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The Cohomology Ring of Weight Varieties and Polygon Spaces 2 0 R. F. Goldin*† 0 2 UniversityofMaryland E-mail: [email protected] n a J 5 We use a theorem of Tolman and Weitsman [23] to find explicit formulæ 1 fortherational cohomology ringsofthesymplectic reductionofflag varieties in Cn, or generic coadjoint orbits of SU(n), by (maximal) torus actions. We G] also calculate the cohomology ring of the moduli space of n points in CPk, whichisisomorphictotheGrassmannianofk planesinCn,byrealizingitas S adegenerate coadjointorbit. . h t Key Words: weightvarieties, symplecticreduction, Schubertpolynomials a m [ 1 1. INTRODUCTION v For M a manifold with a Hamiltonian T action and moment map φ : 8 3 M →t∗, the symplectic reduction is defined as 1 1 M//T(µ):=φ−1(µ)/T 0 2 0 foranyregularvalueµofφ. Weightvarietiesareaspecialcaseofsymplectic / reduction. Let M = O , a coadjoint orbit of a compact, semi-simple Lie h λ group G through the point λ∈t∗, and consider the action of the maximal t a torus T ⊂ G on O . If G = SU(n), we identify the set of Hermitian λ m matrices H with g∗ by tr : A → i·Trace(A·) for all A ∈ H. Under this : identification, we can think of λ as a matrix with real diagonal entries v i (λ1,...,λn), and Oλ as an adjoint orbit of G through λ. The moment X map for the T action on O takes a matrix to its diagonal entries. Thus λ r O //T(µ) consists of Hermitian matrices with spectrum λ and diagonal a λ entriesµ,quotientedoutbytheactionofdiagonalmatrices. Thesymplectic reduction O //T(µ) is a weight variety. λ *Supported byaNationalScienceFoundation Postdoctoral Fellowship †TheauthorwouldliketothankA.KnutsonandS.Billeyformanyusefulconversa- tionsaboutthiswork,andtherefereeforsuchdetailedandhelpfulcomments. 1 2 R.F.GOLDIN The generic coadjoint orbit of SU(n) is symplectomorphic to the complete flag variety in Cn with a symplectic structure given by the spectrum of the orbit. Degenerate coadjoint orbits are homeomorphic to flag vari- etieswithvariousdimensionsmissing: forexampleifthespectrumconsists of only two values, then the coadjoint orbit is homeomorphic to Gr(k,n), the Grassmannian of k planes in Cn. Then O //T(µ) is the symplectic λ reduction of a flag variety by a torus; however, the reduction depends on the symplectic structure on the flag variety. Weight varieties have appeared in several different contexts. They were first termedas such by Knutsonin [17] because oftheir relationshipto the weight spaces of representations. Irreducible representations V of complex G are realized as the holomorphic sections of line bundles over the flag varieties. The dimension of the weight spaces, or the irredicible representations of T in V (specified by the weight of the T action on each isotypic component of V) is the quantization of the symplectic reduction of flag varieties [12]; hence these reductions were named weight varieties. Tech- niquestocomputetheirBettinumbershavebeendevelopedbyKirwan[15] and Klyachko [16]. The advantage of the results in this article is that one obtains the cohomology ring structure. For Betti numbers, one may find that the calculations are straightforward(and hence programmable)using Theorems 1.1 and 1.2, but other methods are likely more efficient. In a