Slice theorem

A theorem reducing the description of the action of a transformation group on some neighbourhood $U$ of a given orbit to that of the stabilizer $H$ of a point $x$ of this orbit on some space $S$ which is "normal" to the orbit at $x$. Namely, this theorem claims that $U$ is the homogeneous fibre space over $G / H$ with fibre $S$. Below, the precise setting and formulation are given, together with a counterpart of the theorem for algebraic transformation groups, called the étale slice theorem.

Slice theorem for topological transformation groups.

Let $G$ be a topological transformation group of a Hausdorff space $X$. A subspace $S$ of $X$ is called a slice at a point $x \in S$ if the following conditions hold:

i) $S$ is invariant under the stabilizer $G_{X}$ of $x$;

ii) the union $G ( S )$ of all orbits intersecting $S$ is an open neighbourhood of the orbit $G ( x )$ of $x$;

iii) if $G \times_{ G _ { x }} S$ is the homogeneous fibre space over $G /G_x$ with fibre $S$, then the equivariant mapping $\varphi : G \times _ { G _ { x } } S \rightarrow X$, which is uniquely defined by the condition that its restriction to the fibre $S$ over $G_{X}$ is the identity mapping $S \rightarrow S$ (cf. also Equivariant cohomology), is a homeomorphism of $G \times_{ G _ { x }} S$ onto $G ( S )$. Equivalent definitions are obtained by replacing iii) either by:

iv) there is an equivariant mapping $\pi : G ( S ) \rightarrow G ( x )$ that is the identity on $G ( x )$ and is such that $\pi ^ { - 1 } ( x ) = S$; or by

v) $S$ is closed in $G ( S )$ and $g ( S ) \cap S \neq \emptyset$ implies $g \in G _ { x }$.

The slice theorem claims that if certain conditions hold, then there is a slice at a point $x \in X$. The conditions depend on the case under consideration. The necessity of certain conditions is explained by the following observation.

If there is a slice $S$ at $x$, then there is a neighbourhood $U$ of $x$ (namely, $G ( S )$) such that the stabilizer of every point of $U$ is conjugate to a subgroup of $G_{X}$. In general, this property fails (e.g., take $G = \operatorname{SL} _ { 2 } ( \mathbf C )$ acting on the space of binary forms of degree $3$ in the variables $t_{1}$, $t_2$ by linear substitutions. Then the stabilizer of $x = t _ { 1 } ^ { 2 } t _ { 2 }$ is trivial but every neighbourhood of $x$ contains a point whose stabilizer has order $3$.)

The first case in which the validity of the slice theorem was investigated is that of a compact Lie group $G$. In this case, it has been proved that if $X$ is a fully regular space, then there is a slice at every point $x \in X$. If, moreover, $X$ is a differentiable manifold and $G$ acts smoothly, then at every $x$ there is a differentiable slice of a special kind. Namely, in this case there is an equivariant diffeomorphism $\varphi$, being the identity on $G ( x )$, of the normal vector bundle (cf. also Normal space (to a surface)) of $G ( x )$ onto an open neighbourhood of $G ( x )$ in $X$. The image under $\varphi$ of the fibre of this bundle over $x$ is a slice at $x$ which is a smooth submanifold of $X$ diffeomorphic to a vector space.

A particular case of the slice theorem for compact Lie groups was for the first time ever proven in [a2]. Then the differentiable and general versions, formulated above, were proven, respectively in [a4] and [a8], [a9], [a10].

The slice theorem is an indispensable tool in the theory of transformation groups, which frequently makes it possible to reduce an investigation to simple group actions like linear ones. In particular, the slice theorem is the key ingredient in the proofs of the following two basic facts of the theory:

1) If $G$ is a compact Lie group, $X$ a separable metrizable space and there are only finitely many conjugacy classes of stabilizers of points in $X$, then there is an equivariant embedding of $X$ in a Euclidean vector space endowed with an orthogonal action of $G$.

2) Let $G$ be a compact Lie group acting smoothly on a connected differentiable manifold $X$. Then there are a subgroup $G_{*}$ of $G$ and a dense open subset $\Omega$ of $X$ such that $\text{codim} ( X \backslash \Omega ) \geq 1$ and the stabilizer of every point $x \in X$ is conjugate to a subgroup of $G_{*}$ which coincides with $G_{*}$ if $x \in \Omega$.

There are versions of the slice theorem for non-compact groups. For instance, let $H$ be an algebraic complex reductive group and $H \rightarrow \operatorname{GL} ( V )$ its finite-dimensional algebraic representation, both defined over the real numbers $\mathbf{R}$. Let $H _ { \mathbf{R} }$ be the Lie group of real points of $H$ and $G$ a subgroup of $H _ { \mathbf{R} }$ containing the connected component of identity element. Let $X$ be a closed $H$-invariant differentiable submanifold of $V _ { \mathbf{R} }$, the space of real points of $V$. Then, [a7], for every closed orbit $G ( x )$ in $X$ there is an equivariant diffeomorphism $\varphi$ of a $G$-invariant neighbourhood $\tilde { U }$ of $G ( x )$ in the normal vector bundle of $G ( x )$ onto a $G$-invariant saturated neighbourhood $U$ of $G ( x )$ in $X$ (a neighbourhood $U$ is saturated if the fact fact that the closure of an orbit intersects $U$ implies that this orbit lies in $U$). In this case, the image under $\varphi$ of the fibre of the natural projection $\tilde { U } \rightarrow G ( x )$ over $x$ is a slice at $x$ for the action of $G$ on $X$.

