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Infinitesimal characters and Lafforgue's pseudocharacters

Vytautas Paškūnas, Julian Quast

TL;DR

The paper extends the DPS construction of infinitesimal characters from individual Galois representations to p-adic families encoded by Lafforgue’s $^LG$-pseudocharacters. It constructs a family of infinitesimal characters $ abla_{Y,oldsymbol extΘ}$ varying functorially over rigid analytic bases, using a moduli space $X^{ ext{adm}}_{^LG}$ of admissible pseudocharacters and a universal deformation framework, then patches these from the absolutely irreducible locus via Luna-type arguments and normality after inverting $p$. It generalizes the approach to $C$-groups and develops a robust relative-analytic apparatus to handle analytifications and specialization, enabling the identification of the center-action on locally analytic vectors in Hecke eigenspaces with pseudocharacter data. The results provide a pathway toward a categorical local Langlands program in the locally analytic, $p$-adic setting by tying Galois pseudocharacters to automorphic determinant data and center actions on representation spaces. Applications to definite unitary groups demonstrate the method’s power beyond irreducible residual representations, extending DPS’s reach to residually reducible Galois representations through determinant-law techniques.

Abstract

We associate infinitesimal characters to $p$-adic families of Lafforgue's pseudocharacters of the absolute Galois group of a $p$-adic local field by extending a construction of Dospinescu, Schraen and the first author. We use this construction to study the action of the centre of the universal enveloping algebra on the locally analytic vectors in the Hecke eigenspaces of the completed cohomology.

Infinitesimal characters and Lafforgue's pseudocharacters

TL;DR

The paper extends the DPS construction of infinitesimal characters from individual Galois representations to p-adic families encoded by Lafforgue’s -pseudocharacters. It constructs a family of infinitesimal characters varying functorially over rigid analytic bases, using a moduli space of admissible pseudocharacters and a universal deformation framework, then patches these from the absolutely irreducible locus via Luna-type arguments and normality after inverting . It generalizes the approach to -groups and develops a robust relative-analytic apparatus to handle analytifications and specialization, enabling the identification of the center-action on locally analytic vectors in Hecke eigenspaces with pseudocharacter data. The results provide a pathway toward a categorical local Langlands program in the locally analytic, -adic setting by tying Galois pseudocharacters to automorphic determinant data and center actions on representation spaces. Applications to definite unitary groups demonstrate the method’s power beyond irreducible residual representations, extending DPS’s reach to residually reducible Galois representations through determinant-law techniques.

Abstract

We associate infinitesimal characters to -adic families of Lafforgue's pseudocharacters of the absolute Galois group of a -adic local field by extending a construction of Dospinescu, Schraen and the first author. We use this construction to study the action of the centre of the universal enveloping algebra on the locally analytic vectors in the Hecke eigenspaces of the completed cohomology.
Paper Structure (27 sections, 58 theorems, 91 equations)

This paper contains 27 sections, 58 theorems, 91 equations.

Table of Contents

  1. Introduction
  2. Strategy of the proof
  3. Application
  4. Overview by section
  5. Notation
  6. Acknowledgements
  7. Dimension and codimension
  8. Chevalley--Shephard--Todd theorem
  9. Lafforgue's pseudocharacters
  10. Comparison to the GIT quotient
  11. Deformations of H-pseudocharacters
  12. Chenevier's determinant laws and finitely generated groups
  13. The moduli space X gen Thetabar of condensed representations
  14. Profinite completions
  15. An adequate homeomorphism
  16. ...and 12 more sections

Key Result

Theorem 1.1

There exists a unique family of $\mathbb {Q}_p$-algebra homomorphisms indexed by pairs $Y\in \mathrm{Rig}_{\mathbb {Q}_p}$ and $\Theta\in \tilde{X}^{\mathrm{adm}}_{\leftidx{^L}{G}{}}(Y)$ with the following properties:

Theorems & Definitions (120)

  • Theorem 1.1: \ref{['main_L']}
  • Theorem 1.2: \ref{['adequate']}
  • Proposition 1.3: \ref{['nice_locus']}
  • Example 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 110 more