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<p>For an irreducible smooth representation of a connected reductive <i>p</i>-adic group, two important associated invariants are the wavefront set and the (partly conjectural) Langlands parameter. While a wavefront set consists of <i>p</i>-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 (Ciubotaru et al. in Am J Math <a href="http://arxiv.org/abs/2112.14354v4">arXiv:2112.14354v4</a>, J Reine Angew Math (Crelles J) 823:191–253, 2025). 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 <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2025_3278_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">\(G_2\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>G</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation>.</p>

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The wavefront set: bounds for the Langlands parameter

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年9月9日

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  • Dan Ciubotaru,
  • Ju-Lee Kim

摘要

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 (Ciubotaru et al. in Am J Math arXiv:2112.14354v4, J Reine Angew Math (Crelles J) 823:191–253, 2025). 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 \(G_2\) G 2 .