Slice theorem for algebraic transformation groups.

Let $G$ be an algebraic transformation group of an algebraic variety $X$, all defined over an algebraically closed field $k$. It happens very rarely that the literal analogue of the slice theorem above holds in this setting. Loosely speaking, the reason is that Zariski-open sets are "too big" (see [a12], 6.1). One obtains an algebraic counterpart of a compact Lie group action on a differentiable manifold by taking $G$ to be a reductive group and $X$ an affine variety. In this setting, a counterpart of a slice is given by the notion of an étale slice, defined as follows.

Let $x$ be a point of $X$ such that the orbit $G ( x )$ is closed. Let $S$ be an affine $G_{X}$-invariant subvariety of $X$ containing $x$. As above, one can consider the homogeneous fibre space $G \times_{ G _ { x }} S$ over $G /G_x$ with fibre $S$, and mapping $\varphi : G \times _ { G _ { x } } S \rightarrow X$. In this situation, $G \times_{ G _ { x }} S$ is an affine variety, $\varphi$ is a morphism, there are the categorical quotients $\pi : X \rightarrow X // G$, $\pi _ { G \times_{ Gx } S} : G \times _ { G _ { X } } S \rightarrow ( G \times _ { Gx } S ) / / G$ and an induced morphism $\varphi /\!/ G : ( G \times_{ G _ { x }} S ) / \!/ G \rightarrow X /\! / G$, cf. [a12]. The subvariety $S$ is called an étale slice at $x$ if

i) $\pi _ { G \times_{ G _ { X }} } S$ is obtained from $\pi _ X$ by means of the base change $\varphi H G$; and

ii) $\varphi H G$ is an étale morphism.

The étale slice theorem, proved in [a6], claims that there is an étale slice at every point $x \in X$ such that the orbit $G ( x )$ is closed.

If $k$ is $\mathbf{C}$, the field of complex numbers, and $x$ is a smooth point of $X$, then the étale slice theorem implies that there exists an analytic slice at $x$. More precisely, there is an invariant analytic neighbourhood of $G ( x )$ in $X$ which is analytically isomorphic to an invariant analytic neighbourhood of $G ( x )$ in the normal vector bundle of $G ( x )$, cf. [a7], [a12].

Like the slice theorem for topological transformation groups, the étale slice theorem is an indispensable result in the investigation of algebraic transformation groups; see [a6] for some basic results deduced from this theorem.

References

[a1]	G.E. Bredon, "Introduction to compact transformation groups" , Acad. Press (1972) MR0413144 Zbl 0246.57017
[a2]	A.M. Gleason, "Spaces with a compact Lie group of transformations" Proc. Amer. Math. Soc. , 1 (1950) pp. 35–43 MR0033830 Zbl 0041.36207
[a3]	K. Jänich, "Differenzierbare $G$-Mannigfaltigkeiten" , Lecture Notes Math. , 6 , Springer (1968) MR0202157 Zbl 0153.53703
[a4]	J.L. Koszul, "Sur certains groupes de transformation de Lie" Colloq. Inst. C.N.R.S., Géom. Diff. , 52 (1953) pp. 137–142 MR0059919
[a5]	Koszul, J.L., "Lectures on groups of transformations" , Tata Inst. (1965) MR218485 Zbl 0195.04605
[a6]	Luna,D., "Slices étales" Bull. Soc. Math. France , 33 (1973) pp. 81–105 Zbl 0286.14014
[a7]	Luna, D., "Sur certaines opérations différentiables des groups de Lie" Amer. J. Math. , 97 (1975) pp. 172–181
[a8]	D. Montgomery, C.T. Yang, "The existence of slice" Ann. of Math. , 65 (1957) pp. 108–116 Zbl 0078.16202
[a9]	G.D. Mostow, "On a theorem of Montgomery" Ann. of Math. , 65 (1957) pp. 432–446
[a10]	R. Palais, "Embeddings of compact differentiable transformation groups in orthogonal representations" J. Math. Mech. , 6 (1957) pp. 673–678
[a11]	R.S. Palais, "Slices and equivariant imbeddings" , Sem. Transformation Groups , Princeton Univ. Press (1960)
[a12]	V.L. Popov, E.B. Vinberg, "Invariant theory" , Algebraic Geometry IV , Encycl. Math. Sci. , 55 , Springer (1994) pp. 122–284 MR1456471 Zbl 1099.13012 Zbl 1088.81075 Zbl 1065.82003 Zbl 1053.82006 Zbl 0783.14028 Zbl 0754.13005 Zbl 0736.15019 Zbl 0735.14010 Zbl 0789.14008 Zbl 0679.14024 Zbl 0491.14004 Zbl 0478.14006
[a13]	Wu Yi Hsiang, "Cohomology theory of topological transformation groups" , Ergebn. Math. , 85 , Springer (1979) Zbl 0511.57002