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Local theta correspondences and Langlands parameters for rigid inner twists

Hirotaka Kakuhama

TL;DR

This work develops a general framework that describes local theta correspondences for rigid inner twists in terms of local Langlands parameters, setting up rigid inner twists and linking data to compare theta lifts with Langlands parameters. The core contribution is Conjecture 6, which relates theta lifts between quaternionic dual pairs via an embedding of L-groups and a matching of endoscopic character relations, with verification in Archimedean cases and in the non-Archimedean rank-one scenario. The paper also provides extensive Archimedean computations translating Harish-Chandra parameters into Langlands parameters (via Mezo’s endoscopic transfer factors) and offers detailed appendices clarifying conventions, transfer factors, and special cases. The results bridge local theta correspondence with refined Langlands-Lpacket data in the rigid-inner-twist setting, with implications for the Prasad conjecture and endoscopic phenomena in both real and p-adic contexts.

Abstract

In this paper, we formulate a conjecture that describes the local theta correspondences in terms of the local Langland correspondences for rigid inner twists, which contain the correspondences for quaternionic dual pairs. Moreover, we verify the conjecture holds in some specific cases.

Local theta correspondences and Langlands parameters for rigid inner twists

TL;DR

This work develops a general framework that describes local theta correspondences for rigid inner twists in terms of local Langlands parameters, setting up rigid inner twists and linking data to compare theta lifts with Langlands parameters. The core contribution is Conjecture 6, which relates theta lifts between quaternionic dual pairs via an embedding of L-groups and a matching of endoscopic character relations, with verification in Archimedean cases and in the non-Archimedean rank-one scenario. The paper also provides extensive Archimedean computations translating Harish-Chandra parameters into Langlands parameters (via Mezo’s endoscopic transfer factors) and offers detailed appendices clarifying conventions, transfer factors, and special cases. The results bridge local theta correspondence with refined Langlands-Lpacket data in the rigid-inner-twist setting, with implications for the Prasad conjecture and endoscopic phenomena in both real and p-adic contexts.

Abstract

In this paper, we formulate a conjecture that describes the local theta correspondences in terms of the local Langland correspondences for rigid inner twists, which contain the correspondences for quaternionic dual pairs. Moreover, we verify the conjecture holds in some specific cases.
Paper Structure (52 sections, 40 theorems, 296 equations, 1 table)

This paper contains 52 sections, 40 theorems, 296 equations, 1 table.

Table of Contents

  1. Introduction
  2. Settings
  3. Notations
  4. Spaces and groups
  5. Quasi-split inner forms
  6. Maximal tori of quasi-split inner forms
  7. Extensions by extension fields
  8. Whittaker data
  9. Rigid inner twists
  10. Settings
  11. Special classes of rigid inner twists
  12. Rigid inner twists for Levi subgroups
  13. Local theta correspondences
  14. Local Langlands correspondence
  15. The L-groups
  16. ...and 37 more sections

Key Result

Lemma 2.3

The natural linear map is bijective and isometric. Moreover, the following diagram is commutative. \xymatrix{ \operatorname{Sp}(\mathbb{W}) \ar[rr] & & \operatorname{Sp}(\mathbb{W}^\natural) \\ G(V) \times G(W) \ar[u]^-{\iota_{V,W}} \ar[rr]_-{(\mathfrak{m}_V, \mathfrak{m}_W)} & & G(V^\natural)\times G(W^\natural) \ar[u

Theorems & Definitions (86)

  • Conjecture 1.1
  • Remark 2.1
  • Lemma 2.3
  • proof
  • Remark 2.4
  • Lemma 3.2
  • proof
  • Proposition 3.3
  • Lemma 3.4
  • proof
  • ...and 76 more