A generalization of Cachazo-Douglas-Seiberg-Witten conjecture for symmetric spaces
aa r X i v : . [ m a t h . R T ] M a y A Generalization ofCachazo-Douglas-Seiberg-Witten Conjecturefor Symmetric Spaces
Shrawan KumarDepartment of MathematicsUniversity of North CarolinaChapel Hill, NC 27599–3250November 10, 2018
Let g be a (finite-dimensional) semisimple Lie algebra over the complex num-bers C and let σ be an involution (i.e., an automorphism of order 2) of g . Let k (resp. p ) be the +1 (resp. −
1) eigenspace of σ . Then, k is a Lie subalgebraof g and p is a k -module under the adjoint action. In this paper we onlyconsider those involutions σ such that p is an irreducible k -module.We fix a g -invariant nondegenerate symmetric bilinear form h , i on g .Then, the decomposition g = k ⊕ p is an orthogonal decomposition.Let R := ∧ ( p ⊕ p ) be the exterior algebra on two copies of p . To dis-tinguish, we denote the first copy of p by p and the second copy by p . Itis bigraded by declaring p (resp. p ) to have bidegree (1,0) (resp. (0,1)).Choose any basis { e i } of p and let { f i } be the dual basis of p , i.e., h e i , f j i = δ i,j . Define a k -module map (under the adjoint action) c : k → p ⊗ p , c ( x ) = X i [ x, e i ] ⊗ f i .
1t is easy to see that c does not depend upon the choice of the basis { e i } .Projected onto ∧ ( p ), we get a k -module map k → ∧ ( p ). This map is denotedby c considered as a map k → ∧ ( p ), and similarly for c : k → ∧ ( p ). Wedenote the image of c i by C i . Let J be the (bigraded) ideal of R generatedby C ⊕ C ⊕ C and let us consider the quotient algebra A := R/J.
The algebra A is a k -algebra (induced from the adjoint action of k ) andlet A k be the subalgebra of k -invariants. The algebra A k contains the element S := P e i ⊗ f i in bidegree (1,1). The aim of this paper is to understand the structure of the algebra A k . In the case when g = s ⊕ s for a simple Lie algebra s and σ is theinvolution which switches the two factors, the study of the structure of A k was initiated by Cachazo-Douglas-Seiberg-Witten who made the followingconjecture. (Observe that in this case k and p both can be identified with s and the adjoint action of k on p under this identification is nothing but theadjoint action of s on itself.) We will refer to this as the diagonal case . (1.1). Conjecture [CDSW] (i) The subalgebra A k of k -invariants in A isgenerated, as an algebra, by the element S .(ii) S h = 0 .(iii) S h − = 0 ,where h is the dual Coxeter number of k = s . They proved the conjecture for s = sl N in [CDSW], and Witten provedit for s = sp N in [W]. He also proved parts (i) and (ii) of the conjecture for s = so N in [W]. Subsequently, Etingof-Kac proved the conjecture for s oftype G by using the theory of abelian ideals. Kumar proved part (i) of theconjecture uniformly in [K3] using geometric and topological methods.Returning to the general case of any involution σ , we prove the followinganalogous result (cf. Theorem 4.8) which is the main result of this paper. (1.2). Theorem Let σ be any involution of a simple Lie algebra g such that p is an irreducible module under the adjoint action of k . Then, the subalgebra A k of k -invariants in A is generated, as an algebra, by the element S . Analogous to our proof in the diagonal case, we need to consider thealgebra B := R/ h C ⊕ C i . We show (cf. Theorem 3.1) that the subalgebra B k of k -invariants of B is graded isomorphic with the singular cohomology2ith complex coefficients H ∗ ( Y ) of a certain finite-dimensional projectivesubvariety Y of the twisted affine Grassmannian X σ (cf. Section 2 for thedefinitions of X σ and Y ). The definition of the subvariety Y is motivated fromthe theory of abelian subspaces of p . The main ingredients in our proof ofTheorem 3.1 are: result of Garland-Lepowsky on the Lie algebra cohomologyof the nil-radical ˆ u σ of a maximal parabolic subalgebra of twisted affine Kac-Moody Lie algebras; the ‘diagonal’ cohomology of ˆ u σ introduced by Kostant;certain results of Han and Cellini-Frajria-Papi on abelian subspaces of p and a certain deformation of the singular cohomology of X σ introduced byBelkale-Kumar.Having identified the algebra B k with H ∗ ( Y ), we next use the fact that H ∗ ( X σ ) surjects onto H ∗ ( Y ) under the restriction map. Section 4 is devotedto study the cohomology algebra H ∗ ( X σ ). The results here are more involvedthan in the diagonal case. One mazor difficulty arises from the fact that thefibration Ω σ ( G o ) → Ω σ ( G o ) /K o γ −→ G o /K o , is nontrivial (cf. Section 4 for various notation). To complete the proof ofour Theorem 4.8, we show that all but one of the generators of H ∗ ( X σ ) goto zero under the canonical projection map B k → A k and the remaining onegenerator goes to S .Finally, analogous to the Cachazo-Douglas-Seiberg-Witten Conjecture,we make the following conjecture. (1.3). Conjecture S h = 0 and S h − = 0 in A k , where h = h g − h k ( h g beingthe dual Coxeter number of g ). Unless otherwise stated, by the cohomology H ∗ ( X ) of a topological space X we mean the singular cohomology H ∗ ( X, C ) with complex coefficients. Acknowledgements.
