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Local converse theorems and Langlands parameters

Nadir Matringe

TL;DR

The paper develops a Langlands-parameter perspective on local converse theorems for quasi-split groups by introducing the acceptability of the dual group ^G. It proves that if ^G is acceptable, then the corresponding G is W_F-acceptable, allowing the WD_F parameter to be recovered from twisted γ-factors in the split case, and identifies counterexamples where the statement fails when ^G is unacceptable. It establishes a variant result for G_2(F) and all quasi-split classical groups, which in characteristic zero and under Gross–Prasad and Rallis aligns with Shahidi γ-factors and the generic local Langlands correspondence of Jantzen–Liu. The results clarify the role of acceptability in local converse phenomena, connect to Silberger–Zink decompositions, and provide case-by-case variants that parallel known GL_n and classical-group correspondences. Together, they deepen the link between parameter-level and representation-level converse theorems and illuminate limitations and extensions in non-split and exceptional settings.

Abstract

Let $F$ be a non Archimedean local field, and $G$ be the $F$-points of a connected quasi-split reductive group defined over $F$. In this note we propose a converse theorem statement for generic Langlands parameters of $G$ when the Langlands dual group of $G$ is acceptable. We then prove it when $G$ is $F$-split. We also prove that the statement does not apply to $\mathrm{SO}_{2n}(F)$ for certain choices of $F$, as soon as $n\geq 3$.Then we consider a variant which we prove for $G=\mathrm{G}_2(F)$ and all quasi-split classical groups. When $F$ has characteristic zero and assuming the validity of the Gross-Prasad and Rallis conjecture, this latter variant translates via the generic local Langlands correspondence of Jantzen and Liu, into the usual local converse theorems for classical groups expressed in terms of Shahidi's gamma factors.

Local converse theorems and Langlands parameters

TL;DR

The paper develops a Langlands-parameter perspective on local converse theorems for quasi-split groups by introducing the acceptability of the dual group ^G. It proves that if ^G is acceptable, then the corresponding G is W_F-acceptable, allowing the WD_F parameter to be recovered from twisted γ-factors in the split case, and identifies counterexamples where the statement fails when ^G is unacceptable. It establishes a variant result for G_2(F) and all quasi-split classical groups, which in characteristic zero and under Gross–Prasad and Rallis aligns with Shahidi γ-factors and the generic local Langlands correspondence of Jantzen–Liu. The results clarify the role of acceptability in local converse phenomena, connect to Silberger–Zink decompositions, and provide case-by-case variants that parallel known GL_n and classical-group correspondences. Together, they deepen the link between parameter-level and representation-level converse theorems and illuminate limitations and extensions in non-split and exceptional settings.

Abstract

Let be a non Archimedean local field, and be the -points of a connected quasi-split reductive group defined over . In this note we propose a converse theorem statement for generic Langlands parameters of when the Langlands dual group of is acceptable. We then prove it when is -split. We also prove that the statement does not apply to for certain choices of , as soon as .Then we consider a variant which we prove for and all quasi-split classical groups. When has characteristic zero and assuming the validity of the Gross-Prasad and Rallis conjecture, this latter variant translates via the generic local Langlands correspondence of Jantzen and Liu, into the usual local converse theorems for classical groups expressed in terms of Shahidi's gamma factors.
Paper Structure (10 sections, 17 theorems, 28 equations)

This paper contains 10 sections, 17 theorems, 28 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Langlands parameters and their local constants
  4. Generic parameters
  5. A consequence of the sharp local converse theorem for ${\mathrm {GL}}_n$
  6. Acceptable reductive algebraic groups over ${\mathbb {C}}$
  7. A consequence of results of Silberger and Zink
  8. A local converse theorem for Langlands parameters
  9. Unacceptable parameters
  10. A variant

Key Result

Theorem 1

Suppose that $\widehat{G}$ is acceptable, then $G$ is $\mathrm{W}_F$-acceptable, hence it is generically ${\mathrm{WD}}_F$-acceptable.

Theorems & Definitions (37)

  • Theorem
  • Lemma 2.1
  • proof
  • Lemma 3.1
  • Lemma 3.2
  • proof
  • Theorem 3.3
  • proof
  • Definition 4.1
  • Theorem 4.2
  • ...and 27 more