small number of cases, the spaces have been found explicitly [13]. The symplectic reduction of the Grassmannian of k-planes in Cn also arises as the moduli space of n points on CPk, as we see below; more details for the case when k =1 can be found in the work of Klyachko[16], Hausmann and Knutson [18] and Kapovich and Millson [14]. The integer cohomology ring of the k =1 polygon spaces was computed in [19]. The generic case we consider is the coadjointorbit of SU(n) of matrices with a specified set of distinct eigenvalues. Equivalently, it is the orbit through λ ∈ t∗ := Lie(T)∗, where λ consists of n distinct eigenvalues λ = (λ ,...,λ ) with nλ = 0, and T is the (n−1)-dimensional max- 1 n 1 i imal torus of SU(n). PWe order them such that λ1 > ··· > λn. These coadjoint orbits are diffeomorphic to the complete flag manifold Fl(n) in Cn. The Grassmannian of k-planes in Cn is diffeomorphic to the SU(n) coadjoint orbit throughν ∈t∗ where ν =(ν ,...,ν ,ν ,...,ν ) consists of 1 1 2 2 two distinct eigenvalues ν and ν . The dimension of the ν eigenspace is 1 2 1 k and that of the ν eigenspace is n−k, so we have kν +(n−k)ν = 0. 2 1 2 We will write Gr(k,n) to indicate these degenerate coadjoint orbits. ν We use the notation λ to indicate the point wλw−1 ∈ t∗ for any w permutation w on n letters. This has the unfortunate consequence that λw = (λw−1(1),...,λw−1(n)) but allows us to use left actions consistently throughoutthe paper. Furthermore,let∆(x,u)∈C[x ,...,x ,u ,...,u ] 1 n 1 n WEIGHTVARIETIES 3 be the polynomial ∆(x,u)= (x −u ). i j Yi<j ∆(x,u) is sometimes called the determinant polynomial. Beforewestatethemaintheorems,webrieflyintroducedivideddifference operators, which are explained in detail in Section 2.3. Definition 1.1. Let f(x ,x ) be a polynomial of variables x and i i+1 i x andpossiblyothervariables. Foreach1≤i≤nthedivided difference i+1 operator ∂ associated to i acts on f as follows: i f(x ,x )−f(x ,x ) i i+1 i+1 i ∂ f(x ,x )= . i i i+1 x −x i i+1 These simple divided difference operators take polynomials of degree k to polynomials of degree k−1. For any element w ∈S , the permutation n group on n letters, one can associate a divided difference operator ∂ as w follows. Write w as a product of simple transpositions s ...s where s i1 il ij isthe simpletraspositionthatswitchesi andi +1. Foranysuchproduct j j where l is minimal, the composition ∂ :=∂ ···∂ w i1 il is well-defined (see [22]). We are now ready to state the main theorems of this article. Theorem 1.1. Let O be a generic coadjoint orbit of SU(n). The ra- λ tional cohomology of O //T(µ) is isomorphic to the ring λ C[x ,...,x ,u ,...,u ] 1 n 1 n n n n (1+u )− (1+x ), u ,∂ ∆(x,u ) i=1 i i=1 i i=1 i v τ (cid:0)Q Q P (cid:1) n n for all v,τ ∈ S such that λ < µ for some k = n i=k+1 v(i) i=k+1 τ(i) 1,...,n−1. Here degxi = dPegui = 2, and P(1+ui)− (1+xi) is the graded difference of symmetric functions in tQhe xis and uiQs. ThereisasimilarstatementforthedegeneratecaseoftheGrassmannian, in which there are only two distinct eigenvalues. Theorem 1.2. The rational cohomology of Gr(k,n) //T(µ) is isomor- ν phic to the ring C[σ (x ,...,x ),σ (x ,...,x ),u ,...,u ] i 1 k i k+1 n 1 n n σ (x ,...,x )−σ (u ,...,u ), u ,∂ ∆(x,u ) i 1 n i 1 n i=1 i v τ (cid:0) P (cid:1) 4 R.F.GOLDIN n n forallv,τ suchthat λ < µ forsomek,and∂ ∆(x,u ) i=k+1 v(i) i=k+1 τ(i) v τ is symmetric in x1,P...,xk and in xPk+1,...,xn. Here σi are the symmetric polynomials in the indicated variables and degx = degu = 2. Note i i n that σ (x ,...,x )−σ (u ,...,u ) for all i is equivalent to (1+u )− i 1 n i 1 n i=1 i n i=1(1+xi). Q Q The forgetful map Fl(n) → Gr(k,n) which “remembers” only the k- planes of each flag is a T-equivariant map which induces an injection on equivariant cohomology: H∗(Gr(k,n))֒→H∗(Fl(n)). (1) T T TheclassesinH∗(Fl(n))whicharesymmetricinx ,...,x andinx ,...,x T 1 k k+1 n are precisely those in the image of the map (1). Theorem1.2 allowsone to compute the cohomologyofthe moduli space of n points in CPk−1, as the reduction of the Grassmannianis isomorphic to this moduli space [10], [9]. In [18], Knutson and Hausmann observe thatthisGelfand-MacPhersoncorrespondenceisjustadualpairsymplectic reduction. TheGrassmannianGr(k,n)isrealizedasasymplecticreduction of Cnk by a U(k) action on the right. Then by reducing in stages we have Gr(k,n)//Tn =(Cnk//U(k))//Tn=Cnk//(Tn×U(k)) n =(Cnk//Tn)//U(k)= CPk−1//U(k). iY=1 This last space is exactly the moduli space of n points in CPk. The group U(k) does not act effectively on n CPk−1. The center, consisting of i=1 scalar matrices, acts trivially. WeQquotient this S1 out and find n Gr(k,n)//T = CPk−1//PU(k), iY=1 where T is the (n−1)-dimensional torus used in Theorem 1.2. Putting in the symplectic structure, we make the following statement. Corollary 1.1. Let M be the moduli space of n points in CPk−1, realized as above by a symplectic reduction as follows: M= Cnk//U(k) aI //Tn(µ +a,µ +a,...,µ +a) 1 2 n =G(cid:0) r(k,n) (cid:1)(cid:0) (cid:1) //T(µ ,...,µ ). (ν1,...,ν1,ν2,...,ν2) 1 n WEIGHTVARIETIES 5 where a=(ν −ν ) and I is the identity matrix in u∗(n). Then H∗(M) is 1 2 isomorphic to the ring C[σ (x ,...,x ),σ (x ,...,x ),u ,...,u ] i 1 k i k+1 n 1 n n σ (x ,...,x )−σ (u ,...,u ), u ,∂ ∆(x,u ) i 1 n i 1 n i=1 i v τ (cid:0) P (cid:1) n n forallv,τ suchthat λ < µ forsomek,and∂ ∆(x,u ) i=k+1 v(i) i=k+1 τ(i) v τ is symmetric in x1,.P..,xk and in xkP+1,...,xn. Remark1. 1. TheequivariantChernclassofthetangentbundleTM → M descends to the ordinary (total) Chern class of M//T under symplectic reduction by a torus. We find the Chern class of the tangent bundle to Fl(n) ∼= Oλ. We note that c(CPn) = (1 − x)n+1, where x is the first Chern class of the tautoligical line bundle S over CPn (see [6]). Using an inductive argument on the fibration Fl(n−1) ֒→ Fl(n) → CPn, one can show that in the basis used above, the Chern class of O is (1−x )n(1− λ 1 x )n−1···(1−x )2. Thenthis isalsothetotalChernclassoftheweight 2 n−1 varieties O //T(µ). λ There are two essential facts that come into play in the results pre- sentedhere. ForM asymplecticmanifoldwithaHamiltoniantorusaction, there isarestrictionmapinequivariantcohomologyfromM tothe µ-level set φ−1(µ) of the moment map φ. The (rational) equivariant cohomology of the level set is equal to the regular cohomology of the reduced space M//T(µ) := φ−1(µ)/T. The first theorem is that the resulting map is a surjection. Theorem 1.3 (Kirwan). Let M be a Hamiltonian T space with moment map φ and µ ∈ t∗ a regular value of φ. Then the map induced by restriction to the level set