Let <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="229_2024_1603_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {G}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">G</mi> </math></EquationSource> </InlineEquation> be a connected reductive group over <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="229_2024_1603_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {O}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">O</mi> </math></EquationSource> </InlineEquation>, a complete discrete valuation ring with finite residue field <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="229_2024_1603_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {F}_q$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">F</mi> <mi>q</mi> </msub> </math></EquationSource> </InlineEquation>. Let <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="229_2024_1603_Article_IEq4.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">$R_{T_r,U_r}^{\theta }$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>R</mi> <mrow> <msub> <mi>T</mi> <mi>r</mi> </msub> <mo>,</mo> <msub> <mi>U</mi> <mi>r</mi> </msub> </mrow> <mi>θ</mi> </msubsup> </math></EquationSource> </InlineEquation> be a level r Deligne–Lusztig representation of <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="229_2024_1603_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="39" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {G}(\mathcal {O})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">G</mi> <mo stretchy="false">(</mo> <mi mathvariant="script">O</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, where r is a positive integer. We show that, if q is not small, and if T is Coxeter and <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="229_2024_1603_Article_IEq6.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="TEX">$\theta =1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>θ</mi> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, then <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="229_2024_1603_Article_IEq7.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">$R_{T_r,U_r}^1$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>R</mi> <mrow> <msub> <mi>T</mi> <mi>r</mi> </msub> <mo>,</mo> <msub> <mi>U</mi> <mi>r</mi> </msub> </mrow> <mn>1</mn> </msubsup> </math></EquationSource> </InlineEquation> degenerates to the <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="229_2024_1603_Article_IEq8.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="TEX">$r=1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>r</mi> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> case. For <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="229_2024_1603_Article_IEq9.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="63" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {G}=\textrm{GL}_2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">G</mi> <mo>=</mo> <msub> <mtext>GL</mtext> <mn>2</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> (or <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="229_2024_1603_Article_IEq10.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="27" /> </InlineMediaObject> <EquationSource Format="TEX">$\textrm{SL}_2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mtext>SL</mtext> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation>), as an application we give the dimensions and decompositions of all <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="229_2024_1603_Article_IEq11.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">$R_{T_r,U_r}^{\theta }$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>R</mi> <mrow> <msub> <mi>T</mi> <mi>r</mi> </msub> <mo>,</mo> <msub> <mi>U</mi> <mi>r</mi> </msub> </mrow> <mi>θ</mi> </msubsup> </math></EquationSource> </InlineEquation> for Coxeter T. This in turn leads us to state a conjectural sign formula for <InlineEquation ID="IEq12"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="229_2024_1603_Article_IEq12.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">$R_{T_r,U_r}^{\theta }$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>R</mi> <mrow> <msub> <mi>T</mi> <mi>r</mi> </msub> <mo>,</mo> <msub> <mi>U</mi> <mi>r</mi> </msub> </mrow> <mi>θ</mi> </msubsup> </math></EquationSource> </InlineEquation>, for general <InlineEquation ID="IEq13"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="229_2024_1603_Article_IEq13.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="75" /> </InlineMediaObject> <EquationSource Format="TEX">$(\mathbb {G}, T, \theta ,r)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">G</mi> <mo>,</mo> <mi>T</mi> <mo>,</mo> <mi>θ</mi> <mo>,</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>.

On a stability of higher level Coxeter unipotent representations

期刊论文
发表日期：

2025年1月4日

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Zhe Chen

摘要

Let $\mathbb {G}$ $G$ be a connected reductive group over $\mathcal {O}$ $O$ , a complete discrete valuation ring with finite residue field $\mathbb {F}_q$ $F_{q}$ . Let $R_{T_r,U_r}^{\theta }$ $R_{T_{r}, U_{r}}^{θ}$ be a level r Deligne–Lusztig representation of $\mathbb {G}(\mathcal {O})$ $G (O)$ , where r is a positive integer. We show that, if q is not small, and if T is Coxeter and $\theta =1$ $θ = 1$ , then $R_{T_r,U_r}^1$ $R_{T_{r}, U_{r}}^{1}$ degenerates to the $r=1$ $r = 1$ case. For $\mathbb {G}=\textrm{GL}_2$ $G = {GL}_{2}$ (or $\textrm{SL}_2$ ${SL}_{2}$ ), as an application we give the dimensions and decompositions of all $R_{T_r,U_r}^{\theta }$ $R_{T_{r}, U_{r}}^{θ}$ for Coxeter T. This in turn leads us to state a conjectural sign formula for $R_{T_r,U_r}^{\theta }$ $R_{T_{r}, U_{r}}^{θ}$ , for general $(\mathbb {G}, T, \theta ,r)$ $(G, T, θ, r)$ .