The log symplectic geometry of Poisson slices
aa r X i v : . [ m a t h . S G ] A ug THE LOG SYMPLECTIC GEOMETRY OF POISSON SLICES
PETER CROOKS AND MARKUS R ¨OSER
Abstract.
Our paper develops a theory of Poisson slices and a uniform approach to their partialcompactifications. The theory in question is loosely comparable to that of symplectic cross-sectionsin real symplectic geometry.
Contents
1. Introduction 21.1. Motivation and context 21.2. Summary of Results 31.3. Organization 5Acknowledgements 52. Preliminaries 52.1. Fundamental conventions 62.2. Quotients of K -varieties 62.3. Poisson varieties 82.4. Hamiltonian reduction 92.5. Lie-theoretic conventions 102.6. The wonderful compactification 112.7. Poisson geometry on T ∗ G and T ∗ G (log D ) 113. Poisson slices 133.1. Poisson transversals and Poisson slices 133.2. Poisson slices via Hamiltonian reduction 163.3. Poisson slices in the log cotangent bundle of G X and X τ X and X τ X and X τ X τ and X Key words and phrases.
Poisson slice, log symplectic variety, wonderful compactification. Introduction
Motivation and context.
The Poisson slice construction yields a number of varieties relevantto geometric representation theory and symplectic geometry. One begins with a complex semisimplelinear algebraic group with Lie algebra g . Let us also consider a Hamiltonian G -variety X , i.e. asmooth Poisson variety with a Hamiltonian action of G and moment map ν : X −→ g . Each sl -triple τ = ( ξ, h, η ) ∈ g ⊕ determines a Slodowy slice S τ := ξ + g η ⊆ g , and the preimage X τ := ν − ( S τ )is a Poisson transversal in X . The variety X τ is thereby Poisson, and we call it the Poissonslice determined by X and τ . To a certain extent, Poisson slices are complex Poisson-geometriccounterparts of symplectic cross-sections [27, 28, 31, 38] in real symplectic geometry.Noteworthy examples of Poisson slices include the product G ×S τ , a hyperk¨ahler and Hamiltonian G -variety studied by Bielawski [5, 6], Moore–Tachikawa [41], and several others [1, 11, 13–15]. Asecond example is a Coulomb branch [7] called the universal centralizer Z τ g := { ( g, y ) ∈ G × S τ : Ad g ( y ) = y } , where τ is a fixed principal sl -triple in g . This hyperk¨ahler variety has received considerableattention in the literature [4, 7, 12, 40, 51, 55, 56], and it features prominently in B˘alibanu’s recentpaper [2]. B˘alibanu assumes G to be of adjoint type. She harnesses the geometry of the wonderfulcompactification G and constructs a fibrewise compactification Z τ g −→ S τ of Z τ g −→ S τ , wherethe latter map is projection onto the S τ -factor. She subsequently endows Z τ g with a log symplecticstructure.The preceding discussion gives rise to the following rough questions. • Is there a coherent and systematic approach to the partial compactification of Poisson slicesthat is related to G and specializes to yield Z τ g −→ S τ as a fibrewise compactification of Z τ g −→ S τ ? • If the previous question has an affirmative answer and X τ is symplectic, does the partialcompactification of X τ carry a log symplectic structure?Our inquiry stands to benefit from two observations. One first notes the universal or atomic natureof G × S τ as a Poisson slice, i.e. the existence of a canonical Poisson variety isomorphism X τ ∼ = ( X × ( G × S τ )) (cid:12) G for each Hamiltonian G -variety X and sl -triple τ in g . These atomic Poisson slices have coun-terparts in the theories of symplectic cross-sections [31], symplectic implosion [27], symplectic con-traction [34], hyperk¨ahler implosion [16, 17], and Kronheimer’s hyperk¨ahler quotient with momen-tum [37]. A second observation is that G × S τ sits inside of a larger log symplectic variety G × S τ as the unique open dense symplectic leaf; the construction of G × S τ assumes G to be of adjointtype and exploits the geometry of G .The preceding considerations motivate us to define X τ := ( X × ( G × S τ )) (cid:12) G and conjecture that X τ is the desired partial compactification of X τ . While this naive conjectureneeds to be refined and made more precise, it inspires many of the results in our paper. HE LOG SYMPLECTIC GEOMETRY OF POISSON SLICES 3
Summary of Results.
Our paper develops a detailed theory of Poisson slices and addressesthe questions posed above. The following is a summary of our results. We work exclusively over C and take all Poisson varieties to be smooth. We use the Killing form to freely identify g ∗ with g , aswell as the left trivialization and Killing form to freely identify T ∗ G with G × g .Suppose that X is a Poisson Hamiltonian G -variety with moment map ν : X −→ g . Let τ be an sl -triple in g and consider the Poisson transversal X τ := ν − ( S τ ) ⊆ X. The following are some first properties of the Poisson slice X τ . Such properties are well-known inthe case of a symplectic variety X (see [5]). Proposition 1.1.
Let X be a Poisson variety endowed with a Hamiltonian G -action and momentmap ν : X −→ g . Suppose that τ = ( ξ, h, η ) is an sl -triple in g . The following statements hold. (i) The Poisson slice X τ is transverse to the G -orbits in X . (ii) There are canonical Poisson variety isomorphisms ( X × ( G × S τ )) (cid:12) G ∼ = X τ ∼ = X (cid:12) ξ U τ . The Hamiltonian G -variety structure on G × S τ and meaning of the unipotent subgroup U τ ⊆ G are given in Subsection 3.2. We also consider some special cases of the Poisson slice construction, including the followingwell-known result in the symplectic category.
Observation 1.2.
Let X be a symplectic variety endowed with a Hamiltonian action of G and amoment map ν : X −→ g . Suppose that τ is an sl -triple in g . The Poisson structure on X τ makesit a symplectic subvariety of X .Now suppose that the above-mentioned Poisson variety X is log symplectic [23, 25, 47, 50], bywhich the following is meant: X has a unique open dense symplectic leaf, and the degeneracylocus of the Poisson bivector is a reduced normal crossing divisor. We establish the following logsymplectic counterpart of Observation 1.2. Proposition 1.3.
Let X be a log symplectic variety endowed with a Hamiltonian G -action andmoment map ν : X −→ g . Suppose that τ is any sl -triple in g . Each irreducible component of X τ is then a Poisson subvariety of X τ . The resulting Poisson structure on each component makes thecomponent a log symplectic subvariety of X . Now assume G to be of adjoint type. One may consider the De Concini–Procesi wonderfulcompactification G of G [18], along with the divisor D := G \ G . The data ( G, D ) determine a logcotangent bundle T ∗ G (log( D )), which is known to have a canonical log symplectic structure. Itsunique open dense symplectic leaf is T ∗ G , and the canonical Hamiltonian ( G × G )-action on T ∗ G extends to such an action on T ∗ G (log( D )). The moment maps ρ = ( ρ L , ρ R ) : T ∗ G −→ g ⊕ g and ρ = ( ρ L , ρ R ) : T ∗ G (log( D )) −→ g ⊕ g can be written in explicit terms. This leads to the following straightforward observations, whoseproofs use Observation 1.2 and Proposition 1.3. To this end, recall that a principal sl -triple is an sl -triple consisting of regular elements. Observation 1.4.
Let τ = ( ξ, h, η ) be a principal sl -triple in g , and consider the principal sl -triple( τ, τ ) := (( ξ, ξ ) , ( h, h ) , ( η, η )) in g ⊕ g . One then has( T ∗ G ) ( τ,τ ) = ρ − ( S τ × S τ ) = Z τ g , and ( T ∗ G (log D )) ( τ,τ ) = ρ − ( S τ × S τ ) = Z τ g . The first Poisson slice is symplectic, while the second is log symplectic.
PETER CROOKS AND MARKUS R ¨OSER
Observation 1.5.
Consider the Hamiltonian action of G = { e } × G ⊆ G × G on T ∗ G . If τ is an sl -triple in g , then ( T ∗ G ) τ = ρ − R ( S τ ) = G × S τ . This Poisson slice is symplectic.In light of these observations, it is natural to consider the Poisson slice G × S τ := ρ − R ( S τ ) ⊆ T ∗ G (log D ) . One has an inclusion G × S τ ⊆ G × S τ , while G × S τ and G × S τ carry residual Hamiltonian actionsof G = G × { e } ⊆ G × G . The respective moment maps are ρ τ := ρ L (cid:12)(cid:12) G ×S τ and ρ τ := ρ L (cid:12)(cid:12) G ×S τ , and they feature in the following result. Theorem 1.6.
Let τ be an sl -triple in g . (i) The Poisson slice G × S τ is irreducible and log symplectic. (ii) The inclusion G × S τ −→ G × S τ is a G -equivariant symplectomorphism onto the uniqueopen dense symplectic leaf in G × S τ . (iii) The diagram G × S τ G × S τ g ρ τ ρ τ (1.1) commutes. (iv) If τ is a principal sl -triple, then (1.1) realizes ρ τ as a fibrewise compactification of ρ τ . Our paper subsequently discusses the relation of (1.1) to B˘alibanu’s fibrewise compactification Z τ g Z τ g S τ . (1.2)We next study Hamiltonian reductions of the form X τ := ( X × ( G × S τ )) (cid:12) G, where X is a Hamiltonian G -variety and τ is an sl -triple in g . The special case τ = 0 featuresprominently in our analysis, and we write X for X τ if τ = 0. This amounts to setting X := ( X × T ∗ G (log D )) (cid:12) G, with G acting as G = G × { e } ⊆ G × G on T ∗ G (log D ).One has the preliminary issue of whether the quotients X τ and X exist. The following resultprovides some sufficient conditions. Lemma 1.7.
Let X be a Hamiltonian G -variety. (i) If X is a principal G -bundle, then X exists as a geometric quotient. (ii) If X exists as a geometric quotient, then X τ exists as a geometric quotient for all sl -triples τ in g . (iii) If X = T ∗ Y for an irreducible smooth principal G -bundle Y , then X τ exists as a geometricquotient for all sl -triples τ in g . HE LOG SYMPLECTIC GEOMETRY OF POISSON SLICES 5
The variety X τ enjoys certain Poisson-geometric features. A first step in this direction is to set X ◦ τ := ( X × ( G × S τ )) ◦ (cid:12) G, where ( X × G × S τ ) ◦ is the open set of points in ( X × ( G × S τ )) ◦ with trivial G -stabilizers. Thevariety X ◦ τ exists as a geometric quotient if X exists as a geometric quotient, in which case one hasinclusions X τ ⊆ X ◦ τ ⊆ X τ Theorem 1.8.
Let X be a Hamiltonian G -variety, and suppose that τ is an sl -triple in g . Assumethat X τ exists as a geometric quotient. (i) The coordinate ring C [ X τ ] carries a natural Poisson bracket for which restriction C [ X τ ] −→ C [ X τ ] is a Poisson algebra morphism. (ii) The variety X ◦ τ is smooth and Poisson, and it contains X τ as an open Poisson subvariety. (iii) If X is symplectic, then each irreducible component of X ◦ τ is log symplectic. (iv) If X is symplectic and X τ is irreducible, then X τ is the open dense symplectic leaf in aunique irreducible component of X ◦ τ . Our final main result addresses the extent to which X τ partially compactifies X τ . We begin byassuming that both X τ and X/G exist as geometric quotients. This allows us to construct canonicalmaps π τ : X τ −→ X/G and π τ : X τ −→ X/G.
It is then straightforward to deduce that X τ X τ X/G π τ π τ (1.3)commutes, where the horizontal arrow is inclusion. This leads to the following theorem. Theorem 1.9.
Let X be a Hamiltonian G -variety, and suppose that τ is a principal sl -triple in g .If X τ and X/G exist as geometric quotients, then (4.10) realizes π τ as a fibrewise compactificationof π τ . In the case of a principal sl -triple τ , we realize the fibrewise compactifications (1.1) and (1.2) asspecial instances of Theorem 1.9.1.3. Organization.
In Section 2, we introduce the concepts from Lie theory and Poisson geometrythat form the foundation for our work. Section 3 details the theory of Poisson slices and providescomplete proofs of Propositions 1.1 and 1.3. Section 4 subsequently considers the Poisson sliceenlargements X τ mentioned above, and it contains the proofs of Theorems 1.6, 1.8 and 1.9. Weconclude with Section 5, which is devoted to illustrative examples. A list of recurring notationappears after Section 5. Acknowledgements.
The first author gratefully acknowledges the Natural Sciences and Engineer-ing Research Council of Canada for support [PDF–516638].2.
Preliminaries
This section provides some of the notation, conventions, and basic results used throughout ourpaper.
