The volume of the N-fold reduced product of coadjoint orbits
aa r X i v : . [ m a t h . S G ] A p r THE VOLUME OF THE N -FOLD REDUCEDPRODUCT OF COADJOINT ORBITS LISA C. JEFFREY AND JIA JI
Abstract.
We compute the symplectic volume of the symplecticreduced space of the product of N coadjoint orbits of a compactconnected Lie group G . We compare our result with the result ofSuzuki and Takakura [24], who study this in the case G = SU (3)starting from geometric quantization. Contents
1. Introduction 12. Setup 23. Notations and Conventions 64. The Case G = SU (3) and N = 3 75. Generalizations 13References 141. Introduction
Our paper studies the symplectic volume of the reduced product of N coadjoint orbits. This can be computed by the method of [11] elabo-rated in [15]. It has also appeared in the literature via geometric quan-tization and Riemann-Roch [24]. Coadjoint orbits are among the mostcommonly used examples of Hamiltonian group actions (via the workof Kirillov-Kostant-Souriau who identified the symplectic structure onthem). Through quantization and the Bott-Borel-Weil theorem, theyare also linked to representation theory.Suzuki and Takakura derived a formula for the symplectic volumeof N -fold reduced product of coadjoint orbits in [24]. Their formulais proved for general N and G = SU (3). Their formula is provedonly for a discrete set of weights (rational values of all weights). Their Date : October 15, 2018.2000
Mathematics Subject Classification.
Primary: 55N15; Secondary: 22E67.
Key words and phrases. reduced product, coadjoint orbit, symplectic reduction.The first author is partially supported by NSERC Discovery Grants. derivation proceeds via Riemann-Roch. We generalize the result toarbitrary connected compact Lie groups G , and our formula is true foran open dense set in the space of weights (there is no discrete hypothesison the weights). In the case G = SU (3) and N = 3, our formula agreesnumerically with Suzuki and Takakura’s formula.2. Setup
First, we describe the main object that we are investigating in thispaper.Let G be a semisimple compact connected Lie group. Let g be theLie algebra of G . Let g ∗ be the dual vector space of g .We fix a maximal torus T in G . Let t be the Lie algebra of T . Let t ∗ be the dual vector space of t . Let W = N ( T ) /T be the correspondingWeyl group.Let Ad : G → Aut( g ) be the adjoint representation of G . Let K : G → Aut( g ∗ ) be the coadjoint representation of G . More explicitly,(1) h K ( g ) ξ, X i = (cid:10) ξ, Ad( g − ) X (cid:11) for all g ∈ G , X ∈ g , ξ ∈ g ∗ , where h· , ·i is the natural pairing betweena covector and a vector. Remark.
Note that for all g, h ∈ G , Ad( g ) ◦ Ad( h ) = Ad( gh ) and K ( g ) ◦ K ( h ) = K ( gh ). That is, both Ad and K are left actions. Weshall always be clear about whether an action is left or right.Let ad : g → End( g ) be the adjoint representation of g . Let K ∗ : g → End( g ∗ ) denote the representation of the Lie algebra g on g ∗ inducedby the coadjoint representation of G . Thus, K ∗ ( X ) = − ad( X ) ∗ . Remark.