φ−1(µ) κ :H∗(M)−→H∗(M//T(µ)) µ T is a surjection. Secondly, the restriction map in equivariant cohomology induced by the inclusion of the fixed point set MT into M is an injection. Theorem 1.4 (Chang-Skjelbred, Kirwan). LetM beaHamiltonianT space with fixed point set MT. The natural map r∗ :H∗(M)֒→H∗(MT) T T 6 R.F.GOLDIN is an inclusion. Note how different this is from ordinary cohomology. If M has isolated fixed points, for example, H∗(MT) is zero except in degree 0, yet H∗(M) may have cohomology in higher degree. Theorem 1.4 suggests the following definition. Definition 1.2. Let α ∈H∗(M) be an equivariant cohomology class T on a compact Hamiltonian T space with fixed point set MT. Define the support of α to be supp α={C connected component of MT|r∗(α)6=0} C where r∗ :H∗(M)→H∗(C) isthe restrictiontothe equivariantcohomol- C T T ogy of the fixed component C. By Theorem 1.3, the cohomology of the symplectic reduction can be computed as the quotient of the equivariant cohomology H∗(M) by the T kernelofthe Kirwanmapκ. Theorem1.4indicatesthatthekernelmaybe generated by cohomology classes which have certain properties restricted to the fixed point set. This line of reasoningwas exploited by Tolman and Weitsman who described the kernel κ [23]. µ Theorem 1.5 (Tolman-Weitsman). Let M be a compact symplectic manifold with a Hamiltonian T action. Let φ:M −→t∗ be a moment map such that µ is a regular value. Define Mµ :={m∈M|hφ(m),ξi≤hµ,ξi)} ξ and K :={α∈H∗(M)|supp α⊂Mµ}. ξ ξ Then the kernel of the natural map κ : H∗(M) −→ H∗(M//T(µ)) is the µ T ideal hKi generated by K := K . ξ [ξ The classes in the kernel are generated by classes which have non-zero restriction to fixed points which (under the moment map) lie entirely to one side of a hyperplane ξ⊥ through µ in t∗. Because M is compact, only µ a finite number of hyperplanes will be necessary. The contribution of this article is the application of the work of Tolman and Weitsman to the case where M is a coadjoint orbit of SU(n). Here there is an explicit description of a generating set of classes for the ideal WEIGHTVARIETIES 7 hKi, which allows one to actually compute the cohomology ring of weight varieties. These classes are represented by double Schubert polynomials, permuted by the Weyl group and satisfying certain properties. Double SchubertpolynomialswerefirstintroducedbyLascouxandSchu¨tzenberger [21], [22]. As a corollary,for SU(n) coadjointorbits, the only vectorsξ ∈t needed for Theorem 1.5 are fundamental weights and their permutations (Corollary4.1). Equivalently,thekerneloftheKirwanmapκ isgenerated µ by classes which restrict to zero on one side of hyperplanes ξ⊥ := ann(ξ), translated to contain µ, parallel to codimension-one walls of the moment polytope. 2. THE T-EQUIVARIANT COHOMOLOGY OF FLAG MANIFOLDS AND GRASSMANNIANS There are two descriptions of the equivariant cohomology of the flag manifold thatwe willuse here. Firstis apresentationofthe ringas aquo- tient of a polynomial ring by relations. This is due in the nonequivariant setting to Borel [4] and can be found in [6]. The equivariant version can be obtained from the standard computation by using the Borel construction: By definition, H∗(Fl(Cn)) := H∗(Fl(Cn)× ET) where ET is a T T universal T bundle, and note that Fl(Cn)× ET