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Relative Character Asymptotics Beyond Stability for $\mathrm{PGL}_2 \times \mathrm{GL}_1$

Trajan Hammonds

Abstract

The asymptotics of relative characters for real Lie groups were studied for representations $(π, σ)$ arising from Gan-Gross-Prasad pairs $(G,H)$ by Nelson and Venkatesh. They successfully compute the asymptotics of relative characters whenever the conductor of the associated Rankin-Selberg $L$-function $L(π\boxtimes σ^\vee)$ lies in a stable locus, i.e. away from conductor dropping. In this paper, we express asymptotics for relative characters in the non-archimedean setting for $(\mathrm{PGL}_2, \mathrm{GL}_1)$. The key new innovation is that our method overcomes the stability hypothesis and allows for significant conductor dropping.

Relative Character Asymptotics Beyond Stability for $\mathrm{PGL}_2 \times \mathrm{GL}_1$

Abstract

The asymptotics of relative characters for real Lie groups were studied for representations arising from Gan-Gross-Prasad pairs by Nelson and Venkatesh. They successfully compute the asymptotics of relative characters whenever the conductor of the associated Rankin-Selberg -function lies in a stable locus, i.e. away from conductor dropping. In this paper, we express asymptotics for relative characters in the non-archimedean setting for . The key new innovation is that our method overcomes the stability hypothesis and allows for significant conductor dropping.
Paper Structure (22 sections, 21 theorems, 193 equations, 1 figure, 1 table)

This paper contains 22 sections, 21 theorems, 193 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Sketch of the Proof
  3. Organization of the Paper
  4. Preliminaries
  5. Whittaker and Kirillov Models
  6. $\mathrm{GL}_1$ Iwasawa-Tate Theory
  7. Local Representations
  8. $\mathrm{GL}_2$ Hecke-Jacquet-Langlands Theory
  9. Test Functions and $\operatorname{Op}$ Calculus
  10. Test Functions
  11. $\operatorname{Op}$ Calculus
  12. Invariant Functionals and Relative Characters
  13. Analysis of the Relative Character
  14. The Image of $\operatorname{Op}(a_\tau)$
  15. Negative Unipotent & Functional Equations
  16. ...and 7 more sections

Key Result

Theorem 1.3

Let $\pi$ be a principal series or supercuspidal unitary representation of $G$ and $\chi$ a unitary character of $H$. If $\pi$ is principal series $\pi = \chi_0 \boxplus \chi_0^{-1}$, assume that $\chi \neq \chi_0, \chi_0^{-1}$ or an unramified twist. Let $a \ \colon \mathrm{Lie}(G)^\ast \to \mathbf

Figures (1)

  • Figure 1: Relative Coadjoint Orbit

Theorems & Definitions (46)

  • Example 1.2
  • Theorem 1.3
  • Remark 1.4
  • Lemma 1
  • proof
  • Corollary 2.1
  • Proposition 2.2
  • Remark 2.3
  • Proposition 2.4
  • Remark 2.5
  • ...and 36 more