We investigate Hamiltonian actions of non-compact Lie groups on a homogeneous bounded domain X. As a main result, we point out a Lie-theoretical condition for a closed Lie group H of the automorphism group of X which ensures that the symplectic reduction <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mu ^{-1}(0)/H$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>μ</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">/</mo> <mi>H</mi> </mrow> </math></EquationSource> </InlineEquation> with respect to the momentum map <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mu $</EquationSource> <EquationSource Format="MATHML"><math> <mi>μ</mi> </math></EquationSource> </InlineEquation> at hand, is a Stein manifold. Moreover, for the class of connected subgroups of translations the quotient <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$(H^\mathbb {C}\cdot X)/H^\mathbb {C}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">(</mo> <msup> <mi>H</mi> <mi mathvariant="double-struck">C</mi> </msup> <mo>·</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">/</mo> <msup> <mi>H</mi> <mi mathvariant="double-struck">C</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> is realized as a Siegel domain and we show that the symplectic reduction <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mu ^{-1}(0)/H$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>μ</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">/</mo> <mi>H</mi> </mrow> </math></EquationSource> </InlineEquation> is biholomorphic to such a Stein quotient.

Hamiltonian Actions on Homogeneous Bounded Domains

期刊论文
发表日期：

2026年6月24日

查看全文

Maxim Kukol

摘要

We investigate Hamiltonian actions of non-compact Lie groups on a homogeneous bounded domain X. As a main result, we point out a Lie-theoretical condition for a closed Lie group H of the automorphism group of X which ensures that the symplectic reduction $\mu ^{-1}(0)/H$ $μ^{- 1} (0) / H$ with respect to the momentum map $\mu $ $μ$ at hand, is a Stein manifold. Moreover, for the class of connected subgroups of translations the quotient $(H^\mathbb {C}\cdot X)/H^\mathbb {C}$ $(H^{C} \cdot X) / H^{C}$ is realized as a Siegel domain and we show that the symplectic reduction $\mu ^{-1}(0)/H$ $μ^{- 1} (0) / H$ is biholomorphic to such a Stein quotient.