PETER CROOKS AND MARKUS R ¨OSER
Fundamental conventions.
We work exclusively over C and understand all group actions asbeing left group actions. We also write O X for the structure sheaf of an algebraic variety X , as well as C [ X ] for the coordinate ring O X ( X ). The dimension of X is understood to be the supremum of thedimensions of the irreducible components. We understand X to be smooth if dim( T x X ) = dim( X )for all x ∈ X . Note that this convention forces a smooth variety to be pure-dimensional.2.2. Quotients of K -varieties. Let K be a linear algebraic group. We adopt the term K -variety in reference to a variety X endowed with an algebraic K -action. Definition 2.1.
Suppose that X is a K -variety. A variety morphism π : X −→ Y is called a categorical quotient of X if the following conditions are satisfied:(i) π is K -invariant;(ii) if θ : X −→ Z is a K -invariant variety morphism, then there exists a unique morphism ϕ : Y −→ Z for which XY Z π θϕ commutes.
Definition 2.2.
Suppose that X is a K -variety. A variety morphism π : X −→ Y is called a goodquotient of X if the following conditions are satisfied:(i) π is surjective, affine, and K -invariant;(ii) if U ⊆ Y is open, then the comorphism π ∗ : O Y ( U ) −→ O X ( π − ( U )) is an isomorphismonto O X ( π − ( U )) K ;(iii) if Z ⊆ X is closed and K -invariant, then π ( Z ) is closed in Y ;(iv) if Z , Z ⊆ X are closed, K -invariant, and disjoint, then π ( Z ) and π ( Z ) are disjoint.One calls π : X −→ Y a geometric quotient of X if π is a good quotient and π − ( y ) is a K -orbitfor each y ∈ Y .Let X be a K -variety admitting a geometric quotient π : X −→ Y , and write X/K for the setof K -orbits in X . One then has a canonical bijection Y ∼ = X/K , through which
X/K inherits avariety structure. Any two geometric quotients π : X −→ Y and π ′ : X −→ Y ′ induce the samevariety structure on X/K , and this structure makes the set-theoretic quotient map X −→ X/K ageometric quotient. With this in mind, we shall sometimes write “
X/K exists” or “the geometricquotient X −→ X/K exists” to mean that X admits a geometric quotient. Lemma 2.3.
Assume that K is connected and reductive. Suppose that X is an affine K -variety,and let π : X −→ Y be a variety morphism. If π − ( y ) is a K -orbit for each y ∈ Y , then π is ageometric quotient of X .Proof. One knows that X admits a good quotient θ : X −→ Z (e.g. [52, Theorem 1.4.2.4]), andthat this quotient is also categorical (e.g. [52, Lemma 1.4.1.1]). It follows that XZ Y θ πϕ (2.1)commutes for some morphism ϕ : Z −→ Y . The surjectivity of π then forces ϕ to be surjective.We claim that ϕ is an isomorphism, i.e. that ϕ is also injective. To this end, let z , z ∈ Z besuch that ϕ ( z ) = ϕ ( z ). Choose x , x ∈ X satisfying θ ( x ) = z and θ ( x ) = z . One then has HE LOG SYMPLECTIC GEOMETRY OF POISSON SLICES 7 π ( x ) = π ( x ), so that x and x belong to the same K -orbit in X . The K -invariance of θ nowimplies that θ ( x ) = z and θ ( x ) = z must coincide. We conclude that ϕ is injective, implyingthat ϕ is indeed an isomorphism. This combines with the commutativity of (2.1) to tell us that π is a good quotient. Since π − ( y ) is a K -orbit for all y ∈ Y , one deduces that π is a geometricquotient. (cid:3) Proposition 2.4.
Assume that K is connected and reductive. Suppose that X is a K -varietyadmitting a geometric quotient π : X −→ Y . If Z ⊆ X is a K -invariant locally closed subvariety,then π ( Z ) is locally closed in Y and π (cid:12)(cid:12) Z : Z −→ π ( Z ) is a geometric quotient of Z .Proof. We begin by showing π ( Z ) to be locally closed in Y . To this end, note that the closure Z ⊆ X is K -invariant. It follows that π ( Z ) is a closed subvariety of Y . An application of [54, Lemma 25.3.2]now forces π (cid:12)(cid:12) Z : Z −→ π ( Z )to be an open map. Since Z is open in Z , we conclude that π ( Z ) is open in the closed subvariety π ( Z ) ⊆ Y . This implies that π ( Z ) is locally closed in Y .Now observe that π (cid:12)(cid:12) Z : Z −→ π ( Z ) is the base change of π under the locally closed immersion π ( Z ) ֒ → Y . Since affine morphisms are stable under base change (e.g. [45, Exercise 4.E.9(2)]), itfollows that π (cid:12)(cid:12) Z is affine. We may therefore cover π ( Z ) with affine open subsets { V α } α ∈A in sucha way that { U α := ( π (cid:12)(cid:12) Z ) − ( V α ) } α ∈A is a cover of Z by affine open subsets. By virtue of [52, Exercise 1.4.1.2 ii)], we are reduced toverifying that each morphism π (cid:12)(cid:12) U α : U α −→ V α is a geometric quotient. This reduction combines with Lemma 2.3 to complete the proof. (cid:3) We will need the following algebro-geometric notion of a principal bundle appearing in [8, Defi-nition 2.3.1].
Definition 2.5.
Suppose that X is a K -variety. A K -invariant variety morphism π : X −→ Y iscalled a principal K -bundle if the following conditions hold:(i) π is faithfully flat, i.e. flat and surjective;(ii) the natural map σ : K × X −→ X × Y X, σ ( k, x ) = ( x, k · x )is an isomorphism.A principal K -bundle is necessarily a geometric quotient (e.g. by [8, Proposition 2.3.3]). Weunderstand “ X is a principal K -bundle” as meaning that X admits a geometric quotient π : X −→ X/K , and that π is a principal K -bundle. Lemma 2.6.
Let X be a smooth K -variety with a free K -action and a good quotient π : X −→ Y .The map π is then a principal K -bundle.Proof. Since the K -action is free, all K -orbits are closed in X . It now follows from Definition 2.1(iv)that π separates K -orbits. Each fibre of π is therefore a single K -orbit. This combines with thesmoothness of X and the freeness of the K -action to imply that π is flat. PETER CROOKS AND MARKUS R ¨OSER
It remains only to prove that σ : K × X −→ X × Y X, σ ( k, x ) = ( x, k · x )is an isomorphism. This follows immediately from the freeness of the K -action and the fact thatthe fibres of π are the K -orbits in X . (cid:3) Poisson varieties.
Let X be a smooth variety. Suppose that P is a global section of Λ ( T X ),and consider the bracket operation defined by { f , f } := P ( df ∧ df ) ∈ O X for all f , f ∈ O X . One calls P a Poisson bivector if this bracket renders O X a sheaf of Poissonalgebras. We use the term Poisson variety in reference to a smooth variety X equipped with aPoisson bivector P . In this case, {· , ·} is called the Poisson bracket . Let us also recall that avariety morphism φ : X −→ X between Poisson varieties ( X , P ) and ( X , P ) is called a Poissonmorphism if dφ ( P ( φ ∗ α )) = P ( α )for all one-forms α defined on any open subset of X . Our convention is to have ( X × X , P ⊕ ( − P ))be the Poisson variety product of ( X , P ) and ( X , P ).Let ( X, P ) be a Poisson variety. Contracting the bivector with cotangent vectors allows one toview P as a bundle morphism P : T ∗ X −→ T X, whose image is a holomorphic distribution on X . One refers to the maximal integral submanifoldsof this distribution as the symplectic leaves of X . The symplectic form ω L on a symplectic leaf L ⊆ X is constructed as follows. One defines the Hamiltonian vector field of a locally definedfunction f ∈ O X by H f := − P ( df ) . (2.2)This gives rise to the tangent space description T x L = { ( H f ) x : f ∈ O X } for all x ∈ L , and one has ( ω L ) x (( H f ) x , ( H f ) x ) = { f , f } ( x )for all x ∈ L and f , f ∈ O X defined near x .We conclude by discussing log symplectic varieties , which have received considerable attention inrecent years (e.g. [2, 10, 23–26, 29, 30, 32, 42, 43, 47, 49, 50]). To this end, one calls a Poisson variety( X, P ) log symplectic if the following conditions hold:(i) ( X, P ) has a unique open dense symplectic leaf X ⊆ X ;(ii) the vanishing locus of P n is a reduced normal crossing divisor D ⊆ X , where 2 n = dim( X )and P n ∈ H ( X, Λ n ( T X )) is the top exterior power of P .In this case, we call D the divisor of ( X, P ). One immediate observation is that D = X \ X . Remark 2.7.
Since symplectic leaves are connected, Condition (i) implies that log symplecticvarieties are irreducible.
HE LOG SYMPLECTIC GEOMETRY OF POISSON SLICES 9
Hamiltonian reduction.
We now review the salient aspects of Hamiltonian actions in thePoisson category. To this end, let K be a linear algebraic group with Lie algebra k . Let ( X, P ) be aPoisson variety, and assume that X is also a K -variety. Each y ∈ k then determines a fundamentalvector field V y on X via ( V y ) x = ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 (exp( ty ) · x ) ∈ T x X for all x ∈ X . The K -action on X is called Hamiltonian if P is K -invariant and there exists a K -equivariant morphism ν : X −→ k ∗ satisfying the following condition: H ν y = − V y (2.3)for all y ∈ k , where ν y ∈ C [ X ] is defined by ν y ( x ) = ν ( x )( y ) , x ∈ X. (2.4)One then refers to ν as a moment map and calls ( X, P, ν ) a
Hamiltonian K -variety . The momentmap ν is known to be a Poisson morphism with respect to the Lie–Poisson structure on k ∗ (e.g. [9,Proposition 7.1]).We now briefly recall the process of Hamiltonian reduction for a Hamiltonian K -variety ( X, P, ν ).One begins by observing that ν − (0) is a K -invariant closed subvariety of X . Let us assume X (cid:12) K exists, by which we mean that the geometric quotient π : ν − (0) −→ ν − (0) /K (2.5)exists. Write X (cid:12) K := ν − (0) /K, and note that the comorphism π ∗ : C [ X (cid:12) K ] −→ C [ ν − (0)] induces an algebra isomorphism C [ X (cid:12) K ] ∼ = −→ C [ ν − (0)] K . (2.6)At the same time, the canonical surjection C [ X ] −→ C [ ν − (0)] restricts to a surjection C [ X ] K −→ C [ ν − (0)] K (2.7)if K is connected and reductive. One also knows that C [ X ] K is a Poisson subalgebra of C [ X ],and that the kernel of (2.7) is a Poisson ideal I ⊆ C [ X ] K . It follows that C [ ν − (0)] K inherits thestructure of a Poisson algebra. One may therefore endow C [ X (cid:12) K ] with the unique Poisson bracketfor which (2.6) is an isomorphism of Poisson algebras. We refer to the data of the variety X (cid:12) K and the Poisson algebra C [ X (cid:12) K ] as the Hamiltonian reduction of (
X, P, ν ) if (2.5) exists and K is connected and reductive.The Hamiltonian reduction process will yield a richer geometric object in the presence of certainassumptions about the K -action on X . To this end, let K be a linear algebraic group and supposethat ( X, P, ν ) is a Hamiltonian K -variety. Assume that the geometric quotient (2.5) exists, andthat K acts freely on ν − (0). The closed subvariety ν − (0) ⊆ X is then smooth, and one deducesthat X (cid:12) K is also smooth. One may also define a Poisson bivector P X (cid:12) K on X (cid:12) K as follows.Suppose that x ∈ ν − (0) and let dπ ∗ x : T ∗ π ( x ) ( X (cid:12) K ) −→ T ∗ x ( ν − (0))be the dual of the differential dπ x : T x ( ν − (0)) −→ T π ( x ) ( X (cid:12) K ). Set P π ( x ) ( α ) := dπ x ( P x ( ˜ α ))for all α ∈ T ∗ π ( x ) ( X (cid:12) K ), where ˜ α ∈ T ∗ x X is any element that annihilates T x ( Kx ) and coincides with dπ ∗ x ( α ) on T x ( ν − (0)). The bivector P X (cid:12) K renders O X (cid:12) K a sheaf of Poisson algebras, recovering the above-described Poisson bracket on C [ X (cid:12) K ]. We call the Poisson variety ( X (cid:12) ζ K, P X (cid:12) ζ K )the Hamiltonian reduction of (
X, P, ν ) at level ζ , provided that (2.5) exists and K is known to actfreely on ν − ( ζ ).The preceding construction generalizes to allow for Hamiltonian reduction at an arbitrary level ζ ∈ k ∗ . To this end, let K ζ denote the K -stabilizer of ζ with respect to the coadjoint action. Onesimply sets X (cid:12) ζ K := ν − ( ζ ) /K ζ if the right-hand side exists as a geometric quotient. The definitions of the Poisson bracket on C [ X (cid:12) ζ K ] and Poisson bivector P X (cid:12) ζ K are analogous to their counterparts above.2.5. Lie-theoretic conventions.
