

• To define and characterize families of tempered functions on H\G, a reductive p-adic symmetric space, with controlled exponents and constant terms.
• To extend the theory of wave packets—previously developed for real reductive groups—to the p-adic setting.
• To establish a framework for constructing wave packets in the Schwartz space using Eisenstein integrals as generating families.
• To generalize the notion of 'tempered' functions on symmetric spaces via H-fixed linear forms on admissible G-modules.
• To provide a foundation for computing Fourier transforms in the Schwartz space, particularly in relation to the L2-Plancherel formula for spherical varieties.

Proposed method

• Introduces the space Atemp(H\G) as smooth, tempered functions that are generalized matrix coefficients of H-fixed linear functionals on admissible G-modules.
• Defines weak constant terms for elements of Atemp(H\G) using the theory of constant terms, following [W] and [L], [KT1].
• Classifies families of functions into types I, I', and II' based on exponent conditions and weak constant term constraints, analogous to [BaCD] in the real case.
• Uses Eisenstein integrals—constructed via [CD]—as key examples of such families, providing a concrete source of tempered functions.
• Applies techniques from Harish-Chandra’s theory and the structure of parabolic subgroups to control growth and decay in the Schwartz space.
• Establishes a Cartan decomposition and uses the norm function Nd and the function ΘG to compare growth on H\G and H′\G′, ensuring integrability and decay properties.

Experimental results

Research questions

• RQ1 How can wave packets be systematically constructed in the Schwartz space of a reductive p-adic symmetric space?
• RQ2 What conditions on exponents and weak constant terms ensure that a family of tempered functions supports a well-defined wave packet?
• RQ3 To what extent can the real-differential methods of Baaj–Clement and Harish-Chandra be adapted to the p-adic setting?
• RQ4 How do Eisenstein integrals serve as generating families for such wave packets in the p-adic context?
• RQ5 What is the role of the G′-conjugacy of maximal σ-split tori in ensuring uniformity of the wave packet construction across symmetric spaces?

Key findings

• Wave packets can be constructed in the Schwartz space of a reductive p-adic symmetric space H\G for certain families of tempered functions.
• Families of type II'—defined by exponent and weak constant term conditions—are shown to support wave packets via Theorem 4.6.
• The construction relies on Eisenstein integrals as explicit examples of such families, as established in Theorem 5.1.
• The existence of wave packets is proven using the structure of parabolic subgroups and the Cartan decomposition, particularly through control of the functions Nd and ΘG.
• A key inequality (6.16) is established: ΘL′((H ∩L′)l) is controlled by ΘL((H ∩L)l) and Nd((H ∩L)l), ensuring uniform decay across the symmetric space.
• The map P ↦ P ∩ G′ is a bijection between σ-parabolic subgroups of G and G′, which ensures compatibility of the structure across symmetric spaces and supports the wave packet construction.