It is my pleasure to thank Weiqiang Wang for askingthe question answered in this paper. I also thank Pierluigi Frajria and PaoloPapi for some helpful correspondences. In particular, the proof of Lemma2.4 is due to them. This work was partially supported by the FRG grant noDMS-0554247 from NSF. 3
Preliminaries and Notation (2.1) Twisted affine Lie algebras.
Let g be a (finite-dimensional) simpleLie algebra over C and let σ be an involution of g . Let k ⊂ g be the +1eigenspace of σ (which is a reductive subalgebra of g ) and let p be the − σ , which is a k -module under the adjoint action. As in theintroduction, we only consider those involutions σ such that p is an irreducible k -module. This will be our tacit assumption on σ throughout the paper. Fix a Cartan subalgebra h σ and a Borel subalgebra b σ ⊃ h σ of k . Let n σ be the nil-radical of b σ . Associated to the pair ( g , σ ) we have the twistedaffine Kac-Moody Lie algebra ˆ g σ := X i ∈ Z g i ⊗ t i ⊕ C c ⊕ C d, where g i := k and g i +1 := p for any i ∈ Z . The bracket in ˆ g σ is defined asfollows: (cid:2) x ⊗ t m + λc + µd, x ′ ⊗ t m ′ + λ ′ c + µ ′ d (cid:3) = (cid:0) [ x, x ′ ] ⊗ t m + m ′ + µm ′ x ′ ⊗ t m ′ − µ ′ mx ⊗ t m (cid:1) + m δ m, − m ′ h x, x ′ i c, where h , i is the normalized g -invariant bilinear form on g as in the intro-duction.The Lie algebra ˆ g σ is a subalgebra of the affine Kac-Moody algebraˆ g := X i ∈ Z g ⊗ t i ⊕ C c ⊕ C d with the bracket defined by the same formula as above.We define the following subalgebras of ˆ g σ called the standard Cartan , standard Borel and the standard maximal parabolic subalgebra respectively:ˆ h σ := h σ ⊗ t ⊕ C c ⊕ C d, ˆ b σ := b σ ⊗ t ⊕ X i> g i ⊗ t i ⊕ C c ⊕ C d, andˆ p σ := X i ≥ g i ⊗ t i ⊕ C c ⊕ C d.
4e also have the nil-radicals ˆ n σ of ˆ b σ and ˆ u σ of ˆ p σ and the Levi subalgebraˆ r σ of ˆ p σ defined as follows:ˆ n σ := n σ ⊗ t ⊕ X i> g i ⊗ t i , ˆ u σ := X i> g i ⊗ t i , andˆ r σ := k ⊗ t ⊕ C c ⊕ C d. The evaluation at 1 gives rise to a Lie algebra homomorphism ev : ˆ g σ → g ⊕ C c ⊕ C d, where c and d are central in the right side.Associated to the twisted affine Kac-Moody Lie algebra ˆ g σ and its sub-algebras ˆ p σ and ˆ b σ , we have the twisted affine Kac-Moody group G σ , thestandard maximal parabolic subgroup P σ and the standard Borel subgroup B σ respectively (cf. [K2, Chapter 6]).Let W σ be the (finite) Weyl group of ( k , h σ ) and let W σ be the (affine)Weyl group of (cid:0) ˆ g σ , ˆ h σ (cid:1) . Let ˆ∆ + σ ⊂ (ˆ h σ ) ∗ be the set of positive roots of ˆ g σ ,i.e., the set of roots for the subalgebra ˆ n σ with respect to the adjoint actionof ˆ h σ . We set ˆ∆ − σ = − ˆ∆ + σ . For any w ∈ W σ , defineΦ( w ) := ˆ∆ + σ ∩ w ˆ∆ − σ , andˆ n σ ( w ) := M α ∈ Φ( w ) (cid:0) ˆ g σ (cid:1) α , where (cid:0) ˆ g σ (cid:1) α denotes the root space of ˆ g σ corresponding to the root α . Sinceeach root in Φ( w ) is real, (cid:0) ˆ g σ (cid:1) α is one-dimensional for each α ∈ Φ( w ). (2.2) Abelian subspaces of p . Let W ′ σ ⊂ W σ be the set of minimal cosetrepresentatives in the cosets W σ /W σ .Following [CFP], we call an element w ∈ W σ minuscule ifˆ n σ ( w − ) ⊂ p ⊗ t. Let us denote the set of minuscule elements in W σ by W minu σ . Then, it iseasy to see that W minu σ ⊂ W ′ σ and, clearly, it is a finite set.We recall the following result from [CFP, Theorem 3.1].5 There is a bijection between W minu σ and the set Ξ of b σ -stable abelian subspaces of p given by w ev (ˆ n σ ( w − )) . In