An explicit determination of the K -theoretic structure constants of the affine Grassmannian associated to S L 2
aa r X i v : . [ m a t h . K T ] S e p An explicit determination of the K -theoretic structureconstants of the affine Grassmannian associated to SL Let G := d SL denote the affine Kac-Moody group associated to SL and ¯ X the associatedaffine Grassmannian. We determine an inductive formula for the Schubert basis structureconstants in the torus-equivariant Grothendieck group of ¯ X . In the case of ordinary (non-equivariant) K -theory we find an explicit closed form for the structure constants. We alsodetermine an inductive formula for the structure constants in the torus-equivariant cohomologyring, and use this formula to find closed forms for some of the structure constants. Let G := d SL be the affine Kac-Moody group associated to SL , completed along the negative roots.Let P be the standard maximal parabolic subgroup and ¯ X := G/P the thick affine Grassmannian.Let H denote the standard maximal torus, and let T := H/Z ( G ) where Z ( G ) denotes the centerof G . Then the natural left action of H on ¯ X descends to an action of T on ¯ X . Let R ( T ) denotethe representation ring of T , and let W denote the affine Weyl group. K -theory Let K T ( ¯ X ) denote the Grothendieck group of T -equivariant coherent sheaves on ¯ X . Then thestructure sheaves of the opposite (finite codimension) Schubert varieties in ¯ X form a ‘basis’ (whereinfinite sums are allowed) of K T ( ¯ X ). More precisely, we have K T ( ¯ X ) = Y k ∈ Z ≥ R ( T ) ˆ O k , where ˆ O k denotes the class of the structure sheaf of the (unique) opposite Schubert variety ofcodimension k . The structure constants d kn,m ∈ R ( T ) are defined byˆ O n · ˆ O m = X k ≥ n,m d kn,m ˆ O k . (1)Using a Chevalley formula due to Lenart-Shimozono [LeSh, Corollary 3.7], phrased in theLakshmibai-Seshadri path model, we explicitly compute the structure constants d k ,m correspondingto multiplication by the Schubert divisor ˆ O (see (22)). Next, the structure constants d mn,m (where1 ≤ m ) are computed (see (23)) using a result of Lam-Schilling-Shimozono on the localizationsof Schubert varieties [LSS, Proposition 2.10]. Then, using the associative law in the K -group, wederive an inductive formula (Proposition 4.2) for the structure constants using d mn,m and d k ,m asour base cases.Let ˆ ξ k denote the ideal sheaf ‘basis’ (where we allow infinite sums) dual to the basis ˆ O k (see § b kn,m ∈ R ( T ) in the basis ˆ ξ k byˆ ξ n · ˆ ξ m = X k ≥ n,m b kn,m ˆ ξ k . (2)Then we have (see (17)) b kn,m = X j ≤ k (cid:16) d jn,m − d jn +1 ,m − d jn,m +1 + d jn +1 ,m +1 (cid:17) . Thus, in principle, the b kn,m can be computed from the d kn,m . K -theory Let K ( ¯ X ) denote the Grothendieck group of coherent sheaves on ¯ X . Let O k denote the class ofthe structure sheaf of the (unique) opposite Schubert variety of codimension k . Then, we have K ( ¯ X ) = Y k ∈ Z ≥ Z O k . Further, the structure constants in K ( ¯ X ) are given by evaluating the T -equivariant structureconstants at 1. Thus we denote the structure constants in the basis O k in ordinary K -theory by d kn,m (1), so that we have, in K ( ¯ X ), O n · O m = X k ≥ n + m d kn,m (1) O k . Similarly, letting ξ k denote the ideal sheaf ‘basis’ (where we allow infinite sums) dual to the basis O k , we denote the structure constants in the basis ξ k in ordinary K -theory by b kn,m (1), so that wehave, in K ( ¯ X ), ξ n · ξ m = X k ≥ n + m b kn,m (1) ξ k . Then the following theorem, which we prove using our inductive formula, gives a closed formfor the structure constants in ordinary K -theory (see Theorem 5.1 and Corollary 5.2): Theorem 1.1.