Let G be a connected semisimple linear algebraic group with Liealgebra g . Note that g is a G -module via the adjoint representationAd : G −→ GL( g ) , g −→ Ad g , and a g -module via the other adjoint representationad : g −→ gl ( g ) , y −→ ad y = [ y, · ] . One obtains an induced action of G on the coordinate ring C [ g ] = Sym( g ∗ ), and we write C [ g ] G ⊆ C [ g ] for the subalgebra of all functions fixed by G . The inclusion C [ g ] G ⊆ C [ g ] then determines amorphism of affine varieties χ : g −→ Spec( C [ g ] G ) , often called the adjoint quotient .Define the centralizer subalgebra g y := { z ∈ g : [ y, z ] = 0 } ⊆ g for each y ∈ g . An element y ∈ g is called regular if the dimension of g y coincides with the rank of g . The set of all regular elements is a G -invariant open dense subvariety of g that we denote by g r .Recall that ( ξ, h, η ) ∈ g ⊕ is an sl -triple if the identities[ ξ, η ] = h, [ h, ξ ] = 2 ξ, and [ h, η ] = − η hold in g , and that the associated Slodowy slice is defined by S τ := ξ + g η ⊆ g . Now assume that τ is a principal sl -triple, i.e. an sl -triple for which ξ, h, η ∈ g r . The slice S τ then lies in g r and is a fundamental domain for the G -action on g r [35, Theorem 8]. This slice isalso known to be a section of the adjoint quotient, meaning that the restriction χ (cid:12)(cid:12) S τ : S τ −→ Spec( C [ g ] G )is a variety isomorphism [35, Theorem 7]. Let us write y τ := ( χ (cid:12)(cid:12) S τ ) − ( χ ( y )) ∈ S τ for each y ∈ g . In other words, y τ is the unique point at which S τ meets χ − ( χ ( y )).Let h· , ·i : g ⊗ C g −→ C denote the Killing form on g . This bilinear form is non-degenerate and G -invariant, i.e. the map g −→ g ∗ , y −→ h y, ·i (2.8)is a G -module isomorphism. The canonical Poisson structure on g ∗ thereby corresponds to a Poissonstructure on g , determined by the following condition: { f , f } ( y ) = h y, [( df ) y , ( df ) y ] i HE LOG SYMPLECTIC GEOMETRY OF POISSON SLICES 11 for all f , f ∈ C [ g ] and y ∈ g , where the right-hand side uses (2.8) to regard ( df ) y , ( df ) y ∈ g ∗ aselements of g . By means of (2.8), we shall make no further distinction between g and g ∗ . One alsohas the ( G × G )-module isomorphism g ⊕ g −→ ( g ⊕ g ) ∗ , ( x , x ) −→ ( h x , ·i , −h x , ·i ) , through which we shall identify g ⊕ g with ( g ⊕ g ) ∗ .2.6. The wonderful compactification.
In this subsection, we assume that G is the adjoint groupof g . Let n = dim g and write Gr( n, g ⊕ g ) for the Grassmannian of all n -dimensional subspaces in g ⊕ g . Note that G × G acts on Gr( n, g ⊕ g ) by( g , g ) · γ := { (Ad g ( y ) , Ad g ( y )) : ( y , y ) ∈ γ } , and on G itself by ( g , g ) · h := g hg − . Let g ∆ ⊆ g ⊕ g denote the diagonally embedded copy of g in g ⊕ g , and consider the ( G × G )-equivariant locally closed immersion ϕ : G −→ Gr( n, g ⊕ g ) , g −→ ( g, e ) · g ∆ . (2.9)We thereby view G as a subvariety of Gr( n, g ⊕ g ) and write G for its closure in Gr( n, g ⊕ g ).The closed subvariety G is ( G × G )-invariant, smooth, and called the wonderful compactification of G [18]. The complement D := G \ G is known to be a normal crossing divisor in G .The pair ( G, D ) determines a so-called log cotangent bundle T ∗ G (log D ) −→ G . One may realizethis vector bundle as the pullback of the tautological bundle T −→
Gr( n, g ⊕ g ) along the inclusion G ֒ → Gr( n, g ⊕ g ). This amounts to setting T ∗ G (log D ) := { ( γ, ( y , y )) ∈ G × ( g ⊕ g ) : ( y , y ) ∈ γ } and defining the bundle projection to be T ∗ G (log D ) −→ G, ( γ, ( y , y )) −→ γ. The action of ( G × G ) on G then lifts to the following ( G × G )-action on T ∗ G (log D ):( g , g ) · ( γ, ( y , y )) := (( g , g ) · γ, (Ad g ( y ) , Ad g ( y ))) . (2.10)2.7. Poisson geometry on T ∗ G and T ∗ G (log D ) . Let all objects and notation be as set in 2.5.Note that the left trivialization and Killing form combine to yield a variety isomorphism T ∗ G ∼ = G × g . We shall thereby make no further distinction between T ∗ G and G × g . The canonical symplecticform ω on T ∗ G is then defined as follows on each tangent space T ( g,x ) ( G × g ) = T g G ⊕ g : ω ( g,x ) (cid:18)(cid:0) ( dL g ) e ( y ) , z (cid:1) , (cid:0) ( dL g ) e ( y ) , z (cid:1)(cid:19) = h y , z i − h y , z i + h x, [ y , y ] i for all y , y , z , z ∈ g , where L g : G −→ G denotes left translation by g and ( dL g ) e : g −→ T g G isthe differential of L g at e ∈ G [39, Section 5, Equation (14L)].Now consider the identifications T ( e,x ) ( G × g ) = g ⊕ g and T ∗ ( e,x ) ( G × g ) = ( g ⊕ g ) ∗ = g ∗ ⊕ g ∗ for each x ∈ g . Write P ω for the Poisson bivector on T ∗ G determined by ω , noting that ( P ω ) ( e,x ) isa vector space isomorphism ( P ω ) ( e,x ) : g ∗ ⊕ g ∗ ∼ = −→ g ⊕ g for each x ∈ g . To compute ( P ω ) ( e,x ) , let κ : g ∗ ∼ = −→ g denote the inverse of (2.8). This leads to the following lemma, which will be needed later. Lemma 2.8. If x ∈ g , then ( P ω ) ( e,x ) ( α, β ) = ( κ ( β ) , [ x, κ ( β )] − κ ( α )) for all ( α, β ) ∈ g ∗ ⊕ g ∗ .Proof. Write P ω ( α, β ) = ( y, z ) ∈ g ⊕ g and note that α ( v ) + β ( w ) = ω ( e,x ) (( P ω ) ( e,x ) ( α, β ) , ( v, w ))= ω ( e,x ) (( y, z ) , ( v, w ))= h y, w i − h z, v i + h x, [ y, v ] i = h y, w i + h [ x, y ] − z, v i . for all v, w ∈ g . It follows that κ ( α ) = [ x, y ] − z and κ ( β ) = y , or equivalently y = κ ( β ) and z = [ x, κ ( β )] − κ ( α ) . (cid:3) Now assume that G is the adjoint group of g . The variety T ∗ G (log D ) admits a distinguished logsymplectic structure (e.g. [2]), some aspects of which we now describe. We begin by noting that˜ ϕ : T ∗ G −→ T ∗ G (log D ) , ( g, x ) −→ (( g, e ) · g ∆ , (Ad g ( x ) , x )) . (2.11)is a symplectomorphism onto the unique open dense symplectic leaf in T ∗ G (log D ). This yields thecommutative diagram T ∗ G T ∗ G (log D ) G G ˜ ϕ ϕ , where ϕ : G −→ G is the map (2.9). One also observes ˜ ϕ to be equivariant with respect to (2.10)and the following ( G × G )-action on T ∗ G :( g , g ) · ( h, y ) := ( g hg − , Ad g ( y )) . (2.12)The ( G × G )-actions (2.10) and (2.12) are Hamiltonian with respective moment maps ρ = ( ρ L , ρ R ) : T ∗ G (log D ) −→ g ⊕ g , ( γ, ( y , y )) −→ ( y , y ) (2.13)and ρ = ( ρ L , ρ R ) : T ∗ G −→ g ⊕ g , ( g, y ) −→ (Ad g ( y ) , y ) . (2.14)Now suppose that ( X, P, ν ) is a Hamiltonian G -variety. Endow X with the Hamiltonian ( G × G )-variety structure for which G R := { e } × G acts trivially and G L := G × { e } HE LOG SYMPLECTIC GEOMETRY OF POISSON SLICES 13 acts via the original Hamiltonian G -action and the identification G = G L . It follows that theproduct Poisson varieties X × T ∗ G and X × T ∗ G (log D ) are Hamiltonian ( G × G )-varieties withrespective moment maps µ = ( µ L , µ R ) : X × T ∗ G −→ g ⊕ g , ( x, ( g, y )) −→ ( ν ( x ) − Ad g ( y ) , − y ) (2.15)and µ = ( µ L , µ R ) : X × T ∗ G (log D ) −→ g ⊕ g , ( x, ( γ, ( y , y ))) −→ ( ν ( x ) − y , − y ) . (2.16)We also have a commutative diagram X × T ∗ G X × T ∗ G (log D ) g ⊕ g iµ µ , (2.17)where i : X × T ∗ G −→ X × T ∗ G (log D ) , ( x, ( g, y )) −→ ( x, (( g, e ) · g ∆ , (Ad g ( y ) , y ))) . (2.18)is the ( G × G )-equivariant open Poisson embedding given by the product of (2.11) with the identitymap X −→ X .The Hamiltonian ( G × G )-variety X × T ∗ G warrants some further discussion. One knows thatthe geometric quotient µ − L (0) −→ ( X × T ∗ G ) (cid:12) G L exists, and that the action of G R on µ − L (0) descends to a Hamiltonian action of G on ( X × T ∗ G ) (cid:12) G L .An associated moment map is obtained by descending − µ R (cid:12)(cid:12) µ − L (0) : µ − L (0) −→ g to the quotient variety ( X × T ∗ G ) (cid:12) G L . It is then not difficult to verify that ψ : X ∼ = −→ ( X × T ∗ G ) (cid:12) G L , x −→ [ x : ( e, ν ( x ))] , x ∈ X (2.19)is an isomorphism of Hamiltonian G -varieties.3. Poisson slices
This section develops the general theory of Poisson slices. Some emphasis is placed on propertiesof the Poisson slice G × S τ and a larger log symplectic variety G × S τ .3.1. Poisson transversals and Poisson slices.
Let (
X, P ) be a Poisson variety. Given x ∈ X and a subspace V ⊆ T x X , we write V † for the annihilator of V in T ∗ x X . Our notation suppressesthe dependence of V † on T x X , as the ambient tangent space will always be clear from context. Wewill use an analogous notation for vector subbundles of T X .Recall that a smooth locally closed subvariety Y ⊆ X is called a Poisson transversal (or cosym-plectic subvariety ) if
T X | Y = T Y ⊕ P ( T Y † ) . (3.1)This has the following straightforward implication for every symplectic leaf L ⊆ X : L and Y havea transverse intersection in X , and L ∩ Y is a symplectic submanifold of L .The Poisson transversal Y inherits a Poisson bivector P Y from ( X, P ). To define it, note thatthe decomposition (3.1) gives rise to an inclusion T ∗ Y ⊆ T ∗ X . One can verify that P ( T ∗ Y ) ⊆ T Y, and P Y is then defined to be the restriction P Y := P (cid:12)(cid:12) T ∗ Y : T ∗ Y −→ T Y.
Note that Y need not be a Poisson subvariety of X in the usual sense; restricting functions neednot define a morphism O X −→ j ∗ O Y of sheaves of Poisson algebras, where j : Y ֒ → X is the inclu-sion. This is particularly apparent if X is symplectic; the Poisson transversals are the symplecticsubvarieties, while the Poisson subvarieties are the open subvarieties.We record the following well-known fact for future reference (cf. [21, Example 4]). Proposition 3.1.
Let X be a symplectic variety. If Y ⊆ X is a Poisson transversal, then Y isa symplectic subvariety of X . The resulting symplectic structure on Y coincides with the Poissonstructure Y inherits as a transversal. We need the following refinement in the case of log symplectic varieties.
Proposition 3.2.