particular, thecardinality | W minu σ | = | Ξ | . We recall the Bruhat decomposition (cf. [K2, Corollary 6.1.20]) of theprojective ind-variety X σ := G σ / P σ = G w ∈ W ′ σ B σ w P σ / P σ , where the Bruhat cell C ( w ) := B σ w P σ / P σ is isomorphic to the affine space C ℓ ( w ) ( ℓ ( w ) being the length of w in the Coxeter group W σ ). Moreover, forany w ∈ W ′ σ , the Zariski closure C ( w ) = G v ∈ W ′ σ and v ≤ w C ( v ) . Define a subset Y of G σ / P σ by Y = G w ∈ W minu σ C ( w ) . Then, Y is a (finite-dimensional) projective subvariety of G σ / P σ . Thisfollows from the following. (2.4) Lemma. For w ∈ W minu σ and any u ∈ W ′ σ such that u ≤ w , we have u ∈ W minu σ .Proof (due to P. Frajria and P. Papi) . By the definition, an element u ∈ W σ is minuscule iff β ( d ) = 1 for all β ∈ Φ( u − ). By the L -shellability of theBruhat order in W ′ σ , we can assume that w = us α , where α ∈ ˆ∆ + σ is a realroot and s α is the reflection through α : s α λ = λ − h λ, α ∨ i α for λ ∈ (ˆ h σ ) ∗ .Since u < w , we have wα ∈ ˆ∆ − σ , and hence α ∈ Φ( w − ). In particular, α ( d ) = 1. Since u ∈ W ′ σ , we have β ( d ) = 0 for any β ∈ Φ( u − ). Thus, itsuffices to prove that for any β ∈ ˆ∆ + σ such that β ( d ) >
1, we have uβ ∈ ˆ∆ + σ .Observe that since β ( d ) > wβ ∈ ˆ∆ + σ .There are three cases to consider: Case I : s α β ∈ ˆ∆ − σ .In this case, h β, α ∨ i >
0. Thus, uβ = w ( s α β ) = w ( β − h β, α ∨ i α ) = wβ − h β, α ∨ i wα ∈ ˆ∆ + σ , since wα ∈ ˆ∆ − σ . 6 ase II : s α β ∈ ˆ∆ + σ and s α β ( d ) = 1.In this case, s α β / ∈ Φ( w − ), i.e., uβ = ws α β ∈ ˆ∆ + σ . Case III : s α β ∈ ˆ∆ + σ and s α β ( d ) = 1.In this case, s α β ( d ) = β ( d ) − h β, α ∨ i α ( d )= β ( d ) − h β, α ∨ i = 1 , since α ( d ) = 1 . Thus, h β, α ∨ i = β ( d ) − > β ( d ) >
1) and hence uβ = ws α β = wβ − h β, α ∨ i ( wα ) ∈ ˆ∆ + σ , since wα ∈ ˆ∆ − σ . This proves the lemma. B k Consider the Z + -graded k -algebra B := ∧ ( p ) ⊗ ∧ ( p ) h C ⊕ C i , where C and C are defined in the Introduction.Following is the first main result of this paper. (3.1) Theorem. The singular cohomology H ∗ ( Y , C ) of Y with complex co-efficients is isomorphic as a Z + -graded algebra with the graded algebra of k -invariants B k . Before we come to the proof of the theorem, we need to recall the follow-ing results. The first theorem is a special case of a result due to Garland-Lepowsky and the second theorem is due to Han. (3.2) Theorem. [K2, Theorem 3.2.7 and Identity (3.2.11.3)]
As a modulefor ˆ r σ , H p (ˆ u σ , C ) ≃ M w ∈ W ′ σ ℓ ( w )= p L ( w − ˆ ρ − ˆ ρ ) , where ˆ ρ is any element of (cid:0) ˆ h σ (cid:1) ∗ satisfying ˆ ρ ( α ∨ i ) = 1 for all the simple coroots { α ∨ , . . . , α ∨ ℓ } ⊂ ˆ h σ of ˆ g σ and L ( w − ˆ ρ − ˆ ρ ) denotes the irreducible ˆ r σ -modulewith highest weight w − ˆ ρ − ˆ ρ . Similarly, by [K2, Theorem 3.2.7], H p (ˆ u − σ , C ) ≃ M w ∈ W ′ σ ,ℓ ( w )= p L ( w − ˆ ρ − ˆ ρ ) ∗ , here ˆ u − σ := P i< g i ⊗ t i . For any b σ -stable abelian subspace I ⊂ p of dimension n , ∧ n ( I ) is a b σ -stable line in ∧ n ( p ) and hence generates an irreducible k -submodule V I of ∧ n ( p ) with highest weight space ∧ n ( I ). Thus, we get a k -module map M I ∈ Ξ V I → ∧ ( p ) → ∧ ( p ) / h C i . If I corresponds via Theorem 2.3 to the element w ∈ W minu σ , then V I hashighest weight ( w − ˆ ρ − ˆ ρ ) | h σ . (3.3) Theorem. [H, Theorem 4.7] The above k -module map M I ∈ Ξ V I → ∧ ( p ) / h C i is an isomorphism. Moreover, by [P, Theorem 