Note that both ad : g → End( g ) and K ∗ : g → End( g ∗ ) areLie algebra homomorphisms.Since T is a Lie subgroup of G , t can be naturally viewed as asubspace of g . On the other hand, t ∗ can be viewed as a subspace of g ∗ if we identify t ∗ with { ξ ∈ g ∗ : K ( t ) ξ = ξ , for all t ∈ T } .Let O ( ξ ) denote the coadjoint orbit through ξ ∈ g ∗ . The followingtheorem is well known. Theorem 1 (Kirillov-Kostant-Souriau) . Given any ξ ∈ g ∗ , the coad-joint orbit O ( ξ ) is a smooth compact connected submanifold in g ∗ andthere exists a natural K ( G ) -invariant symplectic structure on O ( ξ ) . Inother words, there exists a closed non-degenerate K ( G ) -invariant real OLUME OF N -FOLD REDUCED PRODUCTS 3 -form ω O ( ξ ) ∈ Ω ( O ( ξ ); R ) on O ( ξ ) . More explicitly, ω O ( ξ ) can beconstructed in the following way.For all η ∈ O ( ξ ) , let B η be an antisymmetric bilinear form on g defined by (2) B η ( X, Y ) := h η, [ X, Y ] i for all X, Y ∈ g . Then ω O ( ξ ) can be defined by (3) ω O ( ξ ) ( η )( K ∗ ( X )( η ) , K ∗ ( Y )( η )) := B η ( X, Y ) for all X, Y ∈ g , η ∈ O ( ξ ) .Note that for all η ∈ O ( ξ ) ⊆ g ∗ , T η O ( ξ ) = { K ∗ ( X )( η ) : X ∈ g } .This natural -form ω O ( ξ ) is sometimes referred to as the Kirillov-Kostant-Souriau symplectic form on the coadjoint orbit O ( ξ ) . Therefore, a coadjoint orbit O ( ξ ) becomes a symplectic manifoldwhen it is equipped with its Kirillov-Kostant-Souriau symplectic form ω O ( ξ ) . In addition, we have the following: Proposition 2.
The coadjoint action of G on ( O ( ξ ) , ω O ( ξ ) ) is a Hamil-tonian G -action with the moment map given by the inclusion map µ O ( ξ ) : O ( ξ ) ֒ → g ∗ . In other words, µ O ( ξ ) is equivariant with respect tothe coadjoint action of G on O ( ξ ) and the coadjoint action of G on g ∗ ,and, for all X ∈ g , (4) dµ X O ( ξ ) = ι ρ ( X ) ω O ( ξ ) where µ X O ( ξ ) : O ( ξ ) → R is defined by µ X O ( ξ ) ( η ) = (cid:10) µ O ( ξ ) ( η ) , X (cid:11) forall η ∈ O ( ξ ) and ρ ( X ) is the vector field on O ( ξ ) such that for all η ∈ O ( ξ ) , the tangent vector ρ ( X )( η ) ∈ T η O ( ξ ) is (5) ddt (cid:12)(cid:12)(cid:12) t =0 ( K (exp( − tX )) η ) . Let O ( ξ ) , · · · , O ( ξ N ) be N coadjoint orbits. Then we can form theirCartesian product:(6) M ( ξ ) := O ( ξ ) × · · · × O ( ξ N )where(7) ξ := ( ξ , · · · , ξ N ) ∈ N z }| { g ∗ × · · · × g ∗ .We assume the following: Assumption 3.
All of O ( ξ ) , · · · , O ( ξ N ) are diffeomorphic to the ho-mogeneous space G/T . This assumption is equivalent to the assump-tion that all of the stabilizer groups Stab G ( ξ ) , · · · , Stab G ( ξ N ) are con-jugate to the chosen maximal torus T . If all of ξ , · · · , ξ N are contained LISA C. JEFFREY AND JIA JI in t ∗ ⊆ g ∗ , then this assumption is saying that Stab G ( ξ ) = · · · =Stab G ( ξ N ) = T . Remark.
Since every coadjoint orbit O ( ξ ) can be written as O ( ξ ′ ) forsome ξ ′ ∈ t ∗ ⊆ g ∗ , we can always assume that ξ = ( ξ , · · · , ξ N ) satisfiesthat ξ j ∈ t ∗ ⊆ g ∗ for all j .The Cartesian product M ( ξ ) = O ( ξ ) × · · · × O ( ξ N ) carries a naturalsymplectic structure ω ξ defined by:(8) ω ξ := pr ∗ ω O ( ξ ) + · · · + pr ∗ N ω O ( ξ N ) where pr j : O ( ξ ) × · · · × O ( ξ N ) → O ( ξ j ) is the projection onto the j -th component.Let G act on M ( ξ ) = O ( ξ ) × · · · × O ( ξ N ) by the diagonal action ∆:(9) ∆( g )( η , · · · , η N ) := ( K ( g )( η ) , · · · , K ( g )( η N ))for all g ∈ G , η j ∈ O ( ξ j ).In addition to that the symplectic form ω ξ is clearly G -invariant, wealso have the following: Proposition 4.
