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<p>We study the compatibility of the formal degree conjecture and the parabolic induction process in the simplest nontrivial case for quasi-split <i>p</i>-adic groups. For a generic discrete series <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="229_2025_1632_Article_IEq1.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">\(\pi \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>π</mi> </math></EquationSource> </InlineEquation> induced from an irreducible supercuspidal <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="229_2025_1632_Article_IEq2.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">\(\sigma \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>σ</mi> </math></EquationSource> </InlineEquation> of a maximal Levi subgroup, we compute the quotient <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="229_2025_1632_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="72" /> </InlineMediaObject> <EquationSource Format="TEX">\(d(\pi )/d(\sigma )\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo stretchy="false">(</mo> <mi>π</mi> <mo stretchy="false">)</mo> <mo stretchy="false">/</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>σ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> of formal degrees with the assumption that the group is unramified. As an application, we verify the conjecture for discrete series of split <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="229_2025_1632_Article_IEq4.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathrm G_2\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">G</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation> supported on maximal Levi subgroups.</p>

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Formal degrees and parabolic induction: the maximal generic case

  • 期刊论文

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  • 发表日期：

    2025年7月15日

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  • Yiyang Wang

摘要

We study the compatibility of the formal degree conjecture and the parabolic induction process in the simplest nontrivial case for quasi-split p-adic groups. For a generic discrete series \(\pi \) π induced from an irreducible supercuspidal \(\sigma \) σ of a maximal Levi subgroup, we compute the quotient \(d(\pi )/d(\sigma )\) d ( π ) / d ( σ ) of formal degrees with the assumption that the group is unramified. As an application, we verify the conjecture for discrete series of split \(\mathrm G_2\) G 2 supported on maximal Levi subgroups.