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On principal series representations of quasi-split reductive p-adic groups

Maarten Solleveld

TL;DR

The paper constructs a canonical local Langlands correspondence for irreducible principal-series representations of quasi-split p-adic groups by linking Bernstein-block realizations to extended affine Hecke algebras and, on the Galois side, to enhanced L-parameters. A Whittaker datum fixes normalization of intertwining operators, yielding a canonical isomorphism End_G(\\Pi_{\\frak s})^{op} \\cong \\mathcal{H}(\\frak s^{\vee}, q_F^{1/2}) and a correspondence Irr(\\mathcal{H}(\\frak s^{\vee}, q_F^{1/2})) \\leftrightarrow Φ_e(G)^{\\frak s^{\vee}}. The authors develop a robust framework: (i) a genericity test via Steinberg modules in Hecke algebras, (ii) a reduction to graded Hecke algebras with geometric parametrization of representations via triples (\\sigma, y, ρ), and (iii) a canonical bijection between Irr(G)^{\\frak s} and enhanced L-parameters that preserves temperedness, discreteness, cuspidal supports, and central characters. The results generalize prior split/quasi-split cases and provide a categorical perspective, paving the way for functorial LLC statements and deeper connections to derived categories of Langlands parameters.

Abstract

Let G be a quasi-split reductive group over a non-archimedean local field. We establish a local Langlands correspondence for all irreducible smooth complex G-representations in the principal series. The parametrization map is injective, and its image is an explicitly described set of enhanced L-parameters. Our correspondence is determined by the choice of a Whittaker datum for G, and it is canonical given that choice. We show that our parametrization satisfies many expected properties, among others with respect to the enhanced L-parameters of generic representations, temperedness, cuspidal supports and central characters. Along the way we characterize genericity in terms of representations of an affine Hecke algebra.

On principal series representations of quasi-split reductive p-adic groups

TL;DR

The paper constructs a canonical local Langlands correspondence for irreducible principal-series representations of quasi-split p-adic groups by linking Bernstein-block realizations to extended affine Hecke algebras and, on the Galois side, to enhanced L-parameters. A Whittaker datum fixes normalization of intertwining operators, yielding a canonical isomorphism End_G(\\Pi_{\\frak s})^{op} \\cong \\mathcal{H}(\\frak s^{\vee}, q_F^{1/2}) and a correspondence Irr(\\mathcal{H}(\\frak s^{\vee}, q_F^{1/2})) \\leftrightarrow Φ_e(G)^{\\frak s^{\vee}}. The authors develop a robust framework: (i) a genericity test via Steinberg modules in Hecke algebras, (ii) a reduction to graded Hecke algebras with geometric parametrization of representations via triples (\\sigma, y, ρ), and (iii) a canonical bijection between Irr(G)^{\\frak s} and enhanced L-parameters that preserves temperedness, discreteness, cuspidal supports, and central characters. The results generalize prior split/quasi-split cases and provide a categorical perspective, paving the way for functorial LLC statements and deeper connections to derived categories of Langlands parameters.

Abstract

Let G be a quasi-split reductive group over a non-archimedean local field. We establish a local Langlands correspondence for all irreducible smooth complex G-representations in the principal series. The parametrization map is injective, and its image is an explicitly described set of enhanced L-parameters. Our correspondence is determined by the choice of a Whittaker datum for G, and it is canonical given that choice. We show that our parametrization satisfies many expected properties, among others with respect to the enhanced L-parameters of generic representations, temperedness, cuspidal supports and central characters. Along the way we characterize genericity in terms of representations of an affine Hecke algebra.
Paper Structure (11 sections, 47 theorems, 245 equations)

This paper contains 11 sections, 47 theorems, 245 equations.

Table of Contents

  1. Hecke algebras for principal series representations
  2. Whittaker normalization
  3. Modules of Whittaker functionals
  4. Normalization of intertwining operators
  5. Characterization of generic representations
  6. Hecke algebras for principal series L-parameters
  7. Comparison of Hecke algebras
  8. Parameters of generic representations
  9. Reduction to graded Hecke algebras
  10. Geometric representations of graded Hecke algebras
  11. A canonical local Langlands correspondence

Key Result

Theorem A

[see Section sec:5] The Whittaker datum $(U,\xi)$ determines a canonical bijection with the following properties: All Borel's desiderata from Bor2 are satisfied. When $\pi$ is given, $\phi_\pi$ is uniquely determined by (a)--(e) and the local Langlands correspondence for tori.

Theorems & Definitions (91)

  • Theorem A
  • Theorem B
  • Theorem C
  • Proposition 1.1
  • proof
  • Theorem 1.2
  • proof
  • Proposition 2.1
  • proof
  • Lemma 2.2
  • ...and 81 more