The diagonal action ∆ of G on ( M ( ξ ) , ω ξ ) is a Hamil-tonian G -action with the moment map µ ξ : M ( ξ ) → g ∗ being: (10) µ ξ ( η ) = N X j =1 η j for all η := ( η , · · · , η N ) ∈ M ( ξ ) . We assume that:
Assumption 5. ∈ g ∗ is a regular value for µ ξ : M ( ξ ) → g ∗ and µ − ξ (0) = ∅ . Remark.
By Sard’s theorem, the set(11) A := ξ ∈ N z }| { t ∗ × · · · × t ∗ : Assumptions 3, 5 hold. is nonempty and has nonempty interior in t ∗ × · · · × t ∗ .Then, the level set M ( ξ ) := µ − ξ (0) is a closed, thus compact, sub-manifold of M ( ξ ) and the diagonal action ∆ of G restricts to an actionon M ( ξ ). Therefore, we can form the quotient space with respect tothis action of G on M ( ξ ):(12) M red ( ξ ) := M ( ξ ) /G . OLUME OF N -FOLD REDUCED PRODUCTS 5 Sometimes, the above quotient space is also denoted by M ( ξ ) //G . Notethat this quotient space is also compact.If the G -action on M ( ξ ) is free and proper (in our situation, proper-ness is automatically satisfied), then the quotient space M red ( ξ ) = M ( ξ ) /G is a smooth manifold. However, in our situation, the G -action on M ( ξ ) is in general not free. Hence, in general the quotientspace is only an orbifold [12]. To avoid this complication, we will as-sume: Assumption 6.
The quotient space M red ( ξ ) = M ( ξ ) /G is a smoothcompact manifold. Remark.
The above assumption will put further restrictions on which ξ ∈ t ∗ × · · · × t ∗ we can choose as initial data. Thus we only chooseinitial data from the following set in this paper:(13) A ′ := ξ ∈ N z }| { t ∗ × · · · × t ∗ : Assumptions 3, 5, and 6 hold. Suzuki and Takakura also made this assumption in their paper [24](in Section 2.3). It seems reasonable to us to assume that even afterAssumption 6 is imposed, the initial data set A ′ is still nonempty andstill has nonempty interior in t ∗ ×· · ·× t ∗ . Notice that since the elementsin the center of G always act trivially on M ( ξ ) and M ( ξ ), Assumption6 is valid if P G = G/Z ( G ) acts freely on M ( ξ ). This happens for G = SU ( n ) if all the coadjoint orbits O ( ξ i ) are generic.Then, we have the following well known theorem: Theorem 7 (Marsden-Weinstein) . The smooth compact manifold M red ( ξ ) = M ( ξ ) /G carries a unique symplectic structure ω red ( ξ ) such that (14) i ∗ ω ξ = π ∗ ω red ( ξ ) where i : M ( ξ ) ֒ → M ( ξ ) is the inclusion map and π : M ( ξ ) →M red ( ξ ) is the associated projection map. Definition 8.
We call this compact symplectic manifold ( M red ( ξ ) , ω red ( ξ ))an N -fold reduced product .Now, it is natural to consider the symplectic volume of an N -foldreduced product. Notice that for a compact symplectic manifold ( M, ω )of real dimension 2 n , its symplectic volume SVol( M ) is defined by(15) SVol( M ) := Z M e ω/ (2 π ) = Z M (cid:0) e ω/ (2 π ) (cid:1) [2 n ] = Z M ω n (2 π ) n n ! LISA C. JEFFREY AND JIA JI where σ [ j ] is the homogeneous component of degree j of the differentialform σ = σ [0] + σ [1] + · · · .Note that in the case of n = 0, namely, when M consists of k discretepoints for some positive integer k , then SVol( M ) = k .Notice that(16) SVol( M ) = 1(2 πi ) n Z M e iω . We want to know how SVol( M red ( ξ )) varies as ξ varies. We want toobtain a formula computing SVol( M red ( ξ )). Remark.
The dimension of an N -fold reduced product is(17) N (dim G − dim T ) − G = ( N −
2) dim G − N dim T when all orbits are generic. In the case G = SU (3) and N = 3 , this isdim G − T = 8 − Notations and Conventions
Notations.
If the initial point ξ is clear in the context, we willsuppress the inclusion of the point ξ in our notations and write, forexample, M , M , M red instead of M ( ξ ) , M ( ξ ) , M red ( ξ ), respectively.Similarly, this is done for the notations of the symplectic structuresand so on.3.2. Left Actions Versus Right Actions.
We shall always be clearabout whether we are dealing with a left action or a right action.Let G be a Lie group. Let g be the Lie algebra of G . Let M be amanifold. Let Φ : G × M → M be a left G -action on M . Then, for all X ∈ g , the associated fundamental vector field ρ ( X ) on M is definedby:(18) ρ ( X ) x := ddt (cid:12)(cid:12)(cid:12) t =0 ((exp( − tX )) · x )for all x ∈ M .Let Ψ : M × G → M be a right G -action on M . Then, for all X ∈ g ,the associated fundamental vector field ρ ( X ) on M is defined by:(19) ρ ( X ) x := ddt (cid:12)(cid:12)(cid:12) t =0 ( x · (exp( tX )))for all x ∈ M . OLUME OF N -FOLD REDUCED PRODUCTS 7 The definitions above for fundamental vector fields of left and rightactions ensure that the following map ρ : g → Γ( M, T M )is a Lie algebra homomorphism from the Lie algebra g to the Lie algebraΓ( M, T M ) of smooth vector fields on M .3.3. The inner product structure on g . Since our compact con-nected Lie group G is semisimple, the Killing form on g is negativedefinite and hence there is a G -invariant inner product structure ( · , · )on g . Let s be the real dimension of the maximal torus T . Then ourconvention for the inner product on g is:(20) ( X, Y ) := − π ) s tr(ad( X ) ◦ ad( Y ))for all X, Y ∈ g .We will use this inner product structure on g to identify g and g ∗ ,and also t and t ∗ . Under this identification, the coadjoint representa-tion is identified with the adjoint representation. Hence we can defineeverything in the adjoint setting. For matrix Lie groups, the adjointsetting is often more convenient for carrying out computations.4. The Case G = SU (3) and N = 3In this section, we study 3-fold reduced products, or triple reducedproducts for G = SU (3). See [17], [14] for recent studies about theseobjects. Our focus is on the symplectic volume of triple reduced prod-ucts.4.1. The Setup for the case G = SU (3) . The setup here is dueto [24].Let G = SU (3) and let T be its standard maximal torus, i.e., T consists of diagonal matrices in SU (3). Let g be the Lie algebra of G and let t be the Lie algebra of T . Let g ∗ be the dual vector space of g and let t ∗ be the dual vector space of t .In this case, we know that the corresponding Weyl group W is iso-morphic to the permutation group S .The Weyl group W acts on t ∗ ∼ = t by permutations of diagonal en-tries.The elements H := 2 πi − , H := 2 πi − LISA C. JEFFREY AND JIA JI in t are generators of the integral lattice exp − ( I ) ⊂ t . In addition, H , H form a basis of t . Let ω , ω be the basis of t ∗ dual to H , H ,i.e., ω i ( H j ) = δ ij . Under the identification t ∗ ∼ = t , ω , ω correspond tothe elementsΩ := 2 πi − − , Ω := 2 πi − in t , respectively.Let t ∗ + := R ≥ ω + R ≥ ω and Λ + := Z ≥ ω + Z ≥ ω . So t ∗ + isa positive Weyl chamber and Λ + is the associated set of dominantintegral weights. Any element ξ of t ∗ + or Λ + can be written as(21) ξ = ( ℓ − m ) ω + mω , ℓ ≥ m ≥ . Under the identification t ∗ ∼ = t , ξ corresponds to the element(22) X = ( ℓ − m )Ω + m Ω . Every coadjoint orbit can be written as O ξ for some ξ ∈ t ∗ + , and inthis case, O ξ ∩ t ∗ is the W -orbit through ξ , and O ξ ∩ t ∗ + = { ξ } .If ξ = ( ℓ − m ) ω + mω ∈ t ∗ + with ℓ > m >
0, then Stab G ( ξ ) = T and O ξ is diffeomorphic to the homogeneous space G/T .Let ξ , ξ , ξ ∈ t ∗ + so that ξ i = ( ℓ i − m i ) ω + m i ω with ℓ i > m i >
0. Let ξ := ( ξ , ξ , ξ ). Then, ξ determines a triple reduced product( M red ( ξ ) , ω red ( ξ )).4.2. Symplectic Volume of a Triple Reduced Product.