The structure constants in ordinary K -theory are given by d n + m + kn,m (1) = ( − k · ( n + m + k − n − m − k ! · n + m + 2 k ( n + k )( m + k ) .b n + m + kn,m (1) = ( − k · ( n + m + k )! n ! m ! k ! . .3 Equivariant cohomology Let G min denote the minimal affine Kac-Moody group associated to SL . Let P be the standardmaximal parabolic subgroup and let X := G min /P denote the standard affine Grassmannian.Then the T -equivariant cohomology ring, H • T ( X ), has a Schubert basis (see [Ku, Theorem11.3.9]), which we denote by { ˆ ε i } ∞ i =0 . Let Z [ α , α ] denote the graded ring of polynomials withintegral coefficients in the simple roots α and α . Further, let Z [ α , α ]( k ) denote the k -th gradedpiece of Z [ α , α ].We define the T -equivariant cohomology structure constants c kn,m ∈ Z [ α , α ]( n + m − k ) (see[Ku, Corollary 11.3.17]) by ˆ ε n · ˆ ε m = n + m X k =max { n,m } c kn,m ˆ ε k . (3)We derive an inductive formula for the structure constants in T -equivarient cohomology (see Propo-sition 6.2). Using this formula, we derive closed forms for c n + mn,m , c n + m − n,m , and c n + m − n,m . The structureconstants c mn,m (where n ≤ m ) are determined by [Ku, Lemma 11.1.10 and Proposition 11.1.11 (1)and (3)]. What follows is a brief summary of the rest of the paper. In section 2 we introduce general notationfor Kac-Moody groups and their flag varieties. In section 3 we summarize the relevant generalitiesabout K -theory. We also state a result due to Lam-Schilling-Shimozono [LSS] on the localizationsof Schubert varieties. Further, we introduce the notion of Lakshmibai-Seshadri path which allowsus to state a Chevalley formula due to Lenart-Shimozono [LeSh]. In section 4 we specialize to thecase of affine SL and determine an explicit closed form for the Chevalley coefficients. Then, wederive an inductive formula for the structure constants in T -equivariant K -theory. In section 5,we use our inductive formula to determine a closed form for the structure constants in ordinary K -theory. Finally in section 6 we move to the case of T -equivariant cohomology, where we derivean inductive formula for the structure constants and determine closed forms for c n + mn,m , c n + m − n,m , c n + m − n,m and when n ≤ m , for c mn,m . Acknowledgements.
The author would like to thank M. Shimozono for providing a Sagepackage (discussed in [LeSh, 1.5]) which the author used to verify some computations related tothe K -theoretic Chevalley formula. We work over the field C of complex numbers. Let G be any symmetrizable Kac-Moody group over C completed along the negative roots (as opposed to completed along the positive roots as in [Ku,Chapter 6]). Further, let G min ⊂ G be the minimal Kac-Moody group as in [Ku, § B bethe standard Borel subgroup, B − the standard opposite Borel subgroup, H := B ∩ B − the standardmaximal torus, T := H/Z ( G min ) the adjoint torus, where Z ( G min ) is the center of G min . Let W denote the Weyl group. Let ¯ X := G/B denote the thick flag variety (introduced by Kashiwara[Ka]), which contains the standard flag variety X = G min /B . When G is infinite dimensional, ¯ X