Suppose that ( X, P ) is a log symplectic variety with divisor Z . Let Y ⊆ X be anirreducible Poisson transversal, and write P tr for the resulting Poisson bivector on Y . The followingstatements hold. (i) The Poisson variety ( Y, P tr ) is log symplectic with divisor Z ∩ Y . (ii) If one equips Y \ Z and X \ Z with the symplectic structures inherited as symplectic leavesof ( Y, P tr ) and ( X, P ) , respectively, then Y \ Z is a symplectic subvariety of X \ Z .Proof. We begin by proving that Y is a log symplectic subvariety of X in the sense of [25, Definition7.16]. To this end, consider the unique open dense symplectic leaf X := X \ Z ⊆ X . Since Y is aPoisson transversal in X , Proposition 3.1 forces Y := Y ∩ X to be a symplectic subvariety of X .Now let Z , . . . , Z k be the irreducible components of Z , and set Z I := \ i ∈ I Z i for each subset I ⊆ { , . . . , k } . Each irreducible component of Z is a union of symplectic leaves in X (cf. [48, Exercise 5.2]), implying that Z I is a union of symplectic leaves for each I ⊆ { , . . . , k } .On the other hand, the Poisson transversal Y is necessarily transverse to the symplectic leaves in X . These last two sentences imply that Y is transverse to Z I for all I ⊆ { , . . . , k } .The previous two paragraphs show Y to be a log symplectic subvariety of X , and we let P log denote the resulting Poisson bivector on Y . It follows that Y is the unique open dense symplecticleaf of ( Y, P log ), and that its symplectic form is the pullback of the symplectic form on X . We alsoknow that P tr is non-degenerate on Y , and that it coincides with the pullback of the symplecticstructure from X to Y (see Proposition 3.1). One concludes that P log and P tr coincide on Y .Since Y is dense in Y , it follows that P log = P tr . This establishes (i) and (ii). (cid:3) The following well-known result concerns the behaviour of Poisson transversals with respect toPoisson morphisms (cf. [21, Lemma 7]).
Proposition 3.3.
Let φ : X −→ X be a Poisson morphism between Poisson varieties X and X .If Y ⊆ X is a Poisson transversal, then φ − ( Y ) is a Poisson transversal in X . The codimensionof φ − ( Y ) in X is equal to the codimension of Y in X . We now consider a concrete application of Proposition 3.3. To this end, recall the Lie-theoreticnotation and setup established in 2.5.
Corollary 3.4.
Suppose that ( X, P, ν ) is a Hamiltonian G -variety. If τ = ( ξ, h, η ) is an sl -triplein g , then ν − ( S τ ) is a Poisson transversal in X . This transversal has codimension dim g − dim( g η ) in X . HE LOG SYMPLECTIC GEOMETRY OF POISSON SLICES 15
Proof.
The moment map ν : X −→ g is necessarily a morphism of Poisson varieties (e.g. [9, Propo-sition 7.1]). At the same time, [22, Section 3.1] explains that S τ is a Poisson transversal in g . Thedesired now result now follows immediately from Proposition 3.3. (cid:3) A consequence of Corollary 3.4 is that ν − ( S τ ) inherits a Poisson bivector P τ from ( X, P ). Thisgives rise to our notion of a
Poisson slice . Definition 3.5.
Suppose that (
X, P, ν ) is a Hamiltonian G -variety, and let τ be an sl -triple in g .We call X τ := ( ν − ( S τ ) , P τ ) the Poisson slice of (
X, P, ν ) with respect to τ .This next proposition explains why we call X τ a Poisson slice; it is a slice for the G -action on X in the following sense. Proposition 3.6.
Let ( X, P, ν ) be a Hamiltonian G -variety. If τ is an sl -triple in g , then X τ istransverse to the G -orbits in X .Proof. Fix x ∈ ν − ( S τ ) and set y := ν ( x ) ∈ S τ . Consider the differential dν x : T x X −→ g and itsdual dν ∗ x : g ∗ −→ T ∗ x X , and let P g be the Poisson bivector on g . Since ν is a morphism of Poissonvarieties, we have ( P g ) y = dν x ◦ P x ◦ dν ∗ x . We also know S τ ⊆ g to be a Poisson transversal (e.g. by Corollary 3.4), so that g = T y S τ ⊕ ( P g ) y (( T y S τ ) † ) = T y S τ ⊕ dν x ( P x ( dν ∗ x (( T y S τ ) † ))) . One immediate conclusion is that ν is transverse to S τ . We also conclude that T x ( ν − ( S τ )) = ker (cid:18) pr ◦ dν x : T x X −→ ( P g ) y (( T y S τ ) † ) (cid:19) , where pr : g = T y S τ ⊕ ( P g ) y (( T y S τ ) † ) −→ ( P g ) y (( T y S τ ) † )is the natural projection. It follows that T x ( ν − ( S τ )) † = image (cid:18) dν ∗ x ◦ pr ∗ : ( P g ) y (( T y S τ ) † ) ∗ −→ T ∗ x X (cid:19) , where pr ∗ : ( P g ) y (( T y S τ ) † ) ∗ −→ g ∗ is the dual of pr . This amounts to the statement that T x ( ν − ( S τ )) † = dν ∗ x ( g † η ) , while we know that the Killing form identifies g † η ⊆ g ∗ with g ⊥ η = [ g , η ] ⊆ g . We conclude that T x ( ν − ( S τ )) † = span { ( dν [ η,b ] ) x : b ∈ g } , where ν [ η,b ] : X −→ C is defined by ν [ η,b ] ( z ) = h ν ( z ) , [ η, b ] i . Equations (2.2) and (2.3) now imply that P x ( T x ( ν − ( S τ )) † ) = span { P x (( dν [ η,b ] ) x ) : b ∈ g } = span { V [ η,b ] x : b ∈ g } ⊆ T x ( Gx ) . This combines with ν − ( S τ ) being a Poisson transversal to yield T x X = T x ( ν − ( S τ )) ⊕ P x ( T x ( ν − ( S τ )) † ) = T x ( ν − ( S τ )) + T x ( Gx ) , completing the proof. (cid:3) Let Y be an irreducible component of X τ . The bivector P τ then restricts to a Poisson bivector P Y,τ on Y . This leads to the following observation. Corollary 3.7.
Suppose that ( X, P, ν ) is a Hamiltonian G -variety. Assume that ( X, P ) is logsymplectic with divisor Z , and let τ be an sl -triple in g . Let Y be an irreducible component of thePoisson slice X τ . (i) The Poisson variety ( Y, P
Y,τ ) is log symplectic with divisor Y ∩ Z . (ii) If one equips Y \ Z and X \ Z with the symplectic structures inherited as symplectic leavesof ( Y, P
Y,τ ) and ( X, P ) , respectively, then Y \ Z is a symplectic subvariety of X \ Z . (iii) If ( X, P ) is symplectic, then ( X τ , P τ ) is symplectic and the symplectic form on ( X, P ) pullsback to the symplectic form on ( X τ , P τ ) .Proof. This follows immediately from Proposition 3.1, Proposition 3.2, and Corollary 3.4. (cid:3)
The following immediate consequence is used extensively in later sections.
Corollary 3.8. If τ is an sl -triple in g , then G × S τ is a symplectic subvariety of T ∗ G = G × g .Proof. Apply Corollary 3.7(iii) to X = T ∗ G with the Hamiltonian action of G R = { e } × G ⊆ G × G . (cid:3) Poisson slices via Hamiltonian reduction.
Recall the Hamiltonian action of G × G on T ∗ G = G × g discussed in Subsection 2.7. The symplectic subvariety G × S τ is invariant under G L = G × { e } ⊆ G × G , and ρ τ := ρ L (cid:12)(cid:12) G ×S τ : G × S τ −→ g , ( g, x ) −→ Ad g ( x ) (3.2)is a corresponding moment map. Now let ( X, P, ν ) be a Hamiltonian G -variety, and consider theproduct Poisson variety X × ( G ×S τ ). The diagonal action of G on X × ( G ×S τ ) is then Hamiltonianwith moment map µ τ : X × ( G × S τ ) −→ g , ( x, ( g, y )) −→ ν ( x ) − Ad g ( y ) . These considerations allow us to realize Poisson slices via Hamiltonian reduction.
Proposition 3.9.
Let ( X, P, ν ) be a Hamiltonian G -variety, and let τ be an sl -triple in g . If weendow X × ( G × S τ ) with the Poisson structure and Hamiltonian G -action described above, thenthere is a Poisson variety isomorphism ψ τ : X τ ∼ = −→ ( X × ( G × S τ )) (cid:12) G, x −→ [ x : ( e, ν ( x ))] . (3.3) Proof.
We begin by noting that µ − τ (0) = { ( x, ( g, y )) ∈ X × ( G × S τ ) : ν ( x ) = Ad g ( y ) } = { ( x, ( g, y )) ∈ X × ( G × S τ ) : ν ( g − · x ) = y } . It follows that the G -invariant map J : X × ( G × S τ ) −→ X, ( x, ( g, y )) −→ g − · x satisfies J ( µ − τ (0)) ⊆ ν − ( S τ ) = X τ , thereby inducing a map π := J (cid:12)(cid:12) µ − τ (0) : µ − τ (0) −→ X τ . One then verifies that π − ( x ) = G · ( x, e, ν ( x )) ⊆ X × ( G × S τ ) HE LOG SYMPLECTIC GEOMETRY OF POISSON SLICES 17 for all x ∈ X τ , where G · ( x, e, ν ( x )) is the G -orbit of ( x, e, ν ( x )) in X × ( G × S τ ). This forces π tobe the geometric quotient of µ − τ (0) by G (e.g. by [54, Proposition 25.3.5]), i.e.( X × ( G × S τ )) (cid:12) G = X τ . We now have two Poisson structures on X τ : the Poisson structure P red from Hamiltonian reduc-tion, and the structure P tr obtained from X τ being a Poisson slice in X . It suffices to show thatthese Poisson structures coincide.Fix x ∈ X τ and α ∈ T ∗ x X τ . Since X τ is a Poisson transversal in X , there is a unique extensionof α to an element ˜ α ∈ (cid:18) P x (cid:0) ( T x X τ ) † (cid:1)(cid:19) † ⊆ T ∗ x X. The discussion of Poisson transversals in Subsection 3.1 then implies that( P tr ) x ( α ) = P x ( ˜ α ) . (3.4)We also have ( P red ) x ( α ) = dπ z (( P τ ) z ( ˜ α ′ )) , (3.5)where z = ( x, e, ν ( x )), ˜ α ′ ∈ T z ( Gz ) † ⊆ T ∗ z ( X × ( G × S τ ))is an extension of dπ ∗ z ( α ), and dπ ∗ z : T ∗ x X τ −→ T ∗ z ( µ − τ (0))is the dual of dπ z : T z ( µ − τ (0)) −→ T x X τ . Since J is G -invariant, we may take ˜ α ′ := dJ ∗ z ( ˜ α ) . We also observe that dJ z ( a, b, c ) = a − ( V b ) x for all ( a, b, c ) ∈ T z ( X × ( G × S τ )) = T x X ⊕ g ⊕ g η , where V b is the fundamental vector field on X associated to b ∈ g . It follows that( dJ ∗ z ( ˜ α ))( a, b, c ) = ˜ α ( a ) − ˜ α (( V b ) x ) = ˜ α ( a ) − ˜ α ( P x (( dν b ) x )) = ˜ α ( a ) + ( dν b ) x ( P x ( ˜ α )) , yielding ˜ α ′ = ( ˜ α, dν x ( P x ( ˜ α )) , ∈ T ∗ z ( X × ( G × S τ )) = T ∗ x X ⊕ g ∗ ⊕ g ∗ η = T ∗ x X ⊕ g ⊕ g ξ , (3.6)where we have made the identifications g ∗ η = ( g / [ g , ξ ]) ∗ = [ g , ξ ] ⊥ = g ξ . Now set w = ( e, ν ( x )) ∈ G × S τ and let Q τ be the Poisson bivector on G × S τ . Lemma 2.8 then gives( Q τ ) w ( dν x ( P x ( ˜ α )) ,
0) = (0 , − dν x ( P x ( ˜ α ))) . This combines with (3.4), (3.5), and (3.6) to yield( P red ) x ( α ) = dπ z ( P x ( ˜ α ) , − ( Q τ ) w ( dν x ( P x ( ˜ α )) , dπ z ( P x ( ˜ α ) , , dν x ( P x ( ˜ α )))= P x ( ˜ α )= ( P tr ) x ( α ) , as desired. (cid:3) Remark 3.10.
