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BZSV Duality for Some Strongly Tempered Spherical Varieties

Zhengyu Mao, Chen Wan, Lei Zhang

Abstract

We propose two families of relative trace formula comparisons in the study of relative Langlands duality conjectured by Ben-Zvi--Sakellaridis--Venkatesh. This allows us to incorporate numerous relative trace formula comparisons studied during the last four decades under the BZSV duality framework. For the proposed relative trace formula comparisons associated to some strongly tempered spherical varieties, we will prove the fundamental lemma and smooth transfer in the $p$-adic case. Moreover, inspired by the BZSV duality conjecture, we propose a conjecture regarding the degenerate Whittaker period, which generalizes Lapid-Mao's conjecture of the Whittaker period

BZSV Duality for Some Strongly Tempered Spherical Varieties

Abstract

We propose two families of relative trace formula comparisons in the study of relative Langlands duality conjectured by Ben-Zvi--Sakellaridis--Venkatesh. This allows us to incorporate numerous relative trace formula comparisons studied during the last four decades under the BZSV duality framework. For the proposed relative trace formula comparisons associated to some strongly tempered spherical varieties, we will prove the fundamental lemma and smooth transfer in the -adic case. Moreover, inspired by the BZSV duality conjecture, we propose a conjecture regarding the degenerate Whittaker period, which generalizes Lapid-Mao's conjecture of the Whittaker period
Paper Structure (18 sections, 6 theorems, 120 equations, 1 figure)

This paper contains 18 sections, 6 theorems, 120 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The relative Langlands duality conjecture
  3. A conjecture for degenerate Whittaker period
  4. A conjectural relative trace formula comparison for quadruples not related to central values
  5. A relative trace formula for rank 1 spherical varieties
  6. Organization of the paper
  7. Acknowledgement
  8. BSV conjecture for the models in Table \ref{['fig:1']}
  9. The relative trace formula on ${\mathrm{SL}}_2$
  10. The relative trace formula for Model 1-4
  11. The relative trace formula
  12. The proof of the fundamental lemma and smooth transfer
  13. The relative trace formula for Model 5 and 6
  14. The relative trace formula
  15. The proof of the fundamental lemma and smooth transfer
  16. ...and 3 more sections

Key Result

Proposition 3.3

Figures (1)

  • Figure :

Theorems & Definitions (31)

  • Conjecture 1.1
  • Remark 1.2
  • Remark 1.3
  • Conjecture 1.4
  • Remark 1.5
  • Remark 1.7
  • Conjecture 1.8
  • Conjecture 1.9
  • Conjecture 1.11
  • Conjecture 1.12
  • ...and 21 more