3s an infinite dimensional non-quasi compact scheme, whereas X is an ind-projective variety [Ku, § H on ¯ X and X descend to actions of T on ¯ X and X .For any w ∈ W we have the Schubert cell C w := BwB/B ⊂ X, the Schubert variety X w := C w = G w ′ ≤ w C w ′ ⊂ X, the opposite Schubert cell C w := B − wB/B ⊂ ¯ X, and the opposite Schubert variety X w := C w = G w ′ ≥ w C w ′ ⊂ ¯ X, all endowed with the reduced subscheme structures. Then, X w is a (finite dimensional) irreducibleprojective subvariety of X and X w is a finite codimensional irreducible subscheme of ¯ X [Ku, § § R ( T ) denote the representation ring of T . For any integral weight λ let C λ denote theone-dimensional representation of T on C given by t · v = λ ( t ) v for t ∈ T, v ∈ C . By extending thisaction to B we may define, for any integral weight λ , the G -equivariant line bundle L ( λ ) on ¯ X by L ( λ ) := G × B C − λ , where for any representation V of B , G × B V := ( G × V ) /B where B acts on G × V via b ( g, v ) =( gb − , bv ) for g ∈ G, v ∈ V, b ∈ B . Then G × B V is the total space of a G -equivariant vector bundleover X , with projection given by ( g, v ) B gB . In this section we introduce the T -equivariant Grothendieck group of ¯ X . We then explain therelationship between the Grothendieck groups of complete and partial flag varieties, and the re-lationship between the T -equivariant and ordinary Grothendieck groups. Next, we state a closedform for the localizations in the Schubert basis due to Lam-Schilling-Shimozono [LSS]. Finally, weintroduce the concept of Lakshmibai-Seshadri path, which allows us to state a Chevalley formuladue to Lenart-Shimozono [LeSh]. Let K T ( ¯ X ) denote the Grothendieck group of T -equivariant coherent O ¯ X -modules. For any u ∈ W , O X u is a coherent O ¯ X -module, by [KaSh, § K T ( pt ) = R ( T ), we have that K T ( ¯ X ) is amodule over R ( T ). Further, from [KaSh, comment after Remark 2.4] we have: Proposition 3.1. { [ O X u ] } forms a ‘basis’ of K T ( ¯ X ) as an R ( T ) -module (where we allow infinitesums), i.e., K T ( ¯ X ) = Y w ∈ W R ( T )[ O X w ] . d wu,v ∈ R ( T ) are defined by[ O X u ] · [ O X v ] = X w ∈ W d wu,v [ O X w ] . (4)Note that for fixed u, v ∈ W , infinitely many of the d wu,v may be nonzero. We also have d wu,v = 0unless w ≥ u, v . G/B and
G/P
Now letting P be any standard parabolic subgroup, we have K T ( G/P ) = Y w ∈ W P R ( T )[ O X wP ] , where W P denotes the set of minimal length representatives of W/W P , and X wP := BwP/P .We define the structure constants d wu,v ( P ) for G/P in the analogous way.[ O X uP ] · [ O X vP ] = X w ∈ W P d wu,v ( P )[ O X wP ] . Let π : G/B → G/P be the standard ( T -equivariant) projection. Then, π is a locally trivialfibration (with fiber the smooth projective variety P/B ) and hence flat (see [Ku, Chapter 7]). Thus,we have π ∗ [ O X wP ] = [ O π − ( X wP ) ] = [ O X w ] . Since π ∗ : K T ( G/P ) → K T ( G/B ) is a ring homomorphism, we have d wu,v = d wu,v ( P )for any u, v, w ∈ W P . Thus we henceforth drop the notation d wu,v ( P ) in favor of the notation d wu,v ,even when working with partial flag varieties. K -theory Let K ( ¯ X ) denote the Grothendieck group of coherent sheaves on ¯ X . Then, we have K ( ¯ X ) = Y w ∈ W Z [ O X w ] , where [ O X w ] denotes the class of O X w in K ( ¯ X ). Further, the map Z ⊗ R ( T ) K T ( ¯ X ) → K ( ¯ X ) , ⊗ [ O X w ] [ O X w ]is an isomorphism, where we view Z as an R ( T )-module via evaluation at 1. Similar results applyto G/P . Hence, we have, in K ( ¯ X ),[ O X u ] · [ O X v ] = X w ∈ W d wu,v (1)[ O X w ] , P , we have, in K ( G/P ),[ O X uP ] · [ O X vP ] = X w ∈ W P d wu,v (1)[ O X wP ] . (5)We also have d wu,v (1) = 0 unless w ≥ u + v . We identify the set { wB } of T -fixed points of ¯ X with the Weyl group W . Given x ∈ W , let i x : { x } ֒ → X denote the inclusion map. Then pullback induces a ring homomorphism i ∗ x : K T ( ¯ X ) → K T ( { x } ) ∼ = R ( T ). For ψ ∈ K T ( ¯ X ) and x ∈ W , the localization of ψ at x is defined as ψ x := i ∗ x ( ψ ) . We are concerned with localizations in the basis [ O X w ]. The following is [LSS, Lemma 2.3]: Lemma 3.2 ([LSS] Lemma 2.3) . [ O X w ] x = 0 unless x ≥ w . Definition 3.3.
