Given a compact semisimple Lie group G and a maximal torus <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$T\subset G$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>T</mi> <mo>⊂</mo> <mi>G</mi> </mrow> </math></EquationSource> </InlineEquation>, we give an explicit description of all left and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\operatorname {Ad}(T)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>Ad</mo> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-invariant pluriclosed Hermitian structures on G in terms of the corresponding root system. They depend on <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$2d+1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2</mn> <mi>d</mi> <mo>+</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> parameters in the irreducible case, where <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\dim {T}=2d$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>dim</mo> <mi>T</mi> <mo>=</mo> <mn>2</mn> <mi>d</mi> </mrow> </math></EquationSource> </InlineEquation>. As applications, we obtain that the only left and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\operatorname {Ad}(T)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>Ad</mo> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-invariant pluriclosed metrics which are also CYT are bi-invariant metrics (i.e., Bismut flat) and study the pluriclosed flow as a neat ODE system.

Pluriclosed metrics on compact semisimple Lie groups

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发表日期：

2026年3月13日

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Jorge Lauret,
Facundo Montedoro

摘要

Given a compact semisimple Lie group G and a maximal torus $T\subset G$ $T \subset G$ , we give an explicit description of all left and $\operatorname {Ad}(T)$ $Ad (T)$ -invariant pluriclosed Hermitian structures on G in terms of the corresponding root system. They depend on $2d+1$ $2 d + 1$ parameters in the irreducible case, where $\dim {T}=2d$ $dim T = 2 d$ . As applications, we obtain that the only left and $\operatorname {Ad}(T)$ $Ad (T)$ -invariant pluriclosed metrics which are also CYT are bi-invariant metrics (i.e., Bismut flat) and study the pluriclosed flow as a neat ODE system.