In the special case τ = 0, we have S τ = g and G × S τ = G × g = T ∗ G . Proposition3.9 is then seen to recover the isomorphism (2.19). Our next result is that Poisson slices can be realized via Hamiltonian reduction with respect tounipotent radicals of parabolic subgroups. To formulate this result, let τ = ( ξ, h, η ) be an sl -triplein g and write g λ ⊆ g for the eigenspace of ad h with eigenvalue λ ∈ Z . The parabolic subalgebra p τ := M λ ≤ g λ then has u τ := M λ< g λ as its nilradical. Now consider the identifications u ∗ τ ∼ = g / u ⊥ τ = g / p τ ∼ = u − τ := M λ> g λ , and thereby regard ξ ∈ u − τ as an element of u ∗ τ . Write U τ ⊆ G for the unipotent subgroup with Liealgebra u τ , and let ( U τ ) ξ be the U τ -stabilizer of ξ under the coadjoint action. Remark 3.11.
The Lie algebra of ( U τ ) ξ is given by( u τ ) ξ = M λ ≤− g λ . It follows that ( U τ ) ξ = U τ if and only if τ is an even sl -triple, i.e. g − = { } . If τ is a principaltriple, then τ is even and ( U τ ) ξ = U τ is a maximal unipotent subgroup of G .Let ( X, P, ν ) be a Hamiltonian G -variety. The action of U τ is also Hamiltonian with momentmap ν τ := p τ ◦ µ , where g = p τ ⊕ u − τ p τ −→ u − τ = u ∗ τ is the projection. One has ν − τ ( ξ ) = ν − ( ξ + p τ ) , while the proof of [5, Lemma 3.2] shows the stabilizer ( U τ ) ξ to act freely on ξ + p τ . It follows that( U τ ) ξ acts freely on ν − τ ( ξ ). This leads us to prove Proposition 3.13, i.e. that the geometric quotient X (cid:12) ξ U τ = ν − τ ( ξ ) / ( U τ ) ξ (3.7)exists and is Poisson-isomorphic to X τ . Remark 3.12.
The type of Hamiltonian reduction performed in (3.7) is particularly well-studiedin the case of a principal triple τ . In this case, one sometimes calls the Poisson variety X (cid:12) ξ U τ a Whittaker reduction (e.g. [3, 19]). The nomenclature reflects Kostant’s result [36, Theorem 1.2].
Proposition 3.13.
Let ( X, P, ν ) be a Hamiltonian G -variety. If τ = ( ξ, h, η ) is an sl -triple in g ,then there is a canonical isomorphism X (cid:12) ξ U τ ∼ = X τ of Poisson varieties.Proof. We begin by exhibiting X τ as the geometric quotient of ν − τ ( ξ ) by ( U τ ) ξ . To this end, theproof of [5, Lemma 3.2] explains that( U τ ) ξ × S τ −→ ξ + p τ , ( u, x ) −→ Ad u ( x )defines a variety isomorphism. Composing the inverse of this isomorphism with the projection( U τ ) ξ × S τ −→ ( U τ ) ξ HE LOG SYMPLECTIC GEOMETRY OF POISSON SLICES 19 then yields a map φ : ξ + p τ −→ ( U τ ) ξ . Note that for y ∈ ξ + p τ , φ ( y ) is the unique element of ( U τ ) ξ satisfyingAd φ ( y ) − ( y ) ∈ S τ . We may therefore define the map ν − τ ( ξ ) = ν − ( ξ + p τ ) θ −→ X τ , x −→ ( φ ( ν ( x ))) − · x. One has θ − ( x ) = ( U τ ) ξ · x for all x ∈ ν − τ ( ξ ), and we deduce that θ is the geometric quotient of ν − τ ( ξ ) by ( U τ ) ξ (e.g. by [54,Proposition 25.3.5]).The previous paragraph establishes the following fact: Hamiltonian reductions of Hamiltonian G -varieties by U τ at level ξ always exist as geometric quotients. We implicitly use this observationin several places below.To see that the Poisson structures on X τ and X (cid:12) ξ U τ coincide, we argue as follows. One has acanonical isomorphism T ∗ G (cid:12) ξ U τ ∼ = G × S τ (3.8)of symplectic varieties, where U τ acts on T ∗ G via (2.12) as the subgroup U τ = { e } × U τ ⊆ G × G (see [5, Lemma 3.2]). Note also that T ∗ G (cid:12) ξ U τ and G × S τ come with Hamiltonian actions of G induced by the action of G L = G × { e } on T ∗ G ∼ = G × g . One then readily verifies that (3.8) is anisomorphism of Hamiltonian G -varieties.Proposition 3.9 gives a canonical isomorphism of Poisson varieties X τ ∼ = ( X × ( G × S τ )) (cid:12) G. The previous paragraph allows us to write this isomorphism as X τ ∼ = ( X × ( T ∗ G (cid:12) ξ U τ )) (cid:12) G = (( X × T ∗ G ) (cid:12) ξ U τ ) (cid:12) G, where U τ acts trivially on X . Since the actions of G and U τ on X × T ∗ G commute with one another,it follows that X τ ∼ = (( X × T ∗ G ) (cid:12) G ) (cid:12) ξ U τ . An application of Remark 3.10 then yields X τ ∼ = X (cid:12) ξ U τ , completing the proof. (cid:3) Poisson slices in the log cotangent bundle of G . Fix an sl -triple τ in g and recall thenotation in Subsection 2.7. In what follows, we study the Poisson slice G × S τ := ρ − R ( S τ ) ⊆ T ∗ G (log D )and its properties. We begin by observing that G × S τ = { ( γ, ( x, y )) ∈ G × ( g ⊕ g ) : ( x, y ) ∈ γ and y ∈ S τ } . (3.9)A few simplifications arise if τ is a principal sl -triple. To this end, recall the adjoint quotient χ : g −→ Spec( C [ g ] G )and the associated concepts and notation discussed in Subsection 2.5. The image of ρ : T ∗ G (log D ) −→ g ⊕ g is known to be image( ρ ) = { ( x, y ) ∈ g ⊕ g : χ ( x ) = χ ( y ) } (3.10) (see [2, Proposition 3.4]). One consequence is that x, y ∈ g lie in the same fibre of χ whenever( x, y ) ∈ γ for some γ ∈ G . Since S τ is a section of χ , this fact combines with (3.9) to yield G × S τ = { ( γ, ( x, x τ )) : γ ∈ G, x ∈ g , and ( x, x τ ) ∈ γ } . (3.11)We now develop some more manifestly geometric properties of G × S τ , beginning with the fol-lowing result. Theorem 3.14. If τ is an sl -triple in g , then G × S τ is irreducible.Proof. Consider the closed subvariety Y := { ( x, y ) ∈ g × S τ : χ ( x ) = χ τ ( y ) } ⊆ g ⊕ g , where χ τ : S τ −→ Spec( C [ g ] G ) denotes the restriction of χ to S τ . It follows from (3.9) and (3.10)that G × S τ −→ Y, ( γ, ( x, y )) −→ ( x, y ) (3.12)is the pullback of ρ : T ∗ G (log D ) −→ g ⊕ g along the inclusion Y ֒ → g ⊕ g , and that (3.12) issurjective. One also knows that ρ is proper, as it results from restricting the natural projection G × ( g ⊕ g ) −→ g ⊕ g to T ∗ G (log D ) ⊆ G × ( g ⊕ g ). The surjection (3.12) is therefore proper, whilethe proof of [2, Proposition 3.11] shows (3.12) to have connected fibres. If Y were connected, thenthe previous sentence would force G × S τ to be connected as well. This would in turn force G × S τ to be irreducible, as Poisson slices are smooth.In light of the previous paragraph, it suffices to prove that Y is irreducible. We begin by decom-posing g into its simple factors, i.e. g = g ⊕ · · · ⊕ g N with each g i a simple Lie algebra. Our sl -triple τ then amounts to having an sl -triple τ i in g i foreach i = 1 , . . . , N , yielding S τ = S τ × · · · × S τ N ⊆ g ⊕ · · · ⊕ g N . It also follows that χ τ decomposes as a product χ τ = ( χ ) τ × · · · × ( χ N ) τ N , where χ i is the adjoint quotient map on g i and ( χ i ) τ i is its restriction to S τ i . The results [53,Corollary 7.4.1] and [46, Theorem 5.4] then imply that each ( χ i ) τ i is faithfully flat with irreduciblefibres of dimension dim( S τ i ) − rank( g i ). These last two sentences imply that χ τ is faithfully flatwith irreducible, equidimensional fibres, and the same argument forces χ to be faithfully flat withirreducible, equidimensional fibres. Since fibred products of faithfully flat morphisms are faithfullyflat, we conclude that ˜ χ : Y −→ Spec( C [ g ] G ) , ( x, y ) χ ( x )is faithfully flat. We also conclude that˜ χ − ( t ) = χ − ( t ) × χ − τ ( t )must be irreducible for all t ∈ Spec( C [ g ] G ), and that its dimension must be independent of t . Inother words, ˜ χ is a faithfully flat morphism with irreducible, equidimensional fibres. This combineswith the irreducibility of Spec( C [ g ] G ) and [33, Corollary 9.6] to imply that Y is pure-dimensional.We may now apply the result in [44] and deduce that Y is irreducible. This completes the proof. (cid:3) Corollary 3.15. If τ is an sl -triple in g , then G × S τ is log symplectic.Proof. This is an immediate consequence of Corollary 3.7(i) and Theorem 3.14. (cid:3)
HE LOG SYMPLECTIC GEOMETRY OF POISSON SLICES 21
Now observe that the Hamiltonian action of G L = G × { e } ⊆ G × G on T ∗ G (log D ) restricts toa Hamiltonian action of G on G × S τ . An associated moment map is given by ρ τ := ρ L (cid:12)(cid:12)(cid:12)(cid:12) G ×S τ : G × S τ −→ g , ( γ, ( x, y )) x. At the same time, recall the Hamiltonian G -variety structure on G × S τ and the moment map ρ τ : G × S τ −→ g discussed in Subsection 3.2. Let us also recall the map ˜ ϕ : T ∗ G −→ T ∗ G (log D )from (2.11). Proposition 3.16.
Let τ be an sl -triple in g . (i) The map ˜ ϕ : T ∗ G −→ T ∗ G (log D ) restricts to a G -equivariant symplectomorphism from G × S τ to the unique open dense symplectic leaf in G × S τ . (ii) The diagram G × S τ G × S τ g ˜ ϕ (cid:12)(cid:12) G ×S τ ρ τ ρ τ (3.13) commutes.Proof. By Corollary 3.7, the open dense symplectic leaf in G × S τ is obtained by intersecting G × S τ with the open dense symplectic leaf in T ∗ G (log D ). The latter leaf is ˜ ϕ ( T ∗ G ), as is explained inSubsection 2.7. It is also straightforward to establish that˜ ϕ ( G × S τ ) = G × S τ ∩ ˜ ϕ ( T ∗ G ) . These last two sentences show ˜ ϕ ( G × S τ ) to be the unique open dense symplectic leaf in G × S τ . Wealso know that ˜ ϕ restricts to a symplectomorphism from G × S τ to ˜ ϕ ( G × S τ ), where the symplecticform on ˜ ϕ ( G × S τ ) is the pullback of the symplectic form on the leaf in T ∗ G (log D ) (see Corollary3.8). It now follows from Corollary 3.7(ii) that˜ ϕ (cid:12)(cid:12) G ×S τ : G × S τ −→ ˜ ϕ ( G × S τ )is a symplectomorphism with respect to this symplectic structure ˜ ϕ ( G × S τ ) inherits as a leafin G × S τ . This symplectomorphism is G -equivariant, as ˜ ϕ : T ∗ G −→ T ∗ G (log D ) is ( G × G )-equivariant. The proof of (i) is therefore complete, while a straightforward calculation yields (ii). (cid:3) Remark 3.17.
Let τ be a principal sl -triple in g . The description (3.11) allows one to define aclosed embedding G × S τ −→ G × g , ( γ, ( x, x τ )) −→ ( γ, x ) . We thereby obtain a commutative diagram G × S τ G × gg , ρ τ where G × g −→ g is projection to the second factor. One immediate consequence is that ρ τ hasprojective fibres, so that (3.13) realizes ρ τ as a fibrewise compactification of ρ τ . It also follows that ρ − τ ( x ) −→ { γ ∈ G : ( x, x τ ) ∈ γ } , ( γ, ( x, x τ )) −→ γ is a variety isomorphism for each x ∈ g . Relation to the universal centralizer and its fibrewise compactification.