By thelocalization technique in [15], R M red e iω red can be expressed as a finitesum of contributions indexed by the fixed point set M T of M underthe action of the maximal torus T :(23) M T = { ( w · ξ , w · ξ , w · ξ ) : w , w , w ∈ W } . More precisely, we have(24) Z M red e iω red = X w ∈ W Z X ∈ t ̟ ( X ) e i h µ ( w · ξ ) ,X i e w · ξ ( X ) dX, where w = ( w , w , w ) ∈ W , ξ = ( ξ , ξ , ξ ) and(25) w · ξ := ( w · ξ , w · ξ , w · ξ ) , and ̟ ( X ) = Q α h α, X i with α running over all positive roots of G = SU (3) and e F ( X ) is the equivariant Euler class of the normal bundleto the fixed point F . In this case,(26) e w · ξ ( X ) = sgn( w ) ̟ ( X ) , OLUME OF N -FOLD REDUCED PRODUCTS 9 where sgn( w ) := sgn( w ) sgn( w ) sgn( w ).Hence, we have: Theorem 9. (27) Z M red e iω red = X w ∈ W sgn( w ) Z X ∈ t e i h µ ( w · ξ ) ,X i ̟ ( X ) dX. On the other hand, the symplectic volume of the reduced space µ − η,T (0) /T of the Hamiltonian system ( O η , ω η , T, µ η,T ), where µ η,T : O η ֒ → t ∗ ⊂ g ∗ is the moment map associated to the Hamiltoniangroup action (in this case, the coadjoint action) on O η by the stan-dard maximal torus T , is expressed by the following formula, knownfrom [11], [15]: Theorem 10. (28) SVol( µ − η,T (0) /T ) = 12 πi X w ∈ W sgn( w ) Z X ∈ t e i h w · η,X i ̟ ( X ) dX. Let(29) f ( η ) := 2 πi SVol( µ − η,T (0) /T ) = X w ∈ W sgn( w ) Z X ∈ t e i h w · η,X i ̟ ( X ) dX. Then, by writing w = w w − w , w = w w − w and letting w ′ = w − w , w ′ = w − w , we obtain(30) Z M red e iω red = X w ′ ∈ W X w ′ ∈ W sgn( w ′ ) sgn( w ′ ) f ( ξ + w ′ · ξ + w ′ · ξ ) . On the other hand, it is known from [15] that
Theorem 11. (31) SVol( µ − η,T (0) /T ) = X w ∈ W sgn( w ) H β ( w · η ) where β = ( β , β , β ) and β , β , β are the positive roots of SU (3) ,and (32) H β ( ξ ) := vol a ( ( s , s , s ) ∈ R ≥ : X i =1 s i β i = ξ ) where vol a here denotes the standard a -dimensional Euclidean volumemultiplied by a normalization constant, and (33) a = r − dim T where r is the number of positive roots of SU (3) . Notice that in thiscase a = 1 . Therefore, R M red e iω red can also be expressed as(34)2 πi X w ′ ∈ W X w ′ ∈ W sgn( w ′ ) sgn( w ′ ) X w ∈ W sgn( w ) H β ( w · ( ξ + w ′ · ξ + w ′ · ξ )) . By letting w ′ = w − w , w ′ = w − w , we then obtain(35) Z M red e iω red = 2 πi X w ∈ W sgn( w ) H β ( µ ( w · ξ )) . Hence, we obtain the volume formula for triple reduced products for G = SU (3): Theorem 12. (36) SVol( M red ( ξ )) = X w ∈ W sgn( w ) H β ( µ ( w · ξ )) . Here, H β : t ∗ → R is called the Duistermaat-Heckman function for G = SU (3). For general semisimple compact connected Lie group G ,it can be defined as follows: Definition 13. (37) H β ( ξ ) = vol a ( ( s , · · · , s r ) : s i ≥ , r X i =1 s i β i = ξ ) where β = ( β , · · · , β r ) and β , · · · , β r ∈ t ∗ are all the positive roots of G and a = r − dim T . Remark.
