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Period integrals of distinguished polarised strongly tempered hyperspherical varieties

Colin Jia Sheng Loh

Abstract

Recent work of Mao, Wan and Zhang \cite{MWZ} has provided a complete list of strongly tempered hyperspherical varieties and they proposed some new period integrals. In this paper, I will present new period integrals of distinguished polarised strongly tempered hyperspherical varieties and discuss the L-functions these integrals represent, as examples of the Relative Langlands Duality.

Period integrals of distinguished polarised strongly tempered hyperspherical varieties

Abstract

Recent work of Mao, Wan and Zhang \cite{MWZ} has provided a complete list of strongly tempered hyperspherical varieties and they proposed some new period integrals. In this paper, I will present new period integrals of distinguished polarised strongly tempered hyperspherical varieties and discuss the L-functions these integrals represent, as examples of the Relative Langlands Duality.
Paper Structure (41 sections, 12 theorems, 206 equations, 3 tables)

This paper contains 41 sections, 12 theorems, 206 equations, 3 tables.

Table of Contents

  1. Introduction and Main Results
  2. Relative Langlands Duality and Automorphic $L$-functions
  3. Period Invariant
  4. Main Result
  5. Organisation of paper
  6. Acknowledgement
  7. Hyperspherical varieties
  8. Distinguished Polarised Hyperspherical Variety
  9. Strongly Tempered Hyperspherical Variety
  10. Rankin-Selberg integrals
  11. Preliminaries
  12. Notations and subgroups
  13. Root datum of $\mathrm{GSpin}_{2n+1}$
  14. Matrix group isomorphism of low rank $\mathrm{GSpin}$ groups
  15. Representations and Eisenstein series
  16. ...and 26 more sections

Key Result

Theorem 1.1

Let $\mathcal{D} = (G,H,\rho_H,\iota)$ is a strongly tempered BZSV quadruple dual to a distinguished polarised BZSV quadruple $\mathcal{D}^\vee =(G^\vee, G^\vee, \tau \oplus \tau^\vee, 1)$ where $(G^\vee, \tau)$ is a multiplicity free representation in Table Table: Multiplicity-free repn.

Theorems & Definitions (22)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 4.1: Corollary 4.3 of LX, Lemma 3 of G and Section 6.1 of ACS
  • Lemma 4.2
  • proof
  • Lemma 4.3
  • Lemma 4.4
  • Theorem 5.1
  • ...and 12 more