4.13(2)], the k -module L I ∈ Ξ V I is multiplicity free. For any w ∈ W ′ σ , define the Schubert cohomology class ε w ∈ H ℓ ( w ) ( X σ , Z )by ε w (cid:0) [ C ( u )] (cid:1) = δ w,u for u ∈ W ′ σ , where [ C ( u )] ∈ H ℓ ( u ) ( X σ , Z ) denotes the fundamental homology class of C ( u ).Following Belkale-Kumar [BK, § ⊙ in H ∗ ( X σ , Z )as follows. Express the standard cup product ε u · ε v = X w ∈ W ′ σ c wu,v ε w . Now, define ε u ⊙ ε v = X c wu,v δ d wu,v , ε w , where d wu,v := (cid:0) u − ˆ ρ + v − ˆ ρ − w − ˆ ρ − ˆ ρ (cid:1) ( d ) . The product ⊙ descends to a product in H ∗ ( Y , Z ) under the restrictionmap H ∗ ( X σ , Z ) → H ∗ ( Y , Z ). (3.4) Lemma. The product ⊙ coincides with the standard cup product in H ∗ ( Y , Z ) . roof. For any w ∈ W σ , by [K2, Corollary 1.3.22], | Φ( w ) | = ˆ ρ − w ˆ ρ, where | Φ( w ) | := X β ∈ Φ( w ) β. Thus, for any w ∈ W minu σ , by its definition(1) ( ˆ ρ − w − ˆ ρ )( d ) = ℓ ( w ) . To prove the lemma, it suffices to show that whenever c wu,v = 0 for u, v, w ∈ W minu σ , d wu,v = 0. But, c wu,v = 0 gives(2) ℓ ( w ) = ℓ ( u ) + ℓ ( v ) . Thus, d wu,v = (cid:16) u − ˆ ρ − ˆ ρ + v − ˆ ρ − ˆ ρ − (cid:0) w − ˆ ρ − ˆ ρ (cid:1)(cid:17) ( d )= − ℓ ( u ) − ℓ ( v ) + ℓ ( w )= 0 by (2) . Proof of Theorem 3.1.
The cohomology modules H p (ˆ u σ ) and H p (ˆ u − σ ) acquirea grading coming from the total degree of t in ∧ p (ˆ u σ ) and ∧ p (ˆ u − σ ) respectively.This decomposes H p (ˆ u σ ) = M m ∈ Z + H p ( − m ) (ˆ u σ ) , where H p ( − m ) (ˆ u σ ) denotes the space of elements of H p (ˆ u σ ) of total t -degree − m . Define the diagonal cohomology H ∗ D (ˆ u σ ) := M p ∈ Z + H p ( − p ) (ˆ u σ ) , which is a subalgebra of H ∗ (ˆ u σ ), and similarly define H ∗ D (ˆ u − σ ).Let φ : ∧ p ( p ) → H p ( − p ) (ˆ u σ ) be the map induced from the map ¯ φ : ∧ p ( p ) → C p ( − p ) (ˆ u σ ), ¯ φ (cid:0) x ∧ · · · ∧ x p (cid:1)(cid:0) y ⊗ t ∧ · · · ∧ y p ⊗ t (cid:1) = det (cid:0) h x i , y j i (cid:1) i,j , x i , y j ∈ p ) by taking the cohomology class of the image. Clearly, ¯ φ ( x ∧· · · ∧ x p ) is a cocycle and, moreover, ¯ φ (and hence φ ) is surjective. It is easyto see that Ker (cid:0) φ | ∧ p ) (cid:1) = C . Now, take any ω ∈ C p − − p ) (ˆ u σ ), where C p − − p ) (ˆ u σ )denotes the space of ( p − u σ with total t -degree − p . We canwrite ω = N X i =1 ω i ∧ ω i , for some ω i ∈ C − (ˆ u σ ) and ω i ∈ C p − − p +2) (ˆ u σ ). Then, δω = N X i =1 ( δω i ) ∧ ω i , since ω i are δ -closed, where δ is the standard differential of the cochaincomplex C ∗ (ˆ u σ ).From this it is easy to see that Ker φ = h C i . Thus, we get a gradedalgebra isomorphism commuting with the k -module structures:(1) ∧ ∗ ( p ) h C i ≃ H ∗ D (ˆ u σ ) . In exactly the same way, we get an isomorphism of graded algebras commut-ing with the k -module structures:(2) ∧ ∗ ( p ) h C i ≃ H ∗ D (ˆ u − σ ) . In particular, ∧ p ( p ) h C i∩∧ p ( p ) is a self-dual k -module for any p ≥ p, q ≥ (cid:20) ∧ p ( p ) h C i ∩ ∧ p ( p ) ⊗ ∧ q ( p ) h C i ∩ ∧ q ( p ) (cid:21) k ≃ h H pD (ˆ u σ ) ⊗ H qD (ˆ u − σ ) i k . Since ∧ ∗ ( p ) h C i is multiplicity free (by Theorem 3.3) and ∧ p ( p ) h C i∩∧ p ( p ) is self-dual forany p ≥
0, the left side of (3) is nonzero only if p = q . Moreover, c actstrivially on H pD (ˆ u σ ) ⊗ H qD (ˆ u − σ ) and d acts via the multiplication by q − p .Thus, we have a graded algebra isomorphism:(4) (cid:20) ∧ ∗ ( p ) h C i ⊗ ∧ ∗ ( p ) h C i (cid:21) k ∼ −→ h H ∗ D (ˆ u σ ) ⊗ H ∗ D (ˆ u − σ ) i ˆ r σ .