Let τ be aprincipal sl -triple in g . It is instructive to examine the relationship between G × S τ and G × S τ inthe context of Balib˘anu’s paper [2]. We begin by recalling that the universal centralizer of g is theclosed subvariety of T ∗ G = G × g defined by Z τ g := { ( g, x ) ∈ G × g : x ∈ S τ and g ∈ G x } , where G x is the G -stabilizer of x ∈ g . At the same time, recall the Hamiltonian action of G × G on T ∗ G and moment map ρ : T ∗ G −→ g ⊕ g discussed in Subsection 2.7. Consider the product S τ × S τ ⊆ g ⊕ g and observe that Z τ g = ρ − ( S τ × S τ ) . Note also that S τ × S τ is the Slodowy associated to the sl -triple (( ξ, ξ ) , ( h, h ) , ( η, η )). It followsthat Z τ g is a Poisson slice in T ∗ G . Corollary 3.7(iii) then forces this Poisson slice to be a symplecticsubvariety of T ∗ G . Remark 3.18.
Some papers realize the symplectic structure on Z τ g via a Whittaker reduction of T ∗ G (e.g. [2]). To this end, let U ⊆ G be the unipotent subgroup with Lie algebra u . Proposition3.13 then gives a canonical isomorphism Z τ g = ρ − ( S τ × S τ ) ∼ = T ∗ G (cid:12) ( ξ,ξ ) U × U of symplectic varieties, where the symplectic structure on Z τ g is as defined in the previous paragraph.One may replace ρ : T ∗ G −→ g ⊕ g with ρ : T ∗ G (log D ) −→ g ⊕ g and proceeed analogously. Inthe interest of being more precise, consider the Poisson slice Z τ g := ρ − ( S τ × S τ ) = { ( γ, ( x, x )) : γ ∈ G, x ∈ S τ , and ( x, x ) ∈ γ } in T ∗ G (log D ). Remark 3.19.
A counterpart of Remark 3.18 is that Proposition 3.13 gives a canonical isomorphism Z τ g = ρ − ( S τ × S τ ) ∼ = T ∗ G (log D ) (cid:12) ( ξ,ξ ) U × U of Poisson varieties. This realization of Z τ g via Whittaker reduction is used to great effect in [2].Now recall the embedding ˜ ϕ : T ∗ G −→ T ∗ G (log D ) discussed in Subsection 2.7. Balib˘anu [2]shows Z τ g to be log symplectic (cf. Corollary 3.7), and that ˜ ϕ restricts to a symplectomorphismfrom Z τ g to the unique open dense symplectic leaf in Z τ g . One also has a commutative diagram Z τ g Z τ g S τ ˜ ϕ (cid:12)(cid:12) Z τ g q τ q τ , (3.14)where q τ ( g, x ) = x and q τ ( γ, ( x, x )) = x. This diagram is seen to be the pullback of (3.13) along the inclusion S τ ֒ → g , and it thereby exhibits q τ as a fibrewise compactification of q τ (see Remark 3.17 and cf. [2, Section 3]). This amounts to(3.14) being the restriction of (3.13) to a morphism between the Poisson slices Z τ g = ρ − ( S τ × S τ ) = ρ − τ ( S τ ) and Z τ g = ρ − ( S τ × S τ ) = ρ − τ ( S τ ) . This present section combines with Subsection 3.3 to yield the following informal comparisonsbetween ( Z τ g , Z τ g ) and ( G × S τ , G × S τ ): HE LOG SYMPLECTIC GEOMETRY OF POISSON SLICES 23 • q τ (resp. ρ τ ) is a fibrewise compactification of π τ (resp. q τ ); • (3.14) is obtained by pulling (3.13) back along the inclusion S τ ֒ → g ; • Z τ g and G × S τ are symplectic; • Z τ g and G × S τ are log symplectic; • ˜ ϕ restricts to a symplectomorphism from Z τ g (resp. G × S τ ) to the unique open densesymplectic leaf in Z τ g (resp. G × S τ ).4. The geometries of X and X τ This section is concerned with constructing partial compactifications of Poisson slices, an issuemotivated in the introduction of our paper. Our approach is to replace a Poisson slice X τ with aslightly larger variety X τ , provided that the latter makes sense. If X τ is well-defined, we show itto enjoy certain Poisson-geometric features and discuss the extent to which it partially compactifies X τ .4.1. Definitions and first properties.
Fix a Hamiltonian G -variety ( X, P, ν ) and an sl -triple τ in g . The product Hamiltonian G -varieties X × ( G × S τ ) and X × ( G × S τ ) then have respectivemoment maps µ τ : X × ( G × S τ ) −→ g , ( x, ( g, y )) −→ ν ( x ) − Ad g ( y )and µ τ : X × ( G × S τ ) −→ g , ( x, ( γ, ( y , y ))) −→ ν ( x ) − y . Note also that taking the product of˜ ϕ (cid:12)(cid:12) G ×S τ : G × S τ −→ G × S τ with the identity X −→ X produces a G -equivariant open Poisson embedding i τ : X × ( G × S τ ) −→ X × ( G × S τ ) (4.1)( x, ( g, y )) −→ ( x, (( g, e ) · g ∆ , (Ad g ( y ) , y ))) . (see Proposition 3.16). One readily verifies that the diagram X × ( G × S τ ) X × ( G × S τ ) g i τ µ τ µ τ (4.2)commutes.Now recall the Hamiltonian ( G × G )-variety X × T ∗ G (log D ) and moment map µ = ( µ L , µ R ) : X × T ∗ G (log D ) −→ g ⊕ g from Subsection 2.7. Let us write X := ( X × T ∗ G (log D )) (cid:12) G L and X τ := ( X × ( G × S )) (cid:12) G, and understand “ X exists” (resp. “ X τ exists”) to mean that ( X × T ∗ G (log D )) (cid:12) G L (resp. ( X × ( G × S )) (cid:12) G ) exists as a geometric quotient. Remark 4.1. If τ = 0, then X × ( G × S τ ) = X × T ∗ G (log D ), µ τ = µ L , and the G -action on X × ( G × S τ ) is the G L -action X × T ∗ G (log D ). One immediate consequence is that X = X . Remark 4.2.
The action of G R on X × T ∗ G (log D ) induces a residual G -action on X , providedthat X exists. This G -action features prominently in what follows. It is reasonable to seek conditions under which X and X τ exist. We defer this matter to Section5, which is largely devoted to examples. In the interim, we assume that X τ exists. Let us also recallthe map i : X × T ∗ G −→ X × T ∗ G (log D ) from (2.18). This map restricts to a G -equivariant openembedding i (cid:12)(cid:12) µ − τ (0) : µ − τ (0) ֒ → µ − τ (0) , (4.3)which in turn descends to a morphism j τ : ( X × ( G × S τ )) (cid:12) G −→ X τ . (4.4)Let us consider the composition k τ := j τ ◦ ψ τ : X τ −→ X τ , (4.5)where ψ τ : X τ −→ ( X × ( G × S τ )) (cid:12) G is the Poisson variety isomorphism from (3.3). It isstraightforward to verify that k τ ( x ) = [ x : ( g ∆ , ( ν ( x ) , ν ( x )))] (4.6)for all x ∈ X τ Proposition 4.3.
Let τ be an sl -triple in g . If X τ exists, then k τ : X τ −→ X τ is an openembedding.Proof. Since ψ τ is a variety isomorphism, it suffices to prove that j τ is an open embedding. Weachieve this by first considering the commutative square µ − τ (0) µ − τ (0)( X × ( G × S τ )) (cid:12) G X τi (cid:12)(cid:12) µ − τ (0) j τ . (4.7)The vertical morphisms are open maps by virtue of being geometric quotients [54, Lemma 25.3.2],and we have explained that the upper horizontal map is open. It follows that j τ is also an openmap. Together with the observation that j τ is injective, this implies that j τ is an open embedding.Our proof is complete. (cid:3) The inclusion X τ −→ X composes with the quotient map X −→ X/G to yield π τ : X τ −→ X/G, (4.8)provided that
X/G exists. We may also consider the morphism π τ : X τ −→ X/G, [ x : ( γ, ( y , y ))] −→ [ x ] (4.9)if both X τ and X/G exist. The following is then an immediate consequence of (4.6).
Proposition 4.4.
Let τ be an sl -triple in g . If X τ and X/G exist, then the diagram X τ X τ X/G k τ π τ π τ (4.10) commutes. This diagram is particularly noteworthy if τ is a principal sl -triple. Theorem 4.5.
Let τ be a principal sl -triple in g . If X τ and X/G exist, then the diagram (4.10) realizes π τ as a fibrewise compactification of π τ . HE LOG SYMPLECTIC GEOMETRY OF POISSON SLICES 25
Proof.
Our objective is to prove that π τ has projective fibres. Let us begin by fixing a point x ∈ X .We then have π − τ ([ x ]) = { [ x : ( γ, ( ν ( x ) , y ))] : γ ∈ G, y ∈ S τ , and ( ν ( x ) , y ) ∈ γ } . (4.11)On the other hand, it is known that y , y ∈ g belong to the same fibre of the adjoint quotient χ : g −→ Spec( C [ g ] G ) whenever ( y , y ) ∈ γ for some γ ∈ G (see Subsection 3.3). The discussionand notation in Subsection 2.5 associated with principal sl -triples then imply the following: if y ∈ g and y ∈ S τ are such that ( y , y ) ∈ γ for some γ ∈ G , then y = ( y ) τ . We may thereforepresent (4.11) as the statement π − τ ([ x ]) = { [ x : ( γ, ( ν ( x ) , ν ( x ) τ ))] : γ ∈ G and ( ν ( x ) , ν ( x ) τ ) ∈ γ } . In other words, π − τ ([ x ]) is the image of the closed subvariety { γ ∈ G : ( ν ( x ) , ν ( x ) τ ) ∈ γ } ⊆ G under the morphism G −→ X τ , γ −→ [ x : ( γ, ( ν ( x ) , ν ( x ) τ ))] . This subvariety is projective by virtue of being closed in G , and we conclude that π − τ ([ x ]) isprojective. This completes the proof. (cid:3) Let us also examine the case τ = 0 in some detail. To this end, assume that X = X exists andconsider the geometric quotient map π L : µ − L (0) −→ X. The G R -action on µ − L (0) then descends under π L to a G -action X . On the other hand, note thatthe restriction of − µ R : X × T ∗ G (log D ) −→ g , ( x, ( γ, ( y , y ))) −→ y to µ − L (0) is G R -equivariant and G L -invariant. This restriction therefore descends under π L to the G -equivariant morphism ν : X −→ g , [ x : ( γ, ( ν ( x ) , y ))] −→ y. (4.12)Let us write k : X −→ X , π : X −→ X/G , and π : X −→ X/G for (4.5), (4.8), and (4.9),respectively, in the case τ = 0. Proposition 4.6. If X exists, then k : X −→ X is a G -equivariant open embedding and X X g kν ν (4.13) commutes. If X/G also exists, then
X XX/G kπ π (4.14) commutes.Proof.
The commutativity of (4.13) follows immediately from (4.12) and (4.6), while Proposition 4.4forces (4.14) to commute. Proposition 4.3 implies that k is an open embedding. Our equivarianceclaim follows from (4.6), the above-given definition of the G -action on X , and a direct calculation.This completes the proof. (cid:3) The Poisson geometries of X and X τ . Let (
X, P, ν ) be a Hamiltonian G -variety andsuppose that τ is an sl -triple in g . In what follows, we show that the Poisson slice X τ endows X τ with certain Poisson-geometric qualities. The most basic such feature is as follows. Proposition 4.7.
