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The wavefront set: bounds for the Langlands parameter.

Dan Ciubotaru1, Ju-Lee Kim2

  • 1Mathematical Institute, University of Oxford, Oxford, OX2 6GG UK.

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Summary

This study introduces a conjecture relating nilpotent orbits in the wavefront set and Langlands parameters for p-adic groups. The conjecture is verified for specific cases, including depth-zero supercuspidal representations and G2 representations.

Keywords:
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Area of Science:

  • Representation theory
  • Harmonic analysis
  • Number theory

Background:

  • Irreducible smooth representations of connected reductive p-adic groups are key objects.
  • Associated invariants include the wavefront set (p-adic nilpotent orbits) and Langlands parameter (complex nilpotent orbits).
  • Lusztig-Spaltenstein duality connects these orbits for unipotent representations.

Purpose of the Study:

  • To formulate a general upper-bound conjecture relating nilpotent orbits in the wavefront set and Langlands parameter.
  • To explore variants of this conjecture.
  • To verify the conjecture in specific cases.

Main Methods:

  • Formulation of a novel conjecture on nilpotent orbit relationships.
  • Verification of the conjecture through detailed case analysis.
  • Leveraging existing Lusztig-Spaltenstein duality for unipotent representations.

Main Results:

  • A general upper-bound conjecture relating wavefront set and Langlands parameter nilpotent orbits is proposed.
  • Several variants of the conjecture are presented.
  • The conjecture is confirmed for depth-zero supercuspidal representations of classical groups and all irreducible representations of G2.

Conclusions:

  • The proposed conjecture provides a framework for understanding the relationship between wavefront sets and Langlands parameters.
  • The verified cases suggest the conjecture's broad applicability in representation theory.
  • This work deepens the understanding of invariants associated with p-adic group representations.