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Asymptotics of Schwartz functions

Chun-Hsien Hsu

TL;DR

This paper builds a comprehensive Fourier-analytic framework for Braverman–Kazhdan spaces X_P associated to maximal parabolics P of G (classical groups or G_2). It proves that the Getz–Hsu–Leslie Schwartz space S(X_P(F)) coincides with the Braverman–Kazhdan definition S_BK(X_P(F)) in both nonarchimedean and archimedean settings, and establishes locality and a robust Fourier theory, including global Poisson summation. A central contribution is the precise asymptotic description A_{X_P(F)} of functions near the origin, expressed via socles of degenerate principal series, with an explicit filtration that encodes the semisimplification process of these series. The analysis unifies nonarchimedean and archimedean aspects through Igusa-type Mellin theory, gives explicit examples (line bundles on Grassmannians, odd cones, and Lagrangian Grassmannians), and connects asymptotics to an algebra of differential operators (Weyl algebra) on X_P, enabling both spectral and geometric interpretations. The results have implications for harmonic analysis on spherical varieties and for Poisson summation programs, offering concrete realizations of L-functions and intertwiners in terms of the geometry of X_P.

Abstract

Let $G$ be a split, simply connected, almost simple algebraic group, and let $P$ be a maximal parabolic subgroup of $G$. Braverman and Kazhdan in \cite{BKnormalized} defined a Schwartz space on the affine closure $X_P$ of $P^{\mathrm{der}}\backslash G$. An alternate, more analytically tractable definition was given in \cite{Getz:Hsu:Leslie}, following several earlier works. When $G$ is a classical group or $G_2$, we show the two definitions coincide and prove several previously conjectured properties of the Schwartz space that will be useful in applications. Along the way, we give an alternative construction of the ring of differential operators on $X_P$ using the Fourier theory. We also establish the Poisson summation formulae in these cases.

Asymptotics of Schwartz functions

TL;DR

This paper builds a comprehensive Fourier-analytic framework for Braverman–Kazhdan spaces X_P associated to maximal parabolics P of G (classical groups or G_2). It proves that the Getz–Hsu–Leslie Schwartz space S(X_P(F)) coincides with the Braverman–Kazhdan definition S_BK(X_P(F)) in both nonarchimedean and archimedean settings, and establishes locality and a robust Fourier theory, including global Poisson summation. A central contribution is the precise asymptotic description A_{X_P(F)} of functions near the origin, expressed via socles of degenerate principal series, with an explicit filtration that encodes the semisimplification process of these series. The analysis unifies nonarchimedean and archimedean aspects through Igusa-type Mellin theory, gives explicit examples (line bundles on Grassmannians, odd cones, and Lagrangian Grassmannians), and connects asymptotics to an algebra of differential operators (Weyl algebra) on X_P, enabling both spectral and geometric interpretations. The results have implications for harmonic analysis on spherical varieties and for Poisson summation programs, offering concrete realizations of L-functions and intertwiners in terms of the geometry of X_P.

Abstract

Let be a split, simply connected, almost simple algebraic group, and let be a maximal parabolic subgroup of . Braverman and Kazhdan in \cite{BKnormalized} defined a Schwartz space on the affine closure of . An alternate, more analytically tractable definition was given in \cite{Getz:Hsu:Leslie}, following several earlier works. When is a classical group or , we show the two definitions coincide and prove several previously conjectured properties of the Schwartz space that will be useful in applications. Along the way, we give an alternative construction of the ring of differential operators on using the Fourier theory. We also establish the Poisson summation formulae in these cases.
Paper Structure (54 sections, 69 theorems, 464 equations)

This paper contains 54 sections, 69 theorems, 464 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Poisson summation formulae
  4. Outline of the paper
  5. Acknowledgments
  6. Preliminaries
  7. Quasi-characters
  8. Measures
  9. Schwartz spaces
  10. Analytic notation
  11. Multisets
  12. Fourier theory on Braverman-Kazhdan spaces
  13. Induced representations and intertwining operators
  14. The Schwartz space
  15. The Fourier transform $\mathcal{F}_{P|P^{\mathrm{op}}}$
  16. ...and 39 more sections

Key Result

Theorem 1.2

Suppose $G$ is classical or $G_2$. We have In particular, $\mathcal{S}(X_P^\circ(F))<\mathcal{S}(X_P(F))$. Moreover, $\mathcal{S}(X_P(F))$ is local.

Theorems & Definitions (132)

  • Conjecture 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 2.1
  • Lemma 2.2
  • proof
  • Definition 2.3
  • ...and 122 more