is a flag bundle over the T classifying space BT := ET/T. Using standard methods, one computes the cohomology of this bundle. Details can be found in [11], [7]. TheseconddescriptionisduetoArabia[1],andsaysthatthereisabasis for H∗(G/T)asa module overH∗ :=H∗(pt)whichhas certainproperties T T T thatweexpanduponbelow. Theconnectionbetweenthesetwopicturesin the case that G=SU(n) is providedby double Schubert polynomials. We show by Theorem2.3 and Theorem2.5 that double Schubert polynomials, and their corresponding classes under a quotient map, have the properties specified by Arabia. Hence the linear basis can be expressed in the presentation due to Bernstein, Gelfand, and Gelfand. 2.1. Presentations of H∗(Fl(Cn)) and H∗(Gr(k,n)) T T TheTn actiononFl(Cn)whichisinducedbytheactiononCn ofweight 1 on each of n copies of C is not effective: a diagonal S1 ⊂ Tn fixes Fl(Cn). We quotient by this circle and let T be the (n−1)-torus which acts effectively. Theorem 2.1. LetT bethemaximal(n−1)-dimensionaltorusinSU(n). The T-equivariant cohomology of the generic SU(n)-coadjoint orbit is C[x ,...,x ,u ,...,u ] H∗(Fl(Cn))= 1 n 1 n , i=1,...,n T n (1+u )− n (1+x ), n u i=1 i i=1 i i=1 i (cid:0)Q Q P (cid:1) 8 R.F.GOLDIN where degu =degx =2, and the u are the image of the classes from the i i i module structure H∗ →H∗(Fl(Cn)). T T It is straightfoward to see that H∗(Fl(Cn)) = H∗(Fl(E)), the flags of T the bundle E :=Cn× ET →BT for a certain action of T on Cn. Fl(E) T can be canonically realized as a tower of projective bundles over BT Fl(E)=P(Qn−π1n)−1 //... π3 // P(Q1) π2 //P(E) π1 // BT where Q := π∗E/S for S the tautological line bundle over P(E). Thus 1 1 1 1 π∗E = Q ⊕ S and the bundle Q → P(E) can to be projectivized 1 1 1 1 to form π : P(Q ) → P(E). We repeat this process to obtain Q := 2 1 k π∗...π∗E/S ⊕ ··· ⊕ S where S is the tautological line bundle over k 1 1 k k P(Q ). The pullback of E to Fl(E) canonically splits into a sum of k−1 line bundles π∗E =S ⊕···⊕S −−−−→ E 1 n    π  Fly(E) −−−−→ ByT By definition, x =c (S ). It follows that i 1 i n H∗(Fl(E))=H∗(BT)[x ,...,x ]/ (1+x )=c(E). T 1 n i iY=1 Using the splitting principle one obtains that the total Chern class of E is n (1+u ), where u is the first Chern class of the tautological line i=1 i i buQndle over the ith copy of S1 in T under a choice of decomposition T = S1 ×···×S1. Then restricting to the n−1 dimensional torus that acts effectively on Fl(E), we get the desired result. Theorem 2.2. LetT bethemaximal(n−1)-dimensionaltorusinSU(n). The T-equivariantcohomology of theGrassmannian of complex k-planes in Cn is a subring of H∗(Fl(Cn)) and can be written T C[σ (x ,...,x ),σ (x ,...,x ),u ,...,u ] H∗(Gr(k,n))= i 1 k i k+1 n 1 n , T σ (x ,...,x )−σ (u ,...,u ), n u i 1 n i 1 n i=1 i (cid:0) P (cid:1) where σ is the ith symmetric function in the indicated variables, i = i 1,...,n. The Chern classes x and u are the same as those in Theorem i i 2.1. WEIGHTVARIETIES 9 We nowexplorethe relationshipbetweenthisquotientdescriptionofthe equivariantcohomology,andthedescriptionasasubringoftheequivariant cohomology of the fixed point set. 2.2. The Restriction of Equivariant Cohomology to Fixed Points By Theorem 1.4, the T-equivariant cohomology of Fl(Cn) can be em- bedded in a direct sum of polynomial