Let τ be an sl -triple in g . If X τ exists, then C [ X τ ] carries a natural Poissonbracket for which k ∗ τ : C [ X τ ] −→ C [ X τ ] is a Poisson algebra morphism.Proof. The definition X τ := ( X × ( G × S τ )) (cid:12) G combines with the discussion in Subsection 2.4 to yield a Poisson bracket on C [ X τ ], as well as thefollowing facts:(i) C [ X × ( G × S τ )] G is a Poisson subalgebra of C [ X × ( G × S τ )];(ii) C [ µ − τ (0)] G has a unique Poisson bracket for which restriction β : C [ X × ( G × S τ )] G −→ C [ µ − τ (0)] G is a Poisson algebra morphism;(iii) the geometric quotient map µ − τ (0) −→ X τ induces a Poisson algebra isomorphism δ : C [ X τ ] ∼ = −→ C [ µ − τ (0)] G . We have a row of Poisson algebra morphisms C [ X × ( G × S τ )] α ←− C [ X × ( G × S τ )] G β −→ C [ µ − τ (0)] G δ ←− C [ X τ ] , where α is the inclusion. An analogous procedure yields a second row C [ X × ( G × S τ )] α ′ ←− C [ X × ( G × S τ )] G β ′ −→ C [ µ − τ (0)] G δ ′ ←− C [ X τ ]of Poisson algebra morphisms. Now recall the G -equivariant open Poisson embedding i τ : X × ( G × S τ ) −→ X × ( G × S τ )from (4.1), as well as the commutative diagram (4.2). It follows that i τ induces the first threevertical arrows in the commutative diagram C [ X × ( G × S τ )] C [ X × ( G × S τ )] G C [ µ − τ (0)] G C [ X τ ] C [ X × ( G × S τ )] C [ X × ( G × S τ )] G C [ µ − τ (0)] G C [ X τ ] α ′′ α β ′′ β δ ′′ k ∗ τ δα ′ β ′ δ ′ . Observe that α ′′ is a Poisson algebra morphism, as follows from i τ being a Poisson morphism. Onededuces that β ′′ must also be a Poisson algebra morphism. This combines with the commutativityof the middle square and the fact that β ′ and β ′′ are surjective Poisson algebra morphisms to implythat δ ′′ is a Poisson algebra morphism. Since δ and δ ′ are Poisson algebra isomorphisms, this forces k ∗ τ to be a Poisson algebra morphism. (cid:3) Some more manifestly geometric features of X τ may be developed as follows. Write ( X × ( G × S τ )) ◦ for the G -invariant open subvariety of points in X × ( G × S τ ) whose G -stabilizers aretrivial.The G -action on ( X × G × S τ ) ◦ is Hamiltonian with respect to the Poisson structure that ( X × G × S τ ) ◦ inherits from ( X × ( G × S τ )), and µ ◦ τ := µ τ (cid:12)(cid:12)(cid:12)(cid:12) ( X × ( G ×S τ )) ◦ : ( X × ( G × S τ )) ◦ −→ g (4.15) HE LOG SYMPLECTIC GEOMETRY OF POISSON SLICES 27 is a moment map.Now assume that X τ exists and consider the geometric quotient map θ τ : µ − τ (0) −→ X τ . The variety ( µ ◦ τ ) − (0) is G -invariant and open in µ − τ (0), and we set X ◦ τ := θ τ (( µ ◦ τ ) − (0)) ⊆ X τ . We also let θ ◦ τ : ( µ ◦ τ ) − (0) −→ X ◦ τ denote the restriction of θ τ to ( µ ◦ τ ) − (0). Lemma 4.8.
Let τ be an sl -triple in g . If X τ exists, then X ◦ τ is an open subvariety of X τ and θ ◦ τ : ( µ ◦ τ ) − (0) −→ X ◦ τ is the geometric quotient of ( µ ◦ τ ) − (0) by G .Proof. The geometric quotient map θ τ : µ − τ (0) −→ X τ is open [54, Lemma 25.3.2]. It followsthat X ◦ τ = θ τ (( µ ◦ τ ) − (0)) is an open subvariety of X τ . The rest of this lemma is an immediateconsequence of Proposition 2.4. (cid:3) Proposition 4.9.
Let τ be an sl -triple in g . If X τ exists, then X ◦ τ is smooth and Poisson.Proof. Recall that ( X × ( G × S τ )) ◦ is a Hamiltonian G -variety with moment map (4.15). Lemma 4.8then implies that X ◦ τ is the Hamiltonian reduction of ( X × ( G × S τ )) ◦ at level zero. The propositionnow follows from generalities about Hamiltonian reductions by free actions, the relevant parts ofwhich are discussed in Subsection 2.4. (cid:3) Now recall the open embedding k τ : X τ −→ X τ defined in (4.5). Proposition 4.10.
Let τ be an sl -triple in g , and assume that X τ exists. The image of k τ : X τ −→ X τ then lies in X ◦ τ , and k τ defines an open embedding of Poisson varieties X τ −→ X ◦ τ .Proof. Recall that k τ = j τ ◦ ψ τ , and that ψ τ is a Poisson variety isomorphism. It therefore sufficesto prove the following:(i) the image of j τ : ( X × ( G × S τ )) (cid:12) G −→ X τ lies in X ◦ τ ;(ii) j τ defines an open embedding of Poisson varieties ( X × ( G × S τ )) (cid:12) G −→ X ◦ τ .Since G acts freely on X × ( G × S τ ), the image of (4.1) lies in ( X × ( G × S τ )) ◦ . We may thereforeinterpret (4.1) as a G -equivariant open Poisson embedding i τ : X × ( G × S τ ) −→ ( X × ( G × S τ )) ◦ and (4.2) as a commutative diagram X × ( G × S τ ) ( X × ( G × S τ )) ◦ g i τ µ τ µ ◦ τ . Such considerations allow one to regard (4.3) and (4.4) as maps i τ (cid:12)(cid:12) µ − τ (0) : µ − τ (0) ֒ → ( µ ◦ τ ) − (0) (4.16)and j τ : ( X × ( G × S τ )) (cid:12) G −→ X ◦ τ , (4.17) respectively. This verifies (i) and yields the commutative square µ − τ (0) ( µ ◦ τ ) − (0)( X × ( G × S τ )) (cid:12) G X ◦ τi τ (cid:12)(cid:12) µ − τ (0) j τ . (4.18)By combining this square with the description of the Poisson structure on a Hamiltonian reduction,we deduce that (4.17) is a Poisson morphism. This morphism is also an open embedding, as followseasily from Proposition 4.3. Our proof is therefore complete. (cid:3) Let us write X ◦ for X ◦ τ if τ = 0. This variety turns out to enjoy some Poisson geometric featuresbeyond those of a general X ◦ τ . To develop these features, assume that X exists and let π L : µ − L (0) −→ X be the geometric quotient map. Write ( X × T ∗ G (log D )) ◦ for the ( G × G )-invariant open subvariety ofpoints in X × T ∗ G (log D ) whose G L -stabilizers are trivial. The ( G × G )-action on ( X × T ∗ G (log D )) ◦ is Hamiltonian with respect to the Poisson structure that ( X × T ∗ G (log D )) ◦ inherits from X × T ∗ G (log D ), and( µ ◦ L , µ ◦ R ) := ( µ L (cid:12)(cid:12)(cid:12)(cid:12) ( X × T ∗ G (log D )) ◦ , µ R (cid:12)(cid:12)(cid:12)(cid:12) ( X × T ∗ G (log D )) ◦ ) : ( X × T ∗ G (log D )) ◦ −→ g ⊕ g (4.19)is a moment map.Now consider the ( G × G )-invariant open subvariety of ( µ ◦ L ) − (0) of µ − L (0), and observe that X ◦ := π L (( µ ◦ L ) − (0)) . Let π ◦ L : ( µ ◦ L ) − (0) −→ X ◦ denote the restriction of π L to ( µ ◦ L ) − (0). At the same time, recall the definition of the G -actionon X . Lemma 4.11.
Assume that X exists. The subset X ◦ is then a G -invariant open subvariety of X ,and π ◦ L : ( µ ◦ L ) − (0) −→ X ◦ is the geometric quotient of ( µ ◦ L ) − (0) by G L .Proof. Observe that π L is equivariant with respect to the action of G R on µ − L (0) and the above-discussed G -action on X . Since ( µ ◦ L ) − (0) is G R -invariant in µ − L (0), this implies that X ◦ = π L (( µ ◦ L ) − (0)) is G -invariant in X . The rest of this lemma is an immediate consequence of Lemma4.8. (cid:3) The G -action that X ◦ inherits from X is compatible with the Poisson variety structure referencedin Proposition 4.9. To formulate this more precisely, recall the map ν : X −→ g in (4.12) and set ν ◦ := ν (cid:12)(cid:12) X ◦ : X ◦ −→ g . Proposition 4.12. If X exists, then the action of G on X ◦ is Hamiltonian with moment map ν ◦ : X ◦ −→ g . HE LOG SYMPLECTIC GEOMETRY OF POISSON SLICES 29
Proof.
Recall that ( X × T ∗ G (log D )) ◦ is a Hamiltonian ( G × G )-variety with moment map (4.19).One deduces that X ◦ = ( µ ◦ L ) − (0) /G L is a Hamiltonian G -variety, and that the correspondingmoment map is obtained by letting − µ ◦ R (cid:12)(cid:12)(cid:12)(cid:12) ( µ ◦ L ) − (0) : ( µ ◦ L ) − (0) −→ g descend to X ◦ . It remains only to observe that this descended moment map and the G -action on X ◦ are restrictions of ν : X −→ g and the G -action on X , respectively. (cid:3) Proposition 4.13.
Assume that X exists. The image of k : X −→ X then lies in X ◦ , and k defines an open embedding of Hamiltonian G -varieties X −→ X ◦ .Proof. This is a direct consequence of Propositions 4.6 and Proposition 4.10. (cid:3)
The log symplectic geometries of X and X τ . We now examine the Poisson geometriesof X and X τ in the special case of a symplectic Hamiltonian G -variety ( X, P, ν ). These Poissongeometries essentially become log symplectic geometries, as is consistent with the following result.Recall the map i τ : X × ( G × S τ ) −→ X × ( G × S τ ) defined in (4.1). Lemma 4.14.
Let ( X, P, ν ) be an irreducible symplectic Hamiltonian G -variety. If τ is an sl -triplein g , then the following statements then hold: (i) X × ( G × S τ ) is log symplectic; (ii) i τ is a G -equivariant symplectomorphism onto the unique open dense symplectic leaf in X × ( G × S τ ) .Proof. Proposition 3.16 tells us that˜ ϕ (cid:12)(cid:12) G ×S τ : G × S τ −→ G × S τ is a G -equivariant symplectomorphism onto the open dense symplectic leaf in the log symplecticvariety G × S τ . We also recall that i τ is the product of ˜ ϕ (cid:12)(cid:12) G ×S τ with the identity X −→ X . Theselast two sentences imply that i τ is a G -equivariant symplectomorphism onto the complement of thedegeneracy locus in X × ( G × S τ ). Since X × ( G × S τ ) and X × ( G × S τ ) are irreducible, this impliesthat the image of i τ is the unique open dense symplectic leaf in X × ( G × S τ ).Now consider the closed subvariety D τ := ( X × ( G × S τ )) \ i τ ( X × ( G × S τ )) (4.20)of X × ( G × S τ ) It remains only to prove the following things:(a) D τ is a normal crossing divisor;(b) the top exterior power of the Poisson bivector on X × ( G × S τ ) has a reduced vanishinglocus;(c) the vanishing locus in (b) coincides with D τ .To this end, we note that D τ = X × ( G × S τ \ ˜ ϕ ( G × S τ )) . We also observe that G × S τ \ ˜ ϕ ( G × S τ ) is a normal crossing divisor in G × S τ , as ˜ ϕ ( G × S τ ) is theunique open dense symplectic leaf in the log symplectic variety G × S τ (see Proposition 3.16). Theprevious two sentences then force D τ to be a normal crossing divisor in G × S τ , i.e. (a) holds. Theassertion (b) follows immediately from X being symplectic and G × S τ being log symplectic. Theassertion (c) follows from our description of the degeneracy locus in X × ( G × S τ ), as provided inthe first paragraph of the proof. Our proof is therefore complete. (cid:3) Now recall the open embedding k τ : X τ −→ X τ in (4.5), as well as the fact that k τ ( X τ ) ⊆ X ◦ τ (seeProposition 4.10). If X τ is irreducible, then k τ ( X τ ) lies in a unique irreducible component ( X ◦ τ ) irr of the Poisson variety X ◦ τ . The log symplectic nature of X τ is then captured by the following result,which relies heavily on the notation of Subsection 4.2. Theorem 4.15.
Let ( X, P, ν ) be an irreducible symplectic Hamiltonian G -variety. Suppose that τ is an sl -triple in g , and that X τ is irreducible. If X τ exists, then the following statements hold. (i) The Poisson variety ( X ◦ τ ) irr is log symplectic. (ii) The morphism k τ : X τ −→ X τ is a symplectomorphism onto the unique open dense sym-plectic leaf in ( X ◦ τ ) irr .Proof. Since G acts freely on ( X × ( G × S τ )) ◦ , the variety ( µ ◦ τ ) − (0) is smooth. The irreduciblecomponents of ( µ ◦ τ ) − (0) are therefore pairwise disjoint, while the connectedness of G forces thesecomponents to be G -invariant. It follows that the irreducible components of X ◦ τ are precisely theimages of the irreducible components of ( µ ◦ τ ) − (0) under the quotient map θ ◦ τ : ( µ ◦ L ) − (0) −→ X ◦ τ . This implies that ( X ◦ τ ) irr = θ ◦ τ ( Y ) for some unique irreducible component Y ⊆ ( µ ◦ τ ) − (0).Now note that the image of (4.3) lies in a unique irreducible component Z of the smooth variety( µ ◦ L ) − (0), as X τ and µ − τ (0) are irreducible. We also note that θ ◦ τ ( Z ) contains the image of k τ ,as follows from the commutativity of (4.7). We conclude that θ ◦ τ ( Z ) = ( X ◦ τ ) irr , and the previousparagraph then implies that Z = Y .In light of the above, (4.3) may be interpreted as an open embedding i (cid:12)(cid:12) µ − τ (0) : µ − τ (0) −→ Y. (4.21)The irreducibility of Y forces the complement of the image to have positive codimension in Y . Thiscomplement is easily checked to be Y ∩ D τ , where D τ ⊆ X × ( G × S τ ) is defined in (4.20). Wealso observe that Y ∩ D τ has codimension at most one in Y , as D τ is a divisor in X × ( G × S τ ).These last three sentences imply that Y ∩ D τ is a divisor in Y . By [2, Proposition 3.6], the Poissonstructure on θ ◦ τ ( Y ) = ( X ◦ τ ) irr is log symplectic with divisor θ ◦ τ ( Y ∩ D τ ). This completes the proofof (i).Now consider the commutative diagram µ − τ (0) YX τ ( X ◦ τ ) irr i (cid:12)(cid:12) µ − τ (0) k τ , where the right vertical map is the restriction of θ ◦ τ . Since Y ∩ D τ is the complement of the imageof (4.21), we deduce that the image of k τ has a complement of θ ◦ τ ( Y ∩ D τ ). This amounts to theimage of k τ being the unique open dense symplectic leaf in X ◦ irr . Proposition 4.10 then implies that k τ is a symplectomorphism onto this leaf. This establishes (ii), completing the proof. (cid:3) It is worth examining this result in the case τ = 0. To this end, recall the open embedding k : X −→ X ◦ from Subsection 4.1 and the fact that k ( X ) ⊆ X ◦ (see Proposition 4.13). If X isirreducible, then k ( X ) lies in a unique irreducible component X ◦ irr of X ◦ . On the other hand, recallthe G -actions on X and X ◦ discussed in Subsection 4.2. Let us also recall the map ν : X −→ g from (4.12). HE LOG SYMPLECTIC GEOMETRY OF POISSON SLICES 31
Corollary 4.16.
Let ( X, P, ν ) be an irreducible symplectic Hamiltonian G -variety. If X exists,then the following statements hold. (i) The Poisson variety X ◦ irr is log symplectic. (ii) The G -action on X restricts to a Hamiltonian G -action on X ◦ irr with moment map ν (cid:12)(cid:12)(cid:12)(cid:12) X ◦ irr : X ◦ irr −→ g . (iii) The morphism k : X −→ X is a G -equivariant symplectomorphism onto the unique opendense symplectic leaf in X ◦ irr . (iv) The symplectomorphism in (iii) is a embedding of Hamiltonian G -varieties.Proof. Note that µ τ = µ L if τ = 0, where µ L : X × T ∗ G −→ g is the moment map for theHamiltonian action of G L = G × { e } ⊆ G × G on T ∗ G . We also observe that the map X × G −→ µ − L (0) , ( x, g ) −→ ( x, ( g, Ad g − ( ν ( x )))) , ( x, g ) ∈ X × G is a variety isomorphism. It follows that µ − τ (0) is irreducible if τ = 0. Theorem 4.15 now impliesthat X ◦ irr is log symplectic, and that k : X −→ X is a symplectomorphism onto the unique opendense symplectic leaf in X ◦ irr . One also knows that k defines an embedding of Hamiltonian G -varieties X −→ X ◦ (see Proposition 4.13), and that the G -action on X ◦ must preserve the component X ◦ irr .These last two sentences serve to verify (i)–(iv). (cid:3) Examples
We now illustrate some of our results in the context of concrete and familiar examples.5.1.
The existence of X τ and X . Our first step is to find sufficient conditions for the existenceof X τ and X . To this end, let ( X, P, ν ) be a Hamiltonian G -variety. Consider the product ( G × G )-variety X × G , where the ( G × G )-action on X is the one described in Subsection 2.7. Let τ be an sl -triple in g and consider the following conditions:(I) X is a principal G -bundle;(II) ( X × G ) /G L exists;(III) X exists;(IV) X τ exists. Lemma 5.1.
Let ( X, P, ν ) be a Hamiltonian G -variety, and suppose that τ is an sl -triple in g .We then have the chain of implications (I) = ⇒ (II) = ⇒ (III) = ⇒ (IV) .Proof. Assume that (I) holds. To verify (II), we consider Proposition 3.3.2 and Corollary 3.3.3from [8]. These results reduce us to proving that G is G L -quasi-projective, i.e. that there exist afinite-dimensional G L -module V and a G L -equivariant locally closed immersion G −→ P ( V ). Wefirst observe that the Pl¨ucker embeddingGr( n, g ⊕ g ) −→ P (Λ n ( g ⊕ g ))is ( G × G )-equivariant. Its restriction to G is therefore a ( G × G )-equivariant closed immersion G −→ P (Λ n ( g ⊕ g )) . It follows that G is indeed G L -quasi-projective, verifying (II).Now assume that (II) is true, and note that µ − L (0) = (cid:26) ( x, ( γ, ( y , y ))) ∈ X × ( G × ( g ⊕ g )) : y = ν ( x ) and ( ν ( x ) , y ) ∈ γ (cid:27) . One readily deduces that µ − L (0) −→ X × G × g , ( x, ( γ, ( y , y ))) −→ ( x, γ, y )is a G L -equivariant closed immersion, where X × G × g is regarded as the product of the G L -variety X × G and the G L -variety g with trivial action. We also know that X × G × g has a geometricquotient by G L , as (II) is assumed to be true. These last two sentences combine with Proposition2.4 and imply that µ − L (0) has a geometric quotient by G L , i.e. (III) is true.Now assume that (III) holds, and consider the geometric quotient map π L : µ − L (0) −→ X. Let us also observe that the G -action on X × ( G × S ) comes from restricting the G L -action on X × T ∗ G (log D ). Proposition 2.4 then implies that π L restricts to a geometric quotient( X × ( G × S )) ∩ µ − L (0) −→ π L (( X × ( G × S )) ∩ µ − L (0))of ( X × ( G × S )) ∩ µ − L (0) by G . On the other hand, ( X × ( G × S )) ∩ µ − L (0) is precisely the fibre of µ τ : X × ( G × S ) −→ g over 0. These last two sentences imply that µ − τ (0) has a geometric quotientby G , i.e. (IV) holds. (cid:3) This lemma turns out to yield a large class of Hamiltonian G -varieties X such that X τ existsfor all sl -triples τ . To obtain this class, let Y be a smooth G -variety. The G -action on Y has acanonical lift to a Hamiltonian G -action on T ∗ Y , and there is a canonical moment map. This leadsto the following result. Corollary 5.2.
Let Y be an irreducible smooth principal G -bundle and set X = T ∗ Y . Suppose that τ is an sl -triple in g . The following statements hold. (i) The variety X τ exists and is a smooth Poisson variety. (ii) If X τ is irreducible, then it is log symplectic and k τ : X τ −→ X τ is a symplectomorphismonto the unique open dense symplectic leaf in X τ .Proof. We begin by establishing that X τ exists. By Lemmas 2.6 and 5.1, it suffices to show that G -action on X admits a good quotient. Since G acts freely on Y , this is implied by [20, Theorem2.3] and the remark that follows it.Now observe that the G -action on X = T ∗ Y is also free, implying that X τ = X ◦ τ . Proposition4.9 then tells us that X τ is smooth and Poisson. This completes the proof of (i). The assertion (ii)follows immediately from Theorem 4.15. (cid:3) The main examples.
We now discuss some of the examples that motivate and best exhibitthe results in this paper. Our first two examples satisfy the hypotheses of Corollary 5.2, while thethird example has a different nature.
Example 5.3.
Suppose that Y = G is endowed with the G -action defined by g · h := hg − , g, h ∈ G. The induced Hamiltonian G -action on X = T ∗ Y = T ∗ G then satisfies X τ ∼ = G × S τ and X τ = ( T ∗ G × ( G × S τ )) (cid:12) G ∼ = G × S τ for any sl -triple τ in g . The fibrewise compactification in Theorem 4.5 becomes the one mentionedin Remark 3.17. HE LOG SYMPLECTIC GEOMETRY OF POISSON SLICES 33
Example 5.4.
A mild generalization of the previous example can be obtained as follows. Supposethat G is a closed subgroup of a connected linear algebraic group Y = H with Lie algebra h . Notethat G then acts on Y via by the formula g · h := hg − , g ∈ G, h ∈ H. The cotangent bundle X = T ∗ Y = T ∗ H is thereby a Hamiltonian G -variety, and the left trivializa-tion gives an identification X ∼ = H × h ∗ . The moment map is given by H × h ∗ → g ∗ , ( h, α )
7→ − α | g . One finds that X τ ∼ = H × (( S τ ) × g † ) under appropriate identifications, where g † denotes theannihilator of g in h ∗ . By Corollary 5.2, X τ exists for all sl -triples τ in g . Example 5.5.
Let τ be a principal sl -triple in g and recall the notation used in Subsection 3.3.Consider the Hamiltonian G -varieties X = G × S τ and G × S τ , as well as the moment maps ρ τ : G × S τ −→ g and ρ τ : G × S τ −→ g . The discussion of Z τ g and Z τ g in Subsection 3.4 combines with Proposition 3.9 to imply that X τ = ρ − τ ( S τ ) = Z τ g and X τ = (( G × S τ ) × ( G × S τ )) (cid:12) G ∼ = ρ − τ ( S τ ) = Z τ g . The fibrewise compactification in Theorem 4.5 becomes B˘alibanu’s fibrewise compactification (3.14).
Notation • O Y — structure sheaf of an algebraic variety Y • C [ Y ] — coordinate ring of an algebraic variety Y • G — complex semisimple linear algebraic group • G L — the subgroup G × { e } ⊆ G × G • G R — the subgroup { e } × G ⊆ G × G • g — Lie algebra of G • Ad : G −→ GL( g ) — adjoint representation • g ∆ — diagonal in g ⊕ g • n — dimension of g • h· , ·i — Killing form on g • τ — sl -triple in g • S τ — Slodowy slice associated to τ . • χ : g −→ Spec( C [ g ] G ) — adjoint quotient • y τ — unique point at which S τ meets χ − ( χ ( y )), if τ is a principal sl -triple • ρ = ( ρ L , ρ R ) : T ∗ G −→ g ⊕ g — moment map for the ( G × G )-action on T ∗ G • ρ τ : G × S τ −→ g — moment map for the G -action on G × S τ • X — Hamiltonian G -variety • ν : X −→ g — moment map for the G -action on X • X τ — the Poisson slice ν − ( S τ ) • X/G — geometric quotient of X by G • µ = ( µ L , µ R ) : X × T ∗ G −→ g ⊕ g — moment map for the ( G × G )-action on X × T ∗ G • µ τ : X × ( G × S τ ) −→ g moment map for the G -action on X × ( G × S τ ) • ψ τ : X τ −→ ( X × ( G × S τ )) (cid:12) G — canonical Poisson variety isomorphism • G — De Concini–Procesi wonderful compactification of G • D — the divisor G \ G • T ∗ G (log( D )) — log cotangent bundle of ( G, D ) • ρ = ( ρ L , ρ R ) : T ∗ G (log( D )) −→ g ⊕ g — moment map for the ( G × G )-action on T ∗ G (log( D )) • G × S τ — the Poisson slice ρ − R ( S τ ) • Z τ g — universal centralizer of g • Z τ g — B˘alibanu’s partial compactification of Z τ g • ρ τ : G × S τ −→ g — moment map for the G -action on G × S τ • µ = ( µ L , µ R ) : X × T ∗ G (log D ) −→ g ⊕ g — moment map for the ( G × G )-action on X × T ∗ G (log D ) • µ τ : X × ( G × S τ ) −→ g — moment map for the G -action on X × ( G × S τ ) • X — the Hamiltonian reduction ( X × T ∗ G (log D )) (cid:12) G L • k : X −→ X — canonical G -equivariant open embedding • ν : X −→ g — equivariant extension of ν to X • X τ — the Hamiltonian reduction ( X × ( G × S τ )) (cid:12) G • k τ : X τ −→ X τ — canonical open embedding • ( X × ( G × S τ )) ◦ — set of points in X × ( G × S τ ) with trivial G -stabilizers • X ◦ τ — the Hamiltonian reduction ( X × ( G × S τ )) ◦ (cid:12) G References [1]
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