For u, v ∈ W , the set { xy : x ≤ u, y ≤ v } has a maximum element, which wedenote by u ∗ v (see [He, Lemma 1.4]). Then s i ∗ s j = s i s j if i = j , while s i ∗ s i = s i .An explicit closed form for the localizations in the Schubert basis is given by [LSS, Proposition2.10] (see also [G, Theorem 3.12] and [W, ] for the same result in the finite case). Theorem 3.4 ([LSS] Proposition 2.10) . Let x ≥ w ∈ W . Fix a reduced decomposition x = s i s i . . . s i m . For ℓ ≤ m , define β ℓ := s i s i . . . s i ℓ − α i ℓ . Then [ O X w ] x = X ( − ℓ ( w ) ( e β j − . . . ( e β jp − , where the summation runs over all ≤ j < · · · < j p ≤ m such that w = s i j ∗ · · · ∗ s i jp . Localizing the equation defining the structure constants (4) at x gives[ O X u ] x · [ O X v ] x = X w ∈ W d wu,v [ O X w ] x . Now if u ≤ v , then since [ O X w ] x = 0 unless x ≥ w , and d wu,v = 0 unless w ≥ u and w ≥ v , letting x = v gives [ O X u ] v · [ O X v ] v = d vu,v [ O X v ] v , which reduces to[ O X u ] v = d vu,v , for u ≤ v. (6) In this subsection we introduce the notion of Lakshmibai-Seshadri paths. We do not attempt to givethis subject a proper treatment, but instead introduce only the notions necessary to understandthe statement of the Chevalley formula in the subsequent subsection.Let S = { s i : i ∈ I } denote the set of simple reflections of W . For any J ⊂ I , define W J to bethe Weyl group generated by the s j where j ∈ J . Let λ be a dominant integral weight. Then itsstabilizer W λ is the parabolic subgroup W J with J = { i ∈ I : s i λ = λ } .6e define the Bruhat ordering on the orbit W λ of λ by taking the transitive closure of therelations s α σ < σ iff h σ, α ∨ i >
0, where α is a positive root and σ ∈ W λ . Note that by thisconvention, for u, v ∈ W λ , we have u < v iff vλ < uλ (where W λ denotes the set of minimal lengthrepresentatives on W/W λ ). For a real number b , we define the b -Bruhat ordering ‘ < b ’ on W λ bydefining µ to cover ν in the b -Bruhat order iff µ covers ν in the normal Bruhat order and b ( µ − ν )is an integer multiple of a root. Definition 3.5 (Lakshmibai-Seshadri path [St]) . A Lakshmibai-Seshadri (LS) path p of shape λ is a pair p = ( σ, b ) where σ : σ > σ > · · · > σ m , σ i ∈ W/W λ b : 0 = b < b < · · · < b m = 1 , b i ∈ Q . We also require that σ λ < b σ λ < b · · · < b m − σ m λ. Denote by T λ the set of all LS paths of shape λ . For p ∈ T λ we define the weight of p = ( σ, b )to be p (1) = σ m λ − m − X i =1 b i ( σ i +1 λ − σ i λ ) . (7) Proposition 3.6 ([LaSe2], Lemma 4.4’) . Let τ ∈ W/W J and w ∈ W be such that wW J ≥ τ in W/W J . Then the set { v ∈ W : w ≥ v, vW J = τ } has a Bruhat-maximum, which will be denoted by dn ( w, τ ) . The symbol dn is an abbreviation for“down”. For p ∈ T λ , we define, with notation as in Definition 3.5,beg( p ) = σ and end( p ) = σ m . Then for w ∈ W such that beg( p ) ≤ wW λ , define dn( w, p ) by w = w ≥ w ≥ · · · ≥ w m = dn( w, p ) , (8)where w i := dn( w i − , σ i ) for i from 1 to m . Here dn( w i − , σ i ) is defined as in Proposition 3.6.For z, w ∈ W we define D λw,z := { p ∈ T λ : beg( p ) ≤ wW λ , dn( w, p ) = z } . (9) For any integral weight λ , define the Chevalley coefficients a wv ( λ ) ∈ R ( T ) by[ L ( λ )] · [ O X v ] = X w ≥ v a wv ( λ )[ O X w ] . (10)Now letting λ be a dominant integral weight, we have the following Chevalley formula from [LeSh](note that L ( λ ) in our notation is written L − λ in the notation of [LeSh]):7 heorem 3.7 ([LeSh], Corollary 3.7) . For any dominant integral weight λ , we have [ L ( λ )] · [ O X v ] = X w ≥ v X p ∈D λw,v ( − ℓ ( w ) − ℓ ( v ) e − p (1) [ O X w ] , where D λw,z is defined in (9) and p (1) is defined in (7). This immediately gives a wv ( λ ) = X p ∈D λw,v ( − ℓ ( w ) − ℓ ( v ) e − p (1) . (11)Further, for any simple reflection s i , we have [ O X si ] = 1 − e Λ i [ L (Λ i )], where Λ i denotes the i -thfundamental weight. Hence, d ws i ,v = ( − e Λ i a wv (Λ i ) w = v − e Λ i a vv (Λ i ) w = v. (12) d SL in T -equivariant K -theory In this section we specialize to the case of d SL . We begin by explicitly determine the Chevalleycoefficients a wv (Λ ) (see (11)) for the affine Grassmannian associated to SL , where Λ is the zerothfundamental weight. We then determine an inductive formula for the structure constants. d SL Let G := d SL be the affine Kac-Moody group associated to SL , completed along the negative roots.Let P be the standard maximal parabolic subgroup and ¯ X := G/P the thick affine Grassmannian.Let T denote the standard maximal torus, R ( T ) its representation ring, and W the affine Weylgroup.Let g := c sl . We denote by h ⊂ g the Cartan subalgebra, and by h ∗ its dual. Then we have thesimple roots α , α ∈ h ∗ , the simple coroots α ∨ , α ∨ ∈ h , the simple reflections s , s ∈ W , and thefundamental weights Λ , Λ ∈ h ∗ . Note that W is isomorphic to the free product h s i ∗ h s i .Let W P denote the set of minimal length representatives of W/W P , where W P denotes the Weylgroup of P . Then W P = { w n } ∞ n =0 where w n := . . . s s s denotes the word of length n in s and s which is alternating and ends in s (so w = e, w = s , w = s s , w = s s s , etc). Further, W/W P is totally ordered under the relative Bruhatordering.We denote by ˆ O k := [ O X wkP ]. Then we have: K T ( ¯ X ) = Y k ∈ Z ≥ R ( T ) ˆ O k . We henceforth denote the structure constants d w k w n ,w m (see (4)) and the Chevalley coefficients a w k w m (Λ ) (see (10)) by d kn,m and a km respectively. Then, in this notation, we have, in K T ( ¯ X ),ˆ O n · ˆ O m = X k ≥ n,m d kn,m ˆ O k . (13)8urther, we have, by (11), a km = X p ∈D Λ0 wk,wm ( − k + m e − p (1) . (14)We also consider the basis dual to the basis ˆ O k , defined as follows. Let ∂X w k P := X w k P \ C w k P be given the reduced subscheme strcture, where C w k P := B − w k P/P . Letˆ ξ k := [ O X wkP ( − ∂X w k P )]denote the class of the ideal sheaf of ∂X w k P in X w k P . Then { ˆ ξ k } ∞ k =0 forms an R ( T )-‘basis’ of K ( ¯ X ),where we allow infinite sums. Further, we haveˆ ξ k = ˆ O k − ˆ O k +1 . (15)We define the structure constants b kn,m ∈ R ( T ) in the basis ˆ ξ k byˆ ξ n · ˆ ξ m = X k ≥ n,m b kn,m ˆ ξ k . (16)Using (15) and looking at the coefficient of ˆ O k in the product ˆ ξ n · ˆ ξ m we obtain b kn,m − b k − n,m = d kn,m − d kn +1 ,m − d kn,m +1 + d kn +1 ,m +1 . Thus by induction, we have the following expression for the b kn,m in terms of the d kn,m b kn,m = X j ≤ k (cid:16) d jn,m − d jn +1 ,m − d jn,m +1 + d jn +1 ,m +1 (cid:17) . (17) a km We begin by determining the Chevalley coefficients a km . By (14), this can be done by determiningthe set D Λ w k ,w m and the weights of the associated paths. Once the a km are known, we obtain d k ,m using (12).The following is a restatement of [Sa, Lemma 1], although we provide a proof as our conventionsare different. Lemma 4.1.
The LS paths of shape Λ are those paths p = ( σ, b ) such that σ : w ℓ > w ℓ − > · · · > w m b : 0 < b ℓ < b ℓ − < · · · < b m +1 < , where ℓ ≥ m and where b j · j ∈ Z for all m + 1 ≤ j ≤ ℓ . roof. It can be easily checked that all such paths are LS paths.To complete the proof, we must show that all LS paths are of this form, i.e. we must show that σ can have no ‘skips’, and that the b j ’s must satisfy the stated condition.One may compute that w i Λ = ( Λ − j α − ( j + j ) α i = 2 j Λ − ( j + 1) α − ( j + j ) α i = 2 j + 1 . (18)It follows that w i − Λ − w i Λ = ( iα i odd iα i even . (19)Recall that by definition, in the b -Bruhat order, µ covers ν iff µ covers ν in the normal Bruhatorder and b ( µ − ν ) is an integer multiple of a root. By (19), for 0 < b <
1, it is not possible that b ( w i Λ − w i +1 Λ ) and b ( w i − Λ − w i Λ ) are both integer multiples of a root. Hence σ can have noskips. The condition on the b j ’s also follows from (19).As noted in [Sa], the condition on the b j ’s can be rephrased b : 0 < i ℓ ℓ < i ℓ − ℓ − < · · · < i m +1 m + 1 < , where i j ∈ Z ≥ for all m + 1 ≤ j ≤ ℓ . These equalities are equivalent to the requirement that1 ≤ i ℓ ≤ i ℓ − ≤ · · · ≤ i m +1 ≤ m .For ℓ ≥ m , let p ( ℓ, m ) denote the set of all LS paths of shape Λ beginning at w ℓ and ending at w m : p ( ℓ, m ) := { p ∈ T Λ : beg( p ) = w ℓ , end( p ) = w m } . For p ∈ p ( ℓ, m ), the weight (7) becomes: p (1) = w m Λ − ℓ X j = m +1 b j ( w j − Λ − w j Λ ) . Thus, by (19), we have p (1) = w m Λ − X j odd ℓ ≥ j ≥ m +1 i j α − X j even ℓ ≥ j ≥ m +1 i j α . (20)Let k ≥ ℓ . Note that W Λ = W s . It is easy to see thatdn( w k , w ℓ W Λ ) = ( w ℓ k = ℓ or ℓ + 1 w ℓ s k ≥ ℓ + 2 , and dn( w k s , w ℓ W Λ ) = w ℓ s , w k , w ℓ W Λ ) is