rings. We calculate this restriction explicitly using geometric means. A purely algebraic proof can be found in [7]. In [3] Billey finds a simple formula for the restriction of Kostant polynomials (generalized double Schubert polynomials) to the fixed point set. ThefixedpointsetFl(Cn)T isindexedbytheWeylgroupW =N(T)/T. TheT actiononV =CnsplitsV intoasumof1-dimensionalvectorspaces, or lines which we order and call l ,...,l . The fixed points of T on Fl(V) 1 n aretheflagswhichcanbewrittenhl i⊂hl ,l i⊂···⊂hl ,...,l i=V. i1 i1 i2 i1 in We call hl i ⊂ hl ,l i ⊂ ··· ⊂ hl ,...,l i = V the base flag and label the 1 1 2 1 n fixed points by the corresponding permutation in W: p :=hl i⊂hl ,l i⊂···⊂hl ,...,l i. w w(1) w(1) w(2) w(1) w(n) The restrictionmapfromthe descriptioninTheorem2.1tothefixedpoint set is as follows: Theorem 2.3. Letp ∈Fl(V)beinthefixedpointsetFl(V)T asabove. w The inclusion r :p →Fl(V) induces a restriction w w r∗ :H∗(Fl(V))−→H∗(p )=S(t∗)=C[u ,...,u ] (2) w T T w 1 n such that r∗ : x 7→ u and r∗ : u 7→ u , where x and u , i = 1,...,n w i w(i) w i i i i are the generators of the equivariant cohomology in Theorem (2.1). In particular, r:Fl(V)T ֒→Fl(V) induces a map r∗ :H∗(Fl(V))−→H∗(Fl(V)T)= C[u ,...,u ] T T 1 n pM∈W whose further restriction to each component in the direct sum is r∗. w Proof. The classes u come fromthe module structure of H∗(M) over i T H∗. For any p ֒→ M, the induced map H∗(M) → H∗(p) is a module T T T map, and hence r∗u = u for all w,i. A fixed point p ∈ M in the Borel w i i construction corresponds to a fixed copy of BT ∼= p×T ET in M ×T ET. We can restrictthe x to the fixed points of Fl(V), andthen use the Borel i construction to make the restriction equivariant. 10 R.F.GOLDIN We first compute r∗(x ) for any w ∈ W, and proceed inductively to w 1 obtainr∗(x )foralli. LetE :=V × Cn andnotethatFl(E)=Fl(V)× w i T T ET. The class x = c (S ) ∈ H∗(Fl(E)) by definition. The equivariant i 1 i restriction to p is the usual restriction to p × ET in Fl(E). S is the w w T 1 (pullback of) the tautoligical line bundle over P(E). Under the projection toP(E),p × ET projectstol × ET,orasections ofP(E)→BT. w T w(1) T w Then r∗(x )=c (S )| . w 1 1 1 lw(1)×TET As a line bundle, l × ET =s∗S from the commutative diagram w(1) T w 1 s∗S −−−−→ S w 1 1   ByT −−−sw−→ P(yE). As E splits into E =⊕n L , i=1 i with Li = li ×Ti ETi and Ti ∼= S1 acts on li with weight one and on lj trivially for j 6=i, we have s∗S =l × ET =L w 1 w(1) T w(1) so that r∗(x )=c (L ) w 1 1 w(1) by naturalityofthe pullback map. We note that by definition, u =c (L ) i 1 i and hence r∗(x )=c (S )| =c (L )=u . w 1 1 1 lw(1)×TET 1 w(1) w(1) Wecontinueinductively: thepointp alsospecifiesatwo-dimensionalfixed w subspace hl ,l i of V containing l . Under the Borel contruction, w(1) w(2) w(1) this two-space is a section in the projectivization P(Q ) of the quotient 1 Q = π∗E/S , or the restriction of its tautological line bundle S to the 1 1 1 2 copy of BT that is the image of this section. Then r∗(x )=c (S )| =u . w 2 1 2 lw(2)×TET w(2) Itisclearhowtoproceedfromhere. Foreachx ,thereisaprojectionfrom i p to the corresponding i-space, which can (after crossing with ET and w modding out by T) be realized as a line in S . We obtain i r∗(x )=c (S )| =u w i 1 i lw(i)×TET w(i)