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On the Macdonald correspondence

Anne-Marie Aubert

TL;DR

The paper analyzes Macdonald's explicit bijection between irreducible representations of $GL_n(k)$ and inertia classes of tamely ramified $n$-dimensional Weil-Deligne representations of the local Weil group, rooting the construction in Lusztig's finite-field classification. It defines epsilon- and gamma-factors for pairs on finite general linear groups and proves their compatibility with Deligne's factors and the local Langlands program, including depth-zero compatibility. By leveraging the converse theorem of Niën, the work characterizes the Macdonald cusp correspondence as the unique map preserving central characters and pair invariants, linking finite-field representation theory with Weil-Deligne data. The results provide a robust bridge between finite-field representations and $p$-adic Langlands phenomena, and yield a concrete, computable framework for understanding pairwise local factors in this setting.

Abstract

In 1980 Ian G. Macdonald established an explicit bijection between the isomorphism classes of the irreducible representations of ${\mathrm{GL}}_n(k)$, where $k$ is a finite field, and inertia equivalence classes of admissible tamely ramified $n$-dimensional Weil-Deligne representations of $W_F$, where $F$ is a non-archimedean local field with residue field $k$ and $W_F$ the absolute Weil group of $F$. We describe a construction of the Macdonald correspondence based on the specialization to ${\mathrm{GL}}_n(k)$ of Lusztig's classification of irreducible representations of finite groups of Lie type, and review some properties of the correspondence. We define $ε$-factors for pairs of irreducible cuspidal representations of finite general linear groups, and show that they match with the expected Deligne $ε$-factors under the Macdonald correspondence. We use these $ε$-factors for pairs to obtain a characterization of the Macdonald correspondence for the irreducible cuspidal representations

On the Macdonald correspondence

TL;DR

The paper analyzes Macdonald's explicit bijection between irreducible representations of and inertia classes of tamely ramified -dimensional Weil-Deligne representations of the local Weil group, rooting the construction in Lusztig's finite-field classification. It defines epsilon- and gamma-factors for pairs on finite general linear groups and proves their compatibility with Deligne's factors and the local Langlands program, including depth-zero compatibility. By leveraging the converse theorem of Niën, the work characterizes the Macdonald cusp correspondence as the unique map preserving central characters and pair invariants, linking finite-field representation theory with Weil-Deligne data. The results provide a robust bridge between finite-field representations and -adic Langlands phenomena, and yield a concrete, computable framework for understanding pairwise local factors in this setting.

Abstract

In 1980 Ian G. Macdonald established an explicit bijection between the isomorphism classes of the irreducible representations of , where is a finite field, and inertia equivalence classes of admissible tamely ramified -dimensional Weil-Deligne representations of , where is a non-archimedean local field with residue field and the absolute Weil group of . We describe a construction of the Macdonald correspondence based on the specialization to of Lusztig's classification of irreducible representations of finite groups of Lie type, and review some properties of the correspondence. We define -factors for pairs of irreducible cuspidal representations of finite general linear groups, and show that they match with the expected Deligne -factors under the Macdonald correspondence. We use these -factors for pairs to obtain a characterization of the Macdonald correspondence for the irreducible cuspidal representations
Paper Structure (15 sections, 5 theorems, 81 equations)

This paper contains 15 sections, 5 theorems, 81 equations.

Table of Contents

  1. Introduction
  2. Construction of the Macdonald correspondence
  3. The set P(X)^Γ
  4. Weil-Deligne representations
  5. Representations of GL_n(k)
  6. The Harish-Chandra theory
  7. Parametrization of the representations
  8. The Lusztig classification
  9. Definition of the Macdonald correspondence
  10. Definition and preservation of the $\epsilon$-factors
  11. Gamma factors for pairs
  12. More properties of the Macdonald correspondence
  13. Compatibility with the depth-zero local Langlands correspondence
  14. Preservation of the epsilon-factors for pairs
  15. Characterization of the Macdonald correspondence

Key Result

Proposition 2.3.29

Let $M$ be a Levi subgroup of $\mathrm{GL}_n(k)$ such that Then the parabolically induced representation $R_M^{\mathrm{GL}_n}(\pi_1\otimes\cdots\otimes\pi_h)$ is irreducible.

Theorems & Definitions (14)

  • Definition 2.2.7
  • Remark 2.3.28
  • Proposition 2.3.29
  • proof
  • Corollary 2.3.32
  • proof
  • Remark 2.3.35
  • Definition 3.0.6
  • Theorem 1
  • proof
  • ...and 4 more