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<p>Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\((G,\kappa )\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo>,</mo> <mi>κ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> be a compact connected Lie group endowed with a biinvariant Riemannian metric, and let <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\tilde{G}\)</EquationSource> <EquationSource Format="MATHML"><math> <mover accent="true"> <mi>G</mi> <mo stretchy="false">~</mo> </mover> </math></EquationSource> </InlineEquation> be the complexification of <i>G</i>. We apply Grauert tube techniques to the near-diagonal scaling asymptotics of certain operator kernels, which are defined in terms of the matrix elements of an irreducible representation drifting to infinity along a ray in weight space. These kernels are the equivariant components of Poisson and Szegő kernels on a fixed sphere bundle in <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\tilde{G}\)</EquationSource> <EquationSource Format="MATHML"><math> <mover accent="true"> <mi>G</mi> <mo stretchy="false">~</mo> </mover> </math></EquationSource> </InlineEquation>, when the latter is identified with the tangent bundle of <i>G</i> in an appropriate way.</p>

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Eigenfunction Asymptotics in the Complex Domain for a Compact Lie Group

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  • 发表日期：

    2026年4月10日

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  • Simone Gallivanone,
  • Roberto Paoletti

摘要

Let \((G,\kappa )\) ( G , κ ) be a compact connected Lie group endowed with a biinvariant Riemannian metric, and let \(\tilde{G}\) G ~ be the complexification of G. We apply Grauert tube techniques to the near-diagonal scaling asymptotics of certain operator kernels, which are defined in terms of the matrix elements of an irreducible representation drifting to infinity along a ray in weight space. These kernels are the equivariant components of Poisson and Szegő kernels on a fixed sphere bundle in \(\tilde{G}\) G ~ , when the latter is identified with the tangent bundle of G in an appropriate way.