10y Theorem 3.2, we get(5) H pD (ˆ u σ ) ≃ H pD (ˆ u − σ ) ∗ ≃ M w ∈ W minu σ ℓ ( w )= p L (cid:0) w − ˆ ρ − ˆ ρ (cid:1) , as ˆ r σ -modules. Combining (4)–(5), we get the isomorphism(6) (cid:20) ∧ ∗ ( p ) h C i ⊗ ∧ ∗ ( p ) h C i (cid:21) k ≃ M w ∈ W minu σ h L (cid:0) w − ˆ ρ − ˆ ρ (cid:1) ⊗ L (cid:0) w − ˆ ρ − ˆ ρ (cid:1) ∗ i ˆ r σ . Now, by a similar argument to that given in [K3, Section 2.4], the proofof Theorem 3.1 follows. We omit the details. A k Let G be a connected, simply-connected complex algebraic group with Liealgebra g . The involution σ of g , of course, induces an involution of G .Choose a maximal compact subgroup G o of G which is stable under σ andsuch that the subgroup K o := G σo of σ -invariants is a maximal compactsubgroup of K := G σ (cf. [He]). Moreover, as is well known, K is connectedand hence so is K o .Let Ω σ ( G o ) be the space of all continuous maps f : S → G o which are σ -equivariant, i.e., f ( − z ) = σ ( f ( z )) for all z ∈ S . We put the compact-open topology on Ω σ ( G o ). Clearly, the subspace ofconstant loops can be identified with K o . Equivalently, we can view Ω σ ( G o )as the space of continuous maps ¯ f : [0 , π ] → G o such that¯ f ( t + π ) = σ ( ¯ f ( t )) , for all 0 ≤ t ≤ π. In particular, ¯ f (2 π ) = σ ( ¯ f (0)) = ¯ f (0). The correspondence f ¯ f isgiven by ¯ f ( t ) = f ( e it ), for 0 ≤ t ≤ π .Consider the fibrationΩ σ ( G o ) → Ω σ ( G o ) /K o γ −→ G o /K o , γ ( f K o ) = f (1) K o for f ∈ Ω σ ( G o ) and Ω σ ( G o ) is the subspace ofΩ σ ( G o ) consisting of those f such that f (1) = 1.Of course, Ω σ ( G o ) can be identified with the based loop space Ω ( G o ) of G o under f ¯ f | [0 ,π ] .Define the k -module map ¯ c : k ∗ → ∧ ( p ) ∗ by (¯ cf )( x ∧ y ) = f ([ x, y ]), for x, y ∈ p . This gives rise to a map (still denoted by)¯ c : S ( k ∗ ) → ∧ ( p ) ∗ . Consider the restriction of ¯ c to the subring of k -invariants c : S ( k ∗ ) k → C ( g , k ) ≃ [ ∧ ( p ) ∗ ] k . Then, the map c is the Chern-Weil homomorphism with respect to a G o -invariant connection on the G o -equivariant principle K o -bundle G o → G o /K o .Observe that since k is the +1 eigenspace of an involution of g , the dif-ferential δ ≡ C ∗ ( g , k ). Thus, C ∗ ( g , k ) ≃ H ∗ ( g , k ) ≃ H ∗ ( G o /K o ) . Thus, in our case, we can think of c as the map c : S ( k ∗ ) k → H ∗ ( g , k ) ≃ H ∗ ( G o /K o ).We now recall the following result due to H. Cartan on the cohomologyof G o /K o with complex coefficients (cf. [C, § (4.1) Theorem. There exists a finite-dimensional graded subspace V ⊂ H ∗ ( G o /K o ) concentrated in odd degrees such that, as graded algebras, H ∗ ( G o /K o ) ≃ ∧ ( V ) ⊗ Im c. (4.2) Corollary. Consider the map γ : Ω σ ( G o ) /K o → G o /K o defined earlier(obtained from the evaluation at 1). Then, the induced map in cohomology γ ∗ : H ∗ ( G o /K o ) → H ∗ (Ω σ ( G o ) /K o ) under the identification H ∗ ( G o /K o ) ≃ ∧ ( V ) ⊗ Im c of the above theorem, satisfies γ ∗| V ≡ . In particular,
