Motivated by the <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="209_2025_3813_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="46" /> </InlineMediaObject> <EquationSource Format="TEX">$(\mathfrak {g},K)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="fraktur">g</mi> <mo>,</mo> <mi>K</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-cohomology and Dirac cohomology, we determine Dirac series of <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="209_2025_3813_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="66" /> </InlineMediaObject> <EquationSource Format="TEX">$\textrm{GL}(n,\mathbb {H})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>GL</mtext> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi mathvariant="double-struck">H</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, and show that the spin lowest K-type of any Dirac series, which determines the Dirac cohomology, is unique and multiplicity-free for both <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="209_2025_3813_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="66" /> </InlineMediaObject> <EquationSource Format="TEX">$\textrm{GL}(n,\mathbb {H})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>GL</mtext> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi mathvariant="double-struck">H</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="209_2025_3813_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="65" /> </InlineMediaObject> <EquationSource Format="TEX">$\textrm{GL}(n,\mathbb {R})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>GL</mtext> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi mathvariant="double-struck">R</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. This verifies a conjecture about uniqueness of the spin lowest K-type of Dirac series for <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="209_2025_3813_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="65" /> </InlineMediaObject> <EquationSource Format="TEX">$\textrm{GL}(n,\mathbb {R})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>GL</mtext> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi mathvariant="double-struck">R</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> proposed by Dong and Wong (Int Math Res Not IMRN (12):10702–10735, 2022).

Dirac series of GL(n) over an Archimedean field

期刊论文
发表日期：

2025年7月24日

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Yihao Ding,
Hongfeng Zhang

摘要

Motivated by the $(\mathfrak {g},K)$ $(g, K)$ -cohomology and Dirac cohomology, we determine Dirac series of $\textrm{GL}(n,\mathbb {H})$ $GL (n, H)$ , and show that the spin lowest K-type of any Dirac series, which determines the Dirac cohomology, is unique and multiplicity-free for both $\textrm{GL}(n,\mathbb {H})$ $GL (n, H)$ and $\textrm{GL}(n,\mathbb {R})$ $GL (n, R)$ . This verifies a conjecture about uniqueness of the spin lowest K-type of Dirac series for $\textrm{GL}(n,\mathbb {R})$ $GL (n, R)$ proposed by Dong and Wong (Int Math Res Not IMRN (12):10702–10735, 2022).