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edited Mar 17, 2020 at 3:27

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Let $G$ be a compact connected Lie group acting in a Hamiltonian fashion on a symplectic manifold $M$ with momentum map $\mu:M\to \mathfrak{g}^\ast$, where $\mathfrak{g}$ is the Lie algebra of $G$. Then several standard textbooks (for example Ortega-RatiuOrtega–Ratiu: Momentum Maps and Hamiltonian ReductionMomentum Maps and Hamiltonian Reduction (MSN)) treat the stratification of the symplectic quotient $$ M//G:=J^{-1}(0)/G $$ by orbit types $(H)$ of the $G$-action: $$ M//G=\bigsqcup_{(H)}(M//G)_{(H)},\qquad (M//G)_{(H)}=M_{(H)}/G, $$ where $M_{(H)}=\{p\in M: \exists\, g\in G,\; G_p =gHg^{-1}\}$, $G_p$ being the isotropy group of the point $p$.

Now, there is also a decomposition of $M//G$ by infinitesimal orbit types: $$ M//G=\bigsqcup_{(\mathfrak{h})}(M//G)_{(\mathfrak{h})},\qquad (M//G)_{(\mathfrak{h})}=M_{(\mathfrak{h})}/G, $$ where $M_{(\mathfrak{h})}=\{p\in M: \exists\, g\in G,\; \mathfrak{g}_p=\mathrm{Ad}(g)\mathfrak{h}\}$$M_{(\mathfrak{h})}=\{p\in M: \exists\, g\in G,\; \mathfrak{g}_p=\operatorname{Ad}(g)\mathfrak{h}\}$, $\mathfrak{g}_p$ being the isotropy algebra of the point $p$.

If $G$ is the circle group $S^1$, then this decomposition is a stratification and is described, for example, in Lerman-Tolman,Lerman–Tolman: Intersection cohomology of $S^1$ symplectic quotients and small resolutionsIntersection cohomology of $S^1$ symplectic quotients and small resolutions (MSN).

However, I could not find a reference in which the infinitesimal orbit type decomposition (or indeed stratification?) is considered in the general case. Which is a good textbook or paper on this subject?

Let $G$ be a compact connected Lie group acting in a Hamiltonian fashion on a symplectic manifold $M$ with momentum map $\mu:M\to \mathfrak{g}^\ast$, where $\mathfrak{g}$ is the Lie algebra of $G$. Then several standard textbooks (for example Ortega-Ratiu: Momentum Maps and Hamiltonian Reduction) treat the stratification of the symplectic quotient $$ M//G:=J^{-1}(0)/G $$ by orbit types $(H)$ of the $G$-action: $$ M//G=\bigsqcup_{(H)}(M//G)_{(H)},\qquad (M//G)_{(H)}=M_{(H)}/G, $$ where $M_{(H)}=\{p\in M: \exists\, g\in G,\; G_p =gHg^{-1}\}$, $G_p$ being the isotropy group of the point $p$.

Now, there is also a decomposition of $M//G$ by infinitesimal orbit types: $$ M//G=\bigsqcup_{(\mathfrak{h})}(M//G)_{(\mathfrak{h})},\qquad (M//G)_{(\mathfrak{h})}=M_{(\mathfrak{h})}/G, $$ where $M_{(\mathfrak{h})}=\{p\in M: \exists\, g\in G,\; \mathfrak{g}_p=\mathrm{Ad}(g)\mathfrak{h}\}$, $\mathfrak{g}_p$ being the isotropy algebra of the point $p$.

If $G$ is the circle group $S^1$, then this decomposition is a stratification and is described, for example, in Lerman-Tolman, Intersection cohomology of $S^1$ symplectic quotients and small resolutions.

Let $G$ be a compact connected Lie group acting in a Hamiltonian fashion on a symplectic manifold $M$ with momentum map $\mu:M\to \mathfrak{g}^\ast$, where $\mathfrak{g}$ is the Lie algebra of $G$. Then several standard textbooks (for example Ortega–Ratiu: Momentum Maps and Hamiltonian Reduction (MSN)) treat the stratification of the symplectic quotient $$ M//G:=J^{-1}(0)/G $$ by orbit types $(H)$ of the $G$-action: $$ M//G=\bigsqcup_{(H)}(M//G)_{(H)},\qquad (M//G)_{(H)}=M_{(H)}/G, $$ where $M_{(H)}=\{p\in M: \exists\, g\in G,\; G_p =gHg^{-1}\}$, $G_p$ being the isotropy group of the point $p$.

Now, there is also a decomposition of $M//G$ by infinitesimal orbit types: $$ M//G=\bigsqcup_{(\mathfrak{h})}(M//G)_{(\mathfrak{h})},\qquad (M//G)_{(\mathfrak{h})}=M_{(\mathfrak{h})}/G, $$ where $M_{(\mathfrak{h})}=\{p\in M: \exists\, g\in G,\; \mathfrak{g}_p=\operatorname{Ad}(g)\mathfrak{h}\}$, $\mathfrak{g}_p$ being the isotropy algebra of the point $p$.

If $G$ is the circle group $S^1$, then this decomposition is a stratification and is described, for example, in Lerman–Tolman: Intersection cohomology of $S^1$ symplectic quotients and small resolutions (MSN).

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asked Mar 16, 2020 at 18:26

B K

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Infinitesimal orbit type decomposition of Hamiltonian $G$-manifolds

Let $G$ be a compact connected Lie group acting in a Hamiltonian fashion on a symplectic manifold $M$ with momentum map $\mu:M\to \mathfrak{g}^\ast$, where $\mathfrak{g}$ is the Lie algebra of $G$. Then several standard textbooks (for example Ortega-Ratiu: Momentum Maps and Hamiltonian Reduction) treat the stratification of the symplectic quotient $$ M//G:=J^{-1}(0)/G $$ by orbit types $(H)$ of the $G$-action: $$ M//G=\bigsqcup_{(H)}(M//G)_{(H)},\qquad (M//G)_{(H)}=M_{(H)}/G, $$ where $M_{(H)}=\{p\in M: \exists\, g\in G,\; G_p =gHg^{-1}\}$, $G_p$ being the isotropy group of the point $p$.

Now, there is also a decomposition of $M//G$ by infinitesimal orbit types: $$ M//G=\bigsqcup_{(\mathfrak{h})}(M//G)_{(\mathfrak{h})},\qquad (M//G)_{(\mathfrak{h})}=M_{(\mathfrak{h})}/G, $$ where $M_{(\mathfrak{h})}=\{p\in M: \exists\, g\in G,\; \mathfrak{g}_p=\mathrm{Ad}(g)\mathfrak{h}\}$, $\mathfrak{g}_p$ being the isotropy algebra of the point $p$.