Im( γ ∗ ) = γ ∗ (Im c ) . roof. This follows immediately from the fact that H ∗ (Ω σ ( G o ) /K o ) is con-centrated in even degrees only and V lies in odd cohomological degrees.Let L σ ( g ) be the twisted loop algebra L i ∈ Z g i ⊗ t i , i.e., L σ ( g ) is the spaceof all algebraic maps f : C ∗ → g satisfying f ( − z ) = σ ( f ( z )) for all z ∈ C ∗ and the Lie algebra structure is obtained by taking the pointwise bracket.This is a subalgebra of the loop algebra L ( g ) := g ⊗ C [ t, t − ] . Let L σ ( g ) be the kernel of the evaluation map L σ ( g ) → g at 1, x ⊗ a ( t ) a (1) x . Similarly, by L σ ( G o ), we mean the set of algebraic maps f : S → G o with f ( − z ) = σ ( f ( z )) for all z ∈ S and f (1) = 1 (where we call a map f : S → G o algebraic if it extends to an algebraic map ¯ f : C ∗ → G ).We recall the following result from [K1, Theorem 1.6]. (4.3) Theorem. Appropriately defined, the integration map defines an al-gebra isomorphism in cohomology H ∗ ( L σ ( g ) , k ) ≃ H ∗ ( X σ ) . Similarly, we have an algebra isomorphism H ∗ ( L σ ( g )) ≃ H ∗ ( L σ ( G o )) , where L σ ( G o ) is endowed with the Hausdorff topology induced from an ind-variety structure. Analogous to the result of Garland-Raghunathan [GR], we have the fol-lowing. (4.4) Theorem.
The inclusion L σ ( G o ) ֒ → Ω σ ( G o ) is a homotopy equiva-lence, where L σ ( G o ) is endowed with the Hausdorff topology as in the previoustheorem and Ω σ ( G o ) is equipped with the compact-open topology.Similarly, the projective ind-variety X σ under the Hausdorff topology ishomotopically equivalent with the space Ω σ ( G o ) /K o . For any invariant homogeneous polynomial P ∈ S d +1 ( g ∗ ) g of degree d + 1( d ≥ φ P : ∧ d C ( L ( g )) → C φ P ( v ∧ v ∧ · · · ∧ v d − ) = 1 πi Z πθ =0 Φ P ( v ∧ v ∧ · · · ∧ v d − ) , where Φ P : ∧ d C ( L ( g )) → Ω is the map defined byΦ P (cid:0) v ∧ v ∧ · · · ∧ v d − (cid:1) := X µ ∈ S d ε ( µ ) P (cid:16) v µ (0) , (cid:2) v µ (1) , v µ (2) (cid:3) ,. . . , (cid:2) v µ (2 d − , v µ (2 d − (cid:3) , dv µ (2 d − (cid:17) . Here Ω is the space of algebraic 1-forms on C ∗ , d ( x ⊗ a ( t )) = x ⊗ a ′ ( t ) dt (for x ∈ g and a ( t ) ∈ C [ t, t − ]) and in the integral R πθ = o we make the substitution t = e iθ .Let π k : g → k be the projection under the decomposition g = k ⊕ p . Wesimilarly define π p . Define the k -invariant map (for any P ∈ S d +1 ( g ∗ ) g )ˆ φ P : ∧ d C (cid:0) L σ ( g ) / k (cid:1) → C by ˆ φ P (¯ v ∧ · · · ∧ ¯ v d − ) = φ P ( v o ∧ · · · ∧ v o d − ) , where ¯ v i := v i + k ∈ L σ ( g ) / k and v oi := v i − v i (1). Then, ˆ φ P can be viewedas a cochain for the Lie algebra pair ( L σ ( g ) , k ). (4.5) Lemma. Let P ∈ S d +1 ( g ∗ ) g . Then, for the differential δ in the stan-dard cochain complex of the pair ( L σ ( g ) , k ) , δ ˆ φ P descends to a cocycle for theLie algebra pair ( g , k ) under the evaluation map L σ ( g ) → g at 1.Proof. Observe first that, by [FT], the following diagram is commutative upto a nonzero scalar multiple (i.e., d ◦ β P = z − Φ P ∂ , for some z ∈ C ∗ ). ∧ d +1 C ( L ( g )) β P −−−→ Ω y ∂ y d ∧ d C ( L ( g )) −−−→ Φ P Ω , where β P ( v ∧ · · · ∧ v d ) := X µ ∈ S d +1 ε ( µ ) P (cid:16) v µ (0) , (cid:2) v µ (1) , v µ (2) (cid:3) ,. . . , (cid:2) v µ (2 d − , v µ (2 d ) (cid:3)(cid:17) , is the space of algebraic functions on C ∗ , d is the standard deRhamdifferential, and ∂ is the standard differential in the chain complex of the Liealgebra L ( g ). Thus, for v i ∈ L ( g ),( δφ P )( v ∧ v ∧ · · · ∧ v d ) = 1 πi Z πθ =0 Φ P (cid:0) ∂ ( v ∧ v ∧ · · · ∧ v d ) (cid:1) = zπi Z πθ =0 d (cid:0) β P ( v ∧ v ∧ · · · ∧ v d ) (cid:1) = zπi (cid:0) β P ( v ( − ∧ . . . ∧ v d ( − − β P ( v (1) ∧ · · · ∧ v d (1)) (cid:1) . (1)We next show that for any v , . . . , v d ∈ L σ ( g ),(2) ( δ ˆ φ P )(¯ v ∧ · · · ∧ ¯ v d ) = ( δφ P )( v o ∧ · · · ∧ v o d ) , where ¯ v i and v oi are defined above the statement of this lemma. For any x, y ∈ L ( g ),(3) [ x, y ] o − [ x o , y o ] = [ x (1) , y o ] + [ x o , y (1)] . Thus,( δ ˆ φ P )(¯ v ∧ · · · ∧ ¯ v d ) − δφ P ( v o ∧ · · · ∧ v o d )= X i
1) = σ ( v (1)) = 0 . This proves the lemma.By Identity (1) of the above lemma, the restriction ¯ φ P of φ P to ∧ d C ( L σ ( g ))is a cocycle (for the Lie algebra L σ ( g )).As is well known, S ( g ∗ ) g is freely generated by certain homogeneous poly-nomials P , . . . , P ℓ g of degrees m + 1 , m + 1 , . . . , m ℓ g + 1 respectively, where ℓ g is the rank of g and m = 1 < m ≤ · · · ≤ m ℓ g are the exponents of g .The following result is obtained by combining [PS, Proposition 4.11.3]and Theorems 4.3 and 4.4. (4.6) Theorem. The cohomology classes [ ¯ φ P ] , . . . , [ ¯ φ P ℓ g ] ∈ H ∗ ( L σ ( g )) freelygenerate the algebra H ∗ (cid:0) L σ ( g ) (cid:1) ≃ H ∗ (cid:0) L σ ( G o ) (cid:1) ≃ H ∗ (cid:0) Ω σ ( G o ) (cid:1) . Define the differential graded algebra (for short DGA) D = H ∗ ( L σ ( g )) ⊗ C ∗ ( g , k )under the graded tensor product algebra structure. We define the differ-ential d in D as follows: Take d | C ∗ ( g , k ) as the standard differential δ of thecochain complex C ∗ ( g , k ) of the Lie algebra pair ( g , k ) and d ([ ¯ φ P i ]) = δ ˆ φ P i (cf. Lemma 4.5). There is a differential graded algebra homomorphism µ : D → C ∗ ( L σ ( g ) , k ) defined by µ (cid:0) [ ¯ φ P i ] (cid:1) = ˆ φ P i and µ | C ∗ ( g , k ) is the canonical inclusion j : C ∗ ( g , k ) ⊂ C ∗ ( L σ ( g ) , k ) under theevaluation map at 1.Applying the Hirsch lemma to the fibrationΩ σ ( G o ) → Ω σ ( G o ) /K o γ −→ G o /K o , (cf. [DGMS, Lemma 3.1]), and using Theorems 4.3, 4.4 and 4.6, we get thefollowing. 16 The map µ induces a graded algebra isomorphism in coho-mology [ µ ] : H ∗ ( D ) ∼ −→ H ∗ ( X σ ) . In particular, by Corollary 4.2, any cohomology class [ x ] ∈ H ∗ ( X σ ) can berepresented by a cocycle x ∈ C ∗ ( L σ ( g ) , k ) of the form x = X i =( i ,...,i ℓ g ) ∈ Z ℓ g + j (cid:0) c ( Q i ) (cid:1) ( ˆ φ P ) i · · · ( ˆ φ P ℓ g ) i ℓ g , for some Q i ∈ S ( k ∗ ) k , where c : S ( k ∗ ) k → C ( g , k ) is the Chern-Weil homo-morphism defined in the beginning of this section. Finally, we are ready to prove the second main theorem of this paper. (4.8) Theorem.
