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<p>We explicate relations among the Gelfand–Graev modules for central covers of reductive <i>p</i>-adic groups, the Euler–Poincaré polynomial of the Arnold–Brieskorn manifold, and the quantum affine Schur–Weyl duality. These three objects and their relations are dictated by a permutation representation of the Weyl group. Our main result concerns the connection between the Gelfand–Graev module and quantum affine Schur-Weyl duality, which only holds for certain covers of <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="29_2025_1049_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="45" /> </InlineMediaObject> <EquationSource Format="TEX">\(\textrm{GL}(r)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>GL</mtext> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. In this case, the Gelfand–Graev functor is essentially the quantum affine Schur–Weyl functor. This has two significant consequences. First, the commuting algebra of the Iwahori-fixed part of the Gelfand–Graev representation is the quotient of a quantum group. Second, intertwining operators on the <i>p</i>-adic group side are matched with <i>R</i>-matrices on the quantum group side, reproving a result of Brubaker–Buciumas–Bump.</p>

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Genuine Gelfand–Graev functor and the quantum affine Schur–Weyl duality

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  • 发表日期：

    2025年6月7日

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  • Fan Gao,
  • Nadya Gurevich,
  • Edmund Karasiewicz

摘要

We explicate relations among the Gelfand–Graev modules for central covers of reductive p-adic groups, the Euler–Poincaré polynomial of the Arnold–Brieskorn manifold, and the quantum affine Schur–Weyl duality. These three objects and their relations are dictated by a permutation representation of the Weyl group. Our main result concerns the connection between the Gelfand–Graev module and quantum affine Schur-Weyl duality, which only holds for certain covers of \(\textrm{GL}(r)\) GL ( r ) . In this case, the Gelfand–Graev functor is essentially the quantum affine Schur–Weyl functor. This has two significant consequences. First, the commuting algebra of the Iwahori-fixed part of the Gelfand–Graev representation is the quotient of a quantum group. Second, intertwining operators on the p-adic group side are matched with R-matrices on the quantum group side, reproving a result of Brubaker–Buciumas–Bump.