Let U be a finite dimensional vector space over <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathbb R$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">R</mi> </math></EquationSource> </InlineEquation> or <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathbb C$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">C</mi> </math></EquationSource> </InlineEquation>, and let <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\rho :G\rightarrow GL(U)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ρ</mi> <mo>:</mo> <mi>G</mi> <mo stretchy="false">→</mo> <mi>G</mi> <mi>L</mi> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> be a representation of a connected Lie group G. A linear subspace <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$V\subset U$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>V</mi> <mo>⊂</mo> <mi>U</mi> </mrow> </math></EquationSource> </InlineEquation> is called universal if every orbit of G meets V. We study universal subspaces for Lie groups, especially compact Lie groups. Jinpeng and Doković approached universality for compact groups through a certain topological obstruction. They showed that the non-vanishing of the obstruction class is sufficient for the universality of V, and asked whether it is also necessary under certain conditions. In this article, we show that the answer to the question is negative in general, but there are some important situations where the answer is positive. For a complex connected Lie group G, with <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$G/G_V$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>G</mi> <mo stretchy="false">/</mo> <msub> <mi>G</mi> <mi>V</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> is compact, V is universal if and only if the top Chern class of the vector bundle <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$E_W:=(G\times (U/V))/G_V$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>E</mi> <mi>W</mi> </msub> <mo>:</mo> <mo>=</mo> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo>×</mo> <mrow> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">/</mo> <mi>V</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">/</mo> <msub> <mi>G</mi> <mi>V</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> over <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$G/G_V$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>G</mi> <mo stretchy="false">/</mo> <msub> <mi>G</mi> <mi>V</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> is nonzero. Moreover, we prove similar results for both adjoint and complexified adjoint representation of a connected compact group, and give some characterizations for a subalgebra to be universal.

On universal subspaces for Lie groups

期刊论文
发表日期：

2025年11月20日

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Saurav Bhaumik,
Arunava Mandal

摘要

Let U be a finite dimensional vector space over $\mathbb R$ $R$ or $\mathbb C$ $C$ , and let $\rho :G\rightarrow GL(U)$ $ρ : G \to G L (U)$ be a representation of a connected Lie group G. A linear subspace $V\subset U$ $V \subset U$ is called universal if every orbit of G meets V. We study universal subspaces for Lie groups, especially compact Lie groups. Jinpeng and Doković approached universality for compact groups through a certain topological obstruction. They showed that the non-vanishing of the obstruction class is sufficient for the universality of V, and asked whether it is also necessary under certain conditions. In this article, we show that the answer to the question is negative in general, but there are some important situations where the answer is positive. For a complex connected Lie group G, with $G/G_V$ $G / G_{V}$ is compact, V is universal if and only if the top Chern class of the vector bundle $E_W:=(G\times (U/V))/G_V$ $E_{W} : = (G \times (U / V)) / G_{V}$ over $G/G_V$ $G / G_{V}$ is nonzero. Moreover, we prove similar results for both adjoint and complexified adjoint representation of a connected compact group, and give some characterizations for a subalgebra to be universal.