The wavefront set: bounds for the Langlands parameter

2024/03/21 by Dan Ciubotaru, Ciubotaru, Dan, Kim, Ju-Lee
Mathematics · Physics and Astronomy · #22E35 #22E50 #Advanced Differential Geometry Research #Advanced Mathematical Physics Problems #FOS: Mathematics #Geometry and complex manifolds #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2403.14261

openalex publication_date 2024/03/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For an irreducible smooth representation of a connected reductive p-adic group, two important associated invariants are the wavefront set and the (partly conjectural) Langlands parameter. While a wavefront set consists of p-adic nilpotent orbits, one constituent of the Langlands parameter is a complex nilpotent orbit in the dual Lie algebra. For unipotent representations in the sense of Lusztig, the corresponding nilpotent orbits on the two sides are related via the Lusztig--Spaltenstein duality, by the work of Ciubotaru--Mason-Brown--Okada. In this paper, we formulate a general upper-bound conjecture and several variants relating the nilpotent orbits that appear in the wavefront set and in the Langlands parameter. We also verify these expectations in some cases, including the depth-zero supercuspidal representations of classical groups and all the irreducible representations of G₂.

Related

• The (Γ-asymptotic) wavefront sets: GL_n
• Wavefront Sets of Unipotent Representations of Reductive p-adic Groups I
• Wavefront Sets of Unipotent Representations of Reductive p-adic Groups II
• Wave-front sets for p-adic Lie algebras
• On two definitions of wave-front sets for p-adic groups