defined by Proposition 3.6. It follows that for any path p ∈ p ( ℓ, m ) wehave dn( w k , p ) = ( w m k = ℓ or ℓ + 1 w m s k ≥ ℓ + 2 , where by dn( w k , p ) is defined by (8). Thus we have D Λ w k ,w m = p ( k, m ) G p ( k − , m ) . (21)From (21) and (14) we have a km = ( − k + m X p ∈ p ( k,m ) e − p (1) + X p ∈ p ( k − ,m ) e − p (1) . Thus by (12), (20), and (18), we obtain d k ,m = ( − k + m +1 e q m X i =( i k ,...,i m +1 ) ∈ Z k − m ≥ ≤ i k ≤···≤ i m +1 ≤ m e χ ( i ) + X j =( i k − ,...,i m +1 ) ∈ Z k − − m ≥ ≤ i k − ≤···≤ i m +1 ≤ m e χ ( j ) k > m + 1( − k + m +1 e q m X i =( i k ,...,i m +1 ) ∈ Z k − m ≥ ≤ i k ≤···≤ i m +1 ≤ m e χ ( i ) + 1 k = m + 11 − e q m k = m, (22)where q m := l m m α + (cid:18)j m k + j m k(cid:19) α , and χ ( i ℓ , . . . , i m +1 ) := X j odd ℓ ≥ j ≥ m +1 i j α + X j even ℓ ≥ j ≥ m +1 i j α . d kn,m In this subsection, we derive an inductive formula for the structure constants d kn,m . First, we applyTheorem 3.4 to compute the structure constants d mn,m where n ≤ m .Let w m = s i s i . . . s i m be a reduced decomposition. For ℓ ≤ m , define β ℓ := s i s i . . . s i ℓ − α i ℓ Then we have β ℓ = ( ℓα + ( ℓ − α m odd( ℓ − α + ℓα m even . Then combining Theorem 3.4 with (6) gives, for n ≤ m , d mn,m = X ( − n ( e β j − . . . ( e β jp − , (23)11here the summation runs over all 1 ≤ j < · · · < j p ≤ m such that w n = s i j ∗ · · · ∗ s i jp , and theoperation ∗ is defined as in Definition 3.3.Now comparing ˆ O · ( ˆ O n · ˆ O m ) with ( ˆ O · ˆ O n ) · ˆ O m in K T ( ¯ X ), we obtain, for any k , X i d in,m d k ,i = X j d j ,n d kj,m . (24)Let k > n . Then using that d jn,m = 0 unless j ≥ max { n, m } , and solving for d kn,m in (24), we obtainthe following inductive relation for the structure constants: Proposition 4.2.
For k > n , the T -equivariant structure constants in the basis ˆ O k satisfy: d kn,m = 1 (cid:16) d n ,n − d k ,k (cid:17) k − X i =max { n,m } d in,m d k ,i − k X j = n +1 d j ,n d kj,m . (25)Note that all expressions on the right side of (25) may be assumed to be known by inductingon k − n . The base cases are given by (22) and (23). d SL in ordinary K -theory Let K ( ¯ X ) denote the Grothendieck group of coherent sheaves on ¯ X . Denote by O k := [ O X wkP ],where as earlier, P is the standard maximal parabolic subgroup. Then, we have K ( ¯ X ) = Y k ∈ Z ≥ Z O k . Further, by (5), we have, in K ( ¯ X ), O n · O m = X k ≥ n + m d kn,m (1) O k . Consider the set p ( ℓ, m ). By the results from the preceding section, the number of paths p ∈ p ( ℓ, m ) is the same as the number of ( ℓ − m )-tuples ( i ℓ , i ℓ − , . . . , i ℓ − m +1 ) ∈ Z ℓ − m ≥ satisfying 1 ≤ i ℓ ≤ i ℓ − ≤ · · · ≤ i ℓ − m +1 ≤ m . Hence the set p ( ℓ, m ) has cardinality (cid:0) ℓ − m − (cid:1) . Thus, we have, by(21), (cid:12)(cid:12) D Λ w k ,w m (cid:12)(cid:12) = (cid:18) k − m − (cid:19) + (cid:18) k − m − (cid:19) . (26)Evaluating (22) at 1, we have, for k > m , d k ,m (1) = ( − k + m +1 (cid:20)(cid:18) k − m − (cid:19) + (cid:18) k − m − (cid:19)(cid:21) . (27)Comparing O · ( O n − · O m ) with ( O · O n − ) · O m , we obtain, for any ℓ ≥ X i d in − ,m (1) d ℓ ,i (1) = X j d j ,n − (1) d ℓj,m (1) . (28)12etting ℓ = n + m + k in (28), where k ≥
0, using that d jn,m (1) = 0 unless j ≥ n + m , and solvingfor d n + m + kn,m (1), we obtain the following inductive relation for the structure constants: d n + m + kn,m (1) = 1 n n + m + k − X i = n + m − d in − ,m (1) d n + m + k ,i (1) − n + k X j = n +1 d j ,n − (1) d n + m + kj,m (1) . (29)In particular, choosing k = 0 in (29), and using (27) (with k = m + 1), we derive that d n + mn,m (1) = (cid:18) n + mn (cid:19) . (30)Note that all expressions on the right side of (29) may be assumed to be known by inductingon n and k simultaneously. The base cases are covered by (27) and (30). Thus (29) completelydetermines the structure constants. Theorem 5.1.
The structure constants in the basis O k in ordinary K -theory are given by d n + m + kn,m (1) = ( − k · ( n + m + k − n − m − k ! · n + m + 2 k ( n + k )( m + k ) . (31) Proof.
It suffices to show that the above formula (31) satisfies the inductive relation (29) and agreeswith the known formulas for the base cases d k ,m (1) (27) and d n + mn,m (1) (30).The base case d n + mn,m (1) is immediately verified by letting k = 0 in (31). For the base case d k ,m (1), let ℓ = k − − m . Then, according to (31), d k ,m (1) = d m + ℓ ,m (1) = ( − ℓ · ( m + ℓ )!( m − ℓ ! · (1 + m + 2 ℓ )(1 + ℓ )( m + ℓ )= ( − k +1+ m · ( k − m − k − − m )! · k − − m ( k − m )( k − . It is straightforward to verify that this is equal to ( − k +1+ m h(cid:0) k − m − (cid:1) + (cid:0) k − m − (cid:1)i , as desired.Now we must verify the inductive relation (29). We first rewrite (29) as n + m + k − X i = n + m − d in − ,m (1) d n + m + k ,i (1) − n + k X j = n d j ,n − (1) d n + m + kj,m (1) = 0 . (32)Now according to (31), we have d in − ,m (1) = ( − i + n + m +1 · ( i − n − m − i − n − m + 1)! · (2 i + 1 − n − m )( i − m )( i − n + 1) , (33) d n + m + k ,i (1) = ( − n + m + k + i +1 · ( n + m + k − i − n + m + k − i − · (2 n + 2 m + 2 k − − i )( n + m + k − i )( n + m + k − , (34) d j ,n − (1) = ( − j + n · ( j − n − j − n )! · (2 j − n )( j − n + 1)( j − , (35) d n + m + kj,m (1) = ( − n + k + j · ( n + m + k − j − m − n + k − j )! · (2 k + 2 n − j + m )( n + k )( n + m + k − j ) . (36)13hen, plugging in the above formulas (33)-(36), and letting s = i − n − m + 1 and t = j − n ,the left hand side of (32) becomes k X s =0 ( − k · (2 s + n + m − n + m + 2 k − s )( s + n − s + m )( k + 1 − s )( n + m + k − · ( n + m + k − n − m − s !( k − s )! − k X t =0 ( − k · (2 t + n )(2 k + n + m − t )( t + n − t + 1)( n + k )( m + k − t ) · ( n + m + k − n − m − t !( k − t )! . As we wish