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<p>Let <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10468_2025_10318_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">\(\varvec{\widehat{G}}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover accent="true"> <mi mathvariant="bold-italic">G</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </math></EquationSource> </InlineEquation> be a spin group over a locally compact non-archimedean local field <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10468_2025_10318_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">\(\varvec{F}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">F</mi> </mrow> </math></EquationSource> </InlineEquation> of odd residual characteristic. We defined lifted self-dual semisimple characters for <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10468_2025_10318_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">\(\varvec{\widehat{G}}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover accent="true"> <mi mathvariant="bold-italic">G</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </math></EquationSource> </InlineEquation> and from them, we constructed a large class of supercuspidal representations of <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10468_2025_10318_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">\(\varvec{\widehat{G}}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover accent="true"> <mi mathvariant="bold-italic">G</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </math></EquationSource> </InlineEquation> when <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10468_2025_10318_Article_IEq5.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">\(\varvec{F}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">F</mi> </mrow> </math></EquationSource> </InlineEquation> is of characteristic zero. In this paper, we show that any positive level supercuspidal representation of <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10468_2025_10318_Article_IEq6.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">\(\varvec{\widehat{G}}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover accent="true"> <mi mathvariant="bold-italic">G</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </math></EquationSource> </InlineEquation> contains such a character.</p>

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Exhaustion of Supercuspidal Representations of p-adic Spin Groups: Semisimple Characters

  • 期刊论文

  • 发表日期：

    2025年2月4日

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  • Ngô Văn Định

摘要

Let \(\varvec{\widehat{G}}\) G ^ be a spin group over a locally compact non-archimedean local field \(\varvec{F}\) F of odd residual characteristic. We defined lifted self-dual semisimple characters for \(\varvec{\widehat{G}}\) G ^ and from them, we constructed a large class of supercuspidal representations of \(\varvec{\widehat{G}}\) G ^ when \(\varvec{F}\) F is of characteristic zero. In this paper, we show that any positive level supercuspidal representation of \(\varvec{\widehat{G}}\) G ^ contains such a character.