It is clear from the above definition that H β is supported inthe cone(38) C β := ( r X i =1 s i β i : s i ≥ ) ⊆ t ∗ . In the case G = SU (3), we have r = 3 and β = H , β = H , β = H + H . If ξ = ( ℓ − m )Ω + m Ω = ( ℓ − m ) · (2 H + H ) / m · ( H +2 H ) /
3, then we obtain:(39) H β ( ξ ) = κ · max (cid:26) min (cid:26) ℓ − m, ℓ + 13 m (cid:27) , (cid:27) where κ is a normalization constant. Notice that here we have identified g ∗ with g , and thus t ∗ with t , by the inner product structure on g .We fix the basis { Ω , Ω − Ω } for t . Then, each ξ i = ( ℓ i − m i )Ω + m i Ω ∈ t has ( ℓ i , m i ) as its coordinates in this basis. Hence, ξ =( ξ , ξ , ξ ) can be represented in this basis by the vector(40) ( ℓ , ℓ , ℓ , m , m , m ) ∈ R . OLUME OF N -FOLD REDUCED PRODUCTS 11 Also in this basis, the Weyl group elements { v , v , · · · , v } corre-spond to the following matrices: v = σ = Id ∼ = (cid:18) (cid:19) v = σ = (1 2) ∼ = (cid:18) (cid:19) v = σ = (1 2 3) ∼ = (cid:18) − − (cid:19) v = σ = (1 3) ∼ = (cid:18) − − (cid:19) v = σ = (1 3 2) ∼ = (cid:18) − − (cid:19) v = σ = (2 3) ∼ = (cid:18) − − (cid:19) where { σ , · · · , σ } is an enumeration of the Weyl group W used in [24].Hence, the symplectic volume of a triple reduced product for G = SU (3) can be computed explicitly by the following formula: Theorem 14.
SVol( l , l , l , m , m , m ) =(41) κ X i,j,k =0 ( − i + j + k max (cid:26) min (cid:26) ( 23 pr −
13 pr )( P ijk ) , ( 13 pr + 13 pr )( P ijk ) (cid:27) , (cid:27) where (42) P ijk ( l , l , l , m , m , m ) = v i · (cid:18) l m (cid:19) + v j · (cid:18) l m (cid:19) + v k · (cid:18) l m (cid:19) and pr , pr : R → R are the standard projections to the first andsecond coordinates, respectively. A Recent Result of the Symplectic Volume.
In 2008, Suzukiand Takakura [24] gave a result about the symplectic volume of an N -fold reduced product M red ( ξ ) for G = SU (3) and N ≥
3, with ξ lyingin a discrete set.In particular, their result for N = 3 is as follows. Definition 15.
A 6-tuple ( I , · · · , I ), where each I i is a subset of { , , } , is called a 6-partition of { , , } , if I ∪ · · · ∪ I = { , , } and I i ∩ I j = ∅ if i = j .Now let ξ i = ( ℓ i − m i ) ω + m i ω ∈ Λ + with ℓ i > m i > ℓ i , m i ∈ Z ,for i ∈ { , , } .Let L := ℓ + ℓ + ℓ and M := m + m + m .They assume the following condition (in order to apply GIT tech-niques): Assumption 16. ( L + M ) is divisible by 3. Definition 17.
