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A Reduction over finite fields of the tame local Langlands correspondence for SLn

Elena Collacciani

TL;DR

This work proves a version of Vogan’s reduction of the tame local Langlands correspondence for SL_N by constructing a Macdonald–Vogan-type parametrization $\mathcal{M}'_N$ of irreducible representations of $SL_N(k_F)$ via $I_F$-equivalence classes of tame Langlands parameters for $SL_N(F)$. It develops a fiberwise description of these parameterizations, showing that fibers are canonically parameterized by irreducible representations of centralizer component groups, and establishes compatibility with the Local Langlands correspondence for $SL_N(F)$ (via $L'_N$) in the tempered/depth-0 setting. The paper also analyzes the relationship between the GL_N framework and its SL_N reduction through parahoric restriction, defines a refined notion of the head of parahoric restriction, and proves that fiber-level and global parameterizations align when enhanced by the fiber data. The results extend Macdonald–Zink–Gel-type correspondences from GL_N to SL_N and lay groundwork for analogous extensions to other split reductive groups. The approach hinges on deep links between the representation theories of $GL_N$ and $SL_N$, as well as the structure of component groups associated with centralizers in the complex dual group. The findings have potential implications for understanding representations of finite groups of Lie type in terms of p-adic Langlands data.

Abstract

We establish a conjecture formulated by Vogan for SLn. Specifically, we construct a surjection from the set of irreducible representations of SLn(k), where k is a finite field, to the inertia equivalence classes of tame Langlands parameters for SLn(F), where F is a p-adic field with residue field k. Additionally, we provide a parametrization of the fibers of this surjection and examine its compatibility with the Local Langlands Correspondence for SLn. This work extends several results previously established for GLn to the context of SLn.

A Reduction over finite fields of the tame local Langlands correspondence for SLn

TL;DR

This work proves a version of Vogan’s reduction of the tame local Langlands correspondence for SL_N by constructing a Macdonald–Vogan-type parametrization of irreducible representations of via -equivalence classes of tame Langlands parameters for . It develops a fiberwise description of these parameterizations, showing that fibers are canonically parameterized by irreducible representations of centralizer component groups, and establishes compatibility with the Local Langlands correspondence for (via ) in the tempered/depth-0 setting. The paper also analyzes the relationship between the GL_N framework and its SL_N reduction through parahoric restriction, defines a refined notion of the head of parahoric restriction, and proves that fiber-level and global parameterizations align when enhanced by the fiber data. The results extend Macdonald–Zink–Gel-type correspondences from GL_N to SL_N and lay groundwork for analogous extensions to other split reductive groups. The approach hinges on deep links between the representation theories of and , as well as the structure of component groups associated with centralizers in the complex dual group. The findings have potential implications for understanding representations of finite groups of Lie type in terms of p-adic Langlands data.

Abstract

We establish a conjecture formulated by Vogan for SLn. Specifically, we construct a surjection from the set of irreducible representations of SLn(k), where k is a finite field, to the inertia equivalence classes of tame Langlands parameters for SLn(F), where F is a p-adic field with residue field k. Additionally, we provide a parametrization of the fibers of this surjection and examine its compatibility with the Local Langlands Correspondence for SLn. This work extends several results previously established for GLn to the context of SLn.
Paper Structure (18 sections, 16 theorems, 115 equations)

This paper contains 18 sections, 16 theorems, 115 equations.

Table of Contents

  1. Compatibility of Langlands and Macdonald Correspondences for $GL_N$
  2. The Head of Parahoric restriction
  3. Compatibility of Langlands and Macdonald correspondence
  4. Level-$0$ Local Langlands correspondence
  5. Macdonald correspondence
  6. Compatibility of Langlands and Macdonald correspondence
  7. Local Langlands correspondence for $SL_N(F)$
  8. The Macdonald-Vogan correspondence for $SL_N(k_F)$
  9. The $I_F$-equivalence classes
  10. The Macdonald-Vogan correspondence
  11. Parametrization of the fibers of $\mathcal{M}'_N$
  12. Comparison of the fibers of $\mathcal{M}'_N$ and $\mathcal{L}'_N$
  13. Compatibility between Langlands and Macdonald-Vogan parameterizations
  14. Compatibility between $\mathcal{L}'_N$ and $\mathcal{M}'_N$
  15. The head of Parahoric Restriction
  16. ...and 3 more sections

Key Result

Theorem 1.1

Zink For any $\pi\in \Omega_N$, the representation $\overline{\sigma}_{\overline{\Lambda_{\pi}}}$ of $\overline{G_N}$ obtained via head1 is an irreducible constituent of multiplicity $1$ of $\mathcal{P}^{G_N}_{\overline{G}_N}\pi$. If $\overline{\sigma}_{\overline{\Lambda}}$ for $\Lambda$ a partition

Theorems & Definitions (42)

  • Theorem 1.1
  • Remark 1.2
  • Definition 1.3
  • Remark 1.4
  • Theorem 1.5
  • Definition 3.1
  • Remark 3.2
  • Lemma 3.3
  • proof
  • Definition 3.4
  • ...and 32 more