Let g be a simple Lie algebra and let σ be an involution of g with +1 (resp. − ) eigenspace k (resp. p ). Assume that p is an irreducible k -module. Then, the algebra A k of k -invariants of A is generated (as an algebra)by the element S , where A and S are defined in the Introduction.In particular, ( A k ) p,q = 0 if p = q .Proof. By Theorem 3.1, the algebra B k is graded isomorphic with the singularcohomology H ∗ ( Y ), where B := ∧ ( p ) ⊗∧ ( p ) h C ⊕ C i . Moreover, the inclusion a : Y ⊂ X σ induces a surjection in cohomology, since X σ is obtained from Y by attachingreal even-dimensional cells (by virtue of the Bruhat decomposition). Thus,we have H ∗ ( X σ ) a ∗ ։ H ∗ ( Y ) ξ ∼ −→ B k η −→ A k , where η : B k → A k is the standard quotient map.By Theorem 4.7, any cohomology class [ x ] ∈ H ∗ ( X σ ) can be representedby a cocycle x ∈ C ∗ ( L σ ( g ) , k ) of the form x = X i =( i ,...,i ℓ g ) ∈ Z ℓ g + j (cid:0) c ( Q i ) (cid:1) ( ˆ φ P ) i · · · ( ˆ φ P ℓ g ) i ℓ g , for some Q i ∈ S ( k ∗ ) k .If Q i has constant term 0, from the definition of the Chern-Weil homo-morphism c , it is clear that under the composite map y := η ◦ ξ ◦ a ∗ , j ( c ( Q i ))goes to zero. Further, by an argument similar to the proof of Theorem 2.817n [K3], we see that ˆ φ P i goes to zero under y for any 2 ≤ i ≤ ℓ g . We brieflyrecall the main argument here.For any µ ∈ S d and P ∈ S d +1 ( g ∗ ) g ( d ≥ φ P,µ : ⊗ d C (cid:0) L σ ( g ) / k (cid:1) → C , defined byˆ φ P,µ (cid:16) ¯ v ⊗ ¯ v ⊗ · · · ⊗ ¯ v d − (cid:17) = Z πθ =0 P (cid:16) v oµ (0) , (cid:2) v oµ (1) , v oµ (2) (cid:3) , . . . , (cid:2) v oµ (2 d − , v oµ (2 d − (cid:3) , dv oµ (2 d − (cid:17) , where ¯ v i := v i + k . For the notational convenience, assume µ (1) < µ (2). Forany fixed v , v , . . . , ˆ v µ (1) , . . . , ˆ v µ (2) , . . . , v d − ∈ L σ ( g ) , consider the restriction ¯ φ P,µ of the function ˆ φ P,µ to¯ v × ¯ v × · · · × ⊕ p ∈ Z g p +1 ⊗ t p +1 × · · · × ⊕ p ∈ Z g p +1 ⊗ t p +1 × · · · × ¯ v d − , where the two copies of ⊕ p ∈ Z g p +1 ⊗ t p +1 are placed in the µ (1) and µ (2)-thslots. Then, under the identification g p ⊗ t p ∼ = ( g p ⊗ t p ) ∗ induced from thebilinear form h , i , ¯ φ P,µ = X i,j,m,n f i ( n ) ⊗ f j ( m ) Z πθ =0 P (cid:16) v oµ (0) , (cid:2) e i ( n ) o , e j ( m ) o (cid:3) , (cid:2) v oµ (3) , v oµ (4) (cid:3) , . . . , (cid:2) v oµ (2 d − , v oµ (2 d − (cid:3) , dv oµ (2 d − (cid:17) = X i,j,m,n,k ′ f i ( n ) ⊗ f j ( m ) Z πθ =0 P (cid:0) − , h [ e i , e j ] , e ′ k ′ i F k ′ ( n, m ) , − (cid:1) = X i,j,m,n,k ′ h e i , [ e j , e ′ k ′ ] i f i ( n ) ⊗ f j ( m ) Z πθ =0 P (cid:0) − , F k ′ ( n, m ) , − (cid:1) = X j,k ′ ,m,n [ e j , e ′ k ′ ]( n ) ⊗ f j ( m ) Z πθ =0 P (cid:0) − , F k ′ ( n, m ) , − (cid:1) = − X j,k ′ ,m,n [ e ′ k ′ , e j ]( n ) ⊗ f j ( m ) Z πθ =0 P (cid:0) − , F k ′ ( n, m ) , − (cid:1) , { e i } is a basis of p and { f i } is the dual basis; { e ′ k ′ } is a basis of k and { f ′ k ′ } is the dual basis; m, n run over the odd integersand F k ′ ( n, m ) := f ′ k ′ ( n + m ) − f ′ k ′ ( n ) − f ′ k ′ ( m ) + f ′ k ′ .Thus, only the powers of ˆ φ P contribute to the image of y . This completesthe proof of the theorem. (4.9) Remark. It is likely that for the validity of Theorem 4.8 it is enoughto assume that g is semisimple (not necessarily simple). However, we mustassume that p is k -irreducible under the adjoint action since the second gradecomponent ( A ) k has dimension at least equal to the number of irreduciblecomponents of the k -module p . 19 eferences [BK] P. Belkale and S. Kumar, Eigenvalue problem and a new product incohomology of flag varieties, Inventiones Math. (2006), 185–228.[CDSW] F. Cachazo, M.R. Douglas, N. Seiberg and E. Witten, Chiral ringsand anomalies in supersymmetric gauge theory,
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