<p>We study the behavior of Dirac cohomology under Howe’s <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10468_2025_10319_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">$\Theta $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Θ</mi> </math></EquationSource> </InlineEquation>-correspondence in the case of complex reductive dual pairs. More precisely, if <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10468_2025_10319_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="61" /> </InlineMediaObject> <EquationSource Format="TEX">$(G_1,G_2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msub> <mi>G</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>G</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is a complex reductive dual pair with <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10468_2025_10319_Article_IEq6.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">$G_1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>G</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10468_2025_10319_Article_IEq7.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">$G_2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>G</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation> viewed as real groups, we describe those Harish-Chandra modules <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10468_2025_10319_Article_IEq8.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">$\pi _1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>π</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> of <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10468_2025_10319_Article_IEq9.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">$G_1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>G</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> with nonzero Dirac cohomology whose <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10468_2025_10319_Article_IEq10.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">$\Theta $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Θ</mi> </math></EquationSource> </InlineEquation>-liftings <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10468_2025_10319_Article_IEq11.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">$\Theta (\pi _1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Θ</mi> <mo stretchy="false">(</mo> <msub> <mi>π</mi> <mn>1</mn> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> still have nonzero Dirac cohomology. In this case, we compute explicitly the Dirac cohomology of <InlineEquation ID="IEq12"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10468_2025_10319_Article_IEq12.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">$\Theta (\pi _1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Θ</mi> <mo stretchy="false">(</mo> <msub> <mi>π</mi> <mn>1</mn> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>.</p>

Dirac Cohomology and $\Theta $-correspondence for Complex Dual Pairs

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2025年2月8日

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S. Afentoulidis-Almpanis,
G. Liu,
S. Mehdi

摘要

We study the behavior of Dirac cohomology under Howe’s $\Theta $ $Θ$ -correspondence in the case of complex reductive dual pairs. More precisely, if $(G_1,G_2)$ $(G_{1}, G_{2})$ is a complex reductive dual pair with $G_1$ $G_{1}$ and $G_2$ $G_{2}$ viewed as real groups, we describe those Harish-Chandra modules $\pi _1$ $π_{1}$ of $G_1$ $G_{1}$ with nonzero Dirac cohomology whose $\Theta $ $Θ$ -liftings $\Theta (\pi _1)$ $Θ (π_{1})$ still have nonzero Dirac cohomology. In this case, we compute explicitly the Dirac cohomology of $\Theta (\pi _1)$ $Θ (π_{1})$ .

Dirac Cohomology and \(\Theta \)-correspondence for Complex Dual Pairs

摘要