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Parameters and Theta lifts

Zhe Li, Shanwen Wang, Zhiqi Zhu

TL;DR

The paper clarifies how Harish-Chandra parameters and Langlands-Vogan parameters correspond for real symplectic groups and equal-rank orthogonal groups, enabling translation of the Howe dual-pair theta correspondence into the LV framework. It provides explicit recipes for the Langlands-Vogan parameters of theta lifts and shows that the LV parameter of a Sp representation shifts by 1 under the corresponding orthogonal lift, with parabolic induction handling tempered representations. By rephrasing Moeglin’s and Paul’s results in LV terms, the work lays foundational steps toward real GGP conjectures in the Fourier–Jacobi setting and bridges classical HC-parameter descriptions with modern LV-parameter parametrizations for theta correspondences. Overall, it offers a concrete LV-language bridge between theta correspondence and the rich structure of real reductive equal-rank groups, opening pathways for further applications in automorphic representation theory over the reals.

Abstract

In this note, we make explicit the correspondence between Harish-Chandra parameters and Langlands-Vogan parameters for symplectic groups and orthogonal groups of equal rank over reals. As an application, we reformulate Moeglin's results and Paul's work on the Howe correspondence for symplectic-orthogonal dual pairs using Langlands-Vogan parameters.

Parameters and Theta lifts

TL;DR

The paper clarifies how Harish-Chandra parameters and Langlands-Vogan parameters correspond for real symplectic groups and equal-rank orthogonal groups, enabling translation of the Howe dual-pair theta correspondence into the LV framework. It provides explicit recipes for the Langlands-Vogan parameters of theta lifts and shows that the LV parameter of a Sp representation shifts by 1 under the corresponding orthogonal lift, with parabolic induction handling tempered representations. By rephrasing Moeglin’s and Paul’s results in LV terms, the work lays foundational steps toward real GGP conjectures in the Fourier–Jacobi setting and bridges classical HC-parameter descriptions with modern LV-parameter parametrizations for theta correspondences. Overall, it offers a concrete LV-language bridge between theta correspondence and the rich structure of real reductive equal-rank groups, opening pathways for further applications in automorphic representation theory over the reals.

Abstract

In this note, we make explicit the correspondence between Harish-Chandra parameters and Langlands-Vogan parameters for symplectic groups and orthogonal groups of equal rank over reals. As an application, we reformulate Moeglin's results and Paul's work on the Howe correspondence for symplectic-orthogonal dual pairs using Langlands-Vogan parameters.
Paper Structure (29 sections, 12 theorems, 85 equations)

This paper contains 29 sections, 12 theorems, 85 equations.

Table of Contents

  1. Introduction
  2. Langlands-Vogan parameters
  3. Main result
  4. Acknowledgement
  5. Preliminary
  6. Complex reductive groups
  7. Based Root Datum
  8. Automorphisms
  9. Pinning and L-group
  10. Root systems of real reductive groups
  11. Symplectic case
  12. Orthogonal case
  13. Generalities on real forms and Cartan involutions
  14. Parameters for real reductive group of equal rank
  15. Harish-Chandra parameters
  16. ...and 14 more sections

Key Result

Theorem 1.1

Let $V$ be a $2n$-dimensional symplectic space over $\mathbb{R}$. $(1)$ Let $\pi$ be a limit of discrete series representation of $\mathrm{Sp}(V)$ with Langlands-Vogan parameter $(\varphi,\eta)$, where with $p_{\eta,i},q_{\eta,i},z\in\mathbb{N}$, $i=1,\cdots,k$, $z + \sum_{i=1}^k (p_{\eta,i}+q_{\eta,i})=n$ and $\rho_{\lambda_i}$ self-dual irreducible representation of the Weil group $W_{\mathbb{R

Theorems & Definitions (42)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Definition 2.6
  • Remark 2.7
  • ...and 32 more