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<p>Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathfrak g\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="fraktur">g</mi> </math></EquationSource> </InlineEquation> be a reductive Lie algebra and <i>m</i> a positive integer. There is a natural density of irreducible representations of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mathfrak g\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="fraktur">g</mi> </math></EquationSource> </InlineEquation>, whose degrees are not divisible by <i>m</i>. For <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\mathfrak g=\mathfrak {gl}_n\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="fraktur">g</mi> <mo>=</mo> <msub> <mi mathvariant="fraktur">gl</mi> <mi>n</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>, this density decays exponentially to 0 as <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(n \rightarrow \infty \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo stretchy="false">→</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>. Similar results hold for simple Lie algebras and Lie groups, and there are versions for self-dual and orthogonal representations.</p>

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On the divisibility of degrees of representations of Lie groups

  • 期刊论文

  • 发表日期：

    2025年12月15日

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  • Varun Shah,
  • Steven Spallone

摘要

Let \(\mathfrak g\) g be a reductive Lie algebra and m a positive integer. There is a natural density of irreducible representations of \(\mathfrak g\) g , whose degrees are not divisible by m. For \(\mathfrak g=\mathfrak {gl}_n\) g = gl n , this density decays exponentially to 0 as \(n \rightarrow \infty \) n → ∞ . Similar results hold for simple Lie algebras and Lie groups, and there are versions for self-dual and orthogonal representations.