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Anomaly-free Hyperspherical Hamiltonian spaces for simple reductive groups

Guodong Tang, Chen Wan, Lei Zhang

TL;DR

The work addresses the problem of classifying anomaly-free hyperspherical Hamiltonian spaces for simple reductive groups and identifying their conjectural duals under the BZSV duality. It develops and applies a strategy based on the BZSV quadruple framework, leveraging Levi-centralizers, spherical subgroup classifications, and multiplicity-free coisotropic representations to produce a complete Type A and Type E8 analysis, with polarized/vector-space duals and Whittaker induction extending results to general simple G. The paper provides explicit classifications and dual pairs, organized into comprehensive tables, and discusses computational tools (e.g., GAP/SLA) and conjectural mechanisms (Whittaker induction, period integrals) that connect these geometric objects with automorphic and Langlands-theoretic structures. Overall, the results establish a unified geometric framework for anomaly-free hyperspherical spaces and their duals, enabling new perspectives on period integrals and their automorphic significance, while offering a complete catalog for simple groups and a platform for future extensions to non-simple cases.

Abstract

In this paper, we provide a complete list of anomaly-free hyperspherical Hamiltonian spaces for simple reductive groups, as well as their conjectural dual spaces in the sense of BZSV duality.

Anomaly-free Hyperspherical Hamiltonian spaces for simple reductive groups

TL;DR

The work addresses the problem of classifying anomaly-free hyperspherical Hamiltonian spaces for simple reductive groups and identifying their conjectural duals under the BZSV duality. It develops and applies a strategy based on the BZSV quadruple framework, leveraging Levi-centralizers, spherical subgroup classifications, and multiplicity-free coisotropic representations to produce a complete Type A and Type E8 analysis, with polarized/vector-space duals and Whittaker induction extending results to general simple G. The paper provides explicit classifications and dual pairs, organized into comprehensive tables, and discusses computational tools (e.g., GAP/SLA) and conjectural mechanisms (Whittaker induction, period integrals) that connect these geometric objects with automorphic and Langlands-theoretic structures. Overall, the results establish a unified geometric framework for anomaly-free hyperspherical spaces and their duals, enabling new perspectives on period integrals and their automorphic significance, while offering a complete catalog for simple groups and a platform for future extensions to non-simple cases.

Abstract

In this paper, we provide a complete list of anomaly-free hyperspherical Hamiltonian spaces for simple reductive groups, as well as their conjectural dual spaces in the sense of BZSV duality.
Paper Structure (15 sections, 2 theorems, 21 equations, 12 figures, 17 tables)

This paper contains 15 sections, 2 theorems, 21 equations, 12 figures, 17 tables.

Table of Contents

  1. Introduction
  2. Organization of the paper
  3. Acknowledgement
  4. Our strategy
  5. Strategy for classifying the quadruples
  6. Strategy for computing the dual
  7. Type $A$
  8. Classification
  9. Determining $\iota$
  10. Determining $H$ and $\rho_H$
  11. The Dual
  12. Type $E_8$ case
  13. The Classification
  14. The Dual
  15. Tables

Key Result

Proposition 1.2

A BZSV quadruple $\Delta=(G,H,\iota,\rho_H)$ is hyperspherical if it satisfies the following conditions:

Figures (12)

  • Figure 1: Type $B_2=C_2$
  • Figure 2: Type $G_2$
  • Figure 8: Type $B_2=C_2$
  • Figure 9: Type $G_2$
  • Figure 10: Type $F_4$
  • ...and 7 more figures

Theorems & Definitions (15)

  • Definition 1.1
  • Proposition 1.2
  • Definition 1.3
  • Remark 2.1
  • Definition 2.2
  • Conjecture 2.3
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • Lemma 3.1
  • ...and 5 more