For any I ⊂ { , , } , define(43) ℓ I = X i ∈ I ℓ i , m I = X i ∈ I m i . If I and J are disjoint subsets of { , , } , define(44) ℓ I,J = ℓ I + ℓ J = X i ∈ I ∪ J ℓ i , m I,J = m I + m J = X i ∈ I ∪ J m i . Let ξ = ( ξ , ξ , ξ ). Definition 18.
Denote by I ξ the set of all 6-partitions ( I , · · · , I )such that(45) ℓ I ,I + m I ,I <
13 ( L + M ) , ℓ I ,I + m I ,I <
13 ( L + M ) , and denote by J ξ the set of all 6-partitions ( I , · · · , I ) such that(46) ℓ I ,I + m I ,I >
13 ( L + M ) , ℓ I ,I + m I ,I >
13 ( L + M ) . Notice that I ξ and J ξ are disjoint for any ξ . Definition 19.
Define the functions A ξ : I ξ → R and B ξ : J ξ → R asfollows:(47) A ξ ( I , · · · , I ) := − ( − | I | + | I | + | I | L + M − ℓ I ,I − m I ,I ) , (48) B ξ ( I , · · · , I ) := − ( − | I | + | I | + | I | ℓ I ,I + m I ,I − L + M . Then Suzuki and Takakura conclude that the symplectic volume of M red ( ξ ) is given by the following formula: Theorem 20. (49) V ( ξ ) = X ( I , ··· ,I ) ∈I ξ A ξ ( I , · · · , I ) + X ( I , ··· ,I ) ∈J ξ B ξ ( I , · · · , I ) . OLUME OF N -FOLD REDUCED PRODUCTS 13 Since both our formula (41) and their formula (49) describe the sym-plectic volume for the same object, they should agree. In addition,there are numerical evidences that they indeed agree.5.
Generalizations
Symplectic volume of triple reduced products for generalsemisimple compact connected Lie group G . Our method appliesto any semisimple compact connected Lie group G . Therefore, theo-rems 9, 10, 11 and 12 still hold in this more general situation (of course,the set of positive roots is now different and the Duistermaat-Heckmanfunction H β should be replaced by the general one in Definition 13).5.2. Symplectic volume of N-fold reduced products for generalsemisimple compact connected Lie group G . We can also gener-alize our results from the triple reduced product (symplectic quotientof product of three orbits) to the N -fold reduced product (symplecticquotient of product of N orbits). The formulas are similar, althoughwe no longer get a piecewise linear function (the formulas are piecewisepolynomial).In this most general setting, we have the following: Theorem 21. (50) Z M red e iω red = X w ∈ W N sgn( w ) Z X ∈ t e i h µ ( w · ξ ) ,X i ̟ N − ( X ) dX where ξ = ( ξ , · · · , ξ N ) , w = ( w , · · · , w N ) ∈ W N and (51) ̟ ( X ) = Y α h α, X i where α runs over all the positive roots of G .Proof. In this case, the equivariant Euler class is(52) e w · ξ ( X ) = sgn( w ) ̟ N ( X ) . (cid:3) In addition, the symplectic volume of M red can be computed by asimilar formula involving Duistermaat-Heckman functions: Theorem 22. (53) SVol( M red ) = X w ∈ W N sgn( w ) H ( N − · β ( µ ( w · ξ )) where β = ( β , · · · , β r ) with β , · · · , β r being all the positive roots of G and the Duistermaat-Heckman function H ( N − · β is defined as follows: H ( N − · β ( ξ ) := vol a n ( s (1)1 , · · · , s (1) r , · · · , s ( N − , · · · , s ( N − r ) :(54) s ( j ) i ≥ , for all i and j , and N − X j =1 r X i =1 s ( j ) i β i = ξ ) where r is the number of positive roots of G and a = ( N − · r − dim T .Remark. Notice that here the Duistermaat-Heckman function is piece-wise polynomial.
