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Generalised Whittaker models as instances of relative Langlands duality II: Plancherel density and global periods

Wee Teck Gan, Bryan Wang Peng Jun

Abstract

In an earlier paper of the authors, a general family of instances of the relative Langlands duality of Ben-Zvi-Sakellaridis-Venkatesh [BZSV] were proposed and studied in the setting of branching problems for smooth representations. In this paper, we show the numerical conjectures of [BZSV] for the local Plancherel density, as well as an application to their conjectures on global periods, for this general family of instances.

Generalised Whittaker models as instances of relative Langlands duality II: Plancherel density and global periods

Abstract

In an earlier paper of the authors, a general family of instances of the relative Langlands duality of Ben-Zvi-Sakellaridis-Venkatesh [BZSV] were proposed and studied in the setting of branching problems for smooth representations. In this paper, we show the numerical conjectures of [BZSV] for the local Plancherel density, as well as an application to their conjectures on global periods, for this general family of instances.
Paper Structure (16 sections, 12 theorems, 78 equations)

This paper contains 16 sections, 12 theorems, 78 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Results of GW
  4. Main result
  5. Method of proof
  6. Preliminaries
  7. The case under consideration
  8. Notation
  9. Measures
  10. Spectral decomposition
  11. Smooth setting
  12. Plancherel theorems
  13. Key argument
  14. Inner product
  15. Plancherel density of basic function
  16. ...and 1 more sections

Key Result

Theorem 1.2

The numerical conjecture for the local Plancherel density BZSV is true for the hyperspherical dual pair $(M_1,M_2)$. In other words, the local Plancherel density associated to (the quantization of) $M_2$ corresponds to a local $L$-value attached to $M_1$, and vice versa.

Theorems & Definitions (21)

  • Theorem 1.2
  • Remark 2.2
  • Theorem 3.1
  • Proposition 3.2
  • Proposition 3.3
  • Remark 3.4
  • Theorem 3.5
  • Theorem 3.6
  • Remark 3.7
  • Proposition 4.1
  • ...and 11 more