References [1] M. F. Atiyah and R. Bott:
The Moment Map and Equivariant Cohomology ,Topology (1984), 1–28.[2] M. Audin: Torus Actions on Symplectic Manifolds , Second revised edition,Progress in Mathematics, Volume , 2004, Springer Basel AG.[3] N. Berline, E. Getzler and M. Vergne: Heat Kernels and Dirac Operators ,Grundlehren Text Editions, 2004, Springer-Verlag Berlin Heidelberg.[4] N. Berline and M. Vergne:
Classes caract´eristiques ´equivariantes. Formules delocalisation en cohomologie ´equivariante , C. R. Acad. Sci. Paris (1982),539–541.[5] N. Berline and M. Vergne:
Z´eros d’un champ de vecteurs et classes car-act´eristiques ´equivariantes , Duke Math. J. (1983), 539–549.[6] J.-M. Bismut, Localization formulas, superconnections, and the index theoremfor families, Commun. Math. Phys. (1986) 127-166.[7] T. Br¨ocker and T. tom Dieck:
Representations of Compact Lie Groups , Grad-uate Texts in Mathematics , 1985, Springer.[8] A. Cannas da Silva: Lectures on Symplectic Geometry , Corrected 2nd printing,Lecture Notes in Mathematics , 2008, Springer.[9] J. J. Duistermaat and G. J. Heckman:
On the Variation in the Cohomology ofthe Symplectic Form of the Reduced Phase Space , Invent. math. , 259–268(1982)[10] V. Guillemin, J. Kalkman, The Jeffrey-Kirwan localization theorem andresidue operations in equivariant cohomology, J. Reine Angew. Math. (1996) 123-142.[11] V. Guillemin, E. Lerman and S. Sternberg:
Symplectic Fibrations and Multi-plicity Diagrams , 1996, Cambridge University Press.[12] V. Guillemin and S. Sternberg:
Symplectic Techniques in Physics , Reprintedition, 1990, Cambridge University Press.[13] A. Henriques and D. S. Metzler:
Presentations of Noneffective Orbifolds ,Trans. Amer. Math. Soc. 356 (2004), no. 6, 2481–2499.
OLUME OF N -FOLD REDUCED PRODUCTS 15 [14] J. Hurtubise, L. C. Jeffrey, S. Rayan, P. Selick and J. Weitsman: SpectralCurves for the Triple Reduced Product of Coadjoint Orbits for SU (3), arXiv:1708.00752v2.[15] L. C. Jeffrey and F. C. Kirwan: Localization for Nonabelian Group Actions ,Topology (1995), pp. 291–327.[16] L.C. Jeffrey, F.C. Kirwan, Localization and the quantization conjecture, Topol-ogy (1997)[17] L. C. Jeffrey, S. Rayan, G. Seal, P. Selick and J. Weitsman: The Triple Re-duced Product and Hamiltonian Flows , Geometric Methods in Physics, XXXVWorkshop 2016, Trends in Mathematics, 35–49.[18] Y. Kamiyama and M. Tezuka:
Symplectic Volume of the Moduli Space of Spa-tial Polygons , J. Math. Kyoto Univ. (JMKYAZ) 39-3 (1999), 557–575.[19] M. Kapovich and J. J. Millson:
The Symplectic Geometry of Polygons in Eu-clidean Space , J. Differential Geometry (1996), 479–513.[20] A. A. Kirillov: Lectures on the Orbit Method , Graduate Studies in Mathemat-ics, Volume , 2004, American Mathematical Society.[21] B. Kostant: Quantization and unitary representations , Lecture Notes in Math.,vol. 170, Springer-Verlag, Berlin-Hiedelberg-New York, 1970, pp. 87–208.[22] Marsden, J. and Weinstein, A.: Reduction of symplectic manifolds with sym-metry, Rep. Math. Phys. 5 (1974), 121–130.[23] J.-M. Souriau:
Systemes dynamiques , Dunod, Paris, 1970.[24] T. Suzuki and T. Takakura:
Symplectic Volumes of Certain Symplectic Quo-tients Associated with the Special Unitary Group of Degree Three , Tokyo J.Math. (2008), 1–26.[25] M. Vergne, Multiplicity formulas for geometric quantization I, Duke Math. J. (1996) 143-179.[26] E. Witten, Two dimensional gauge theories revisited, J. Geom. Phys. (1992)303-368. Department of Mathematics, University of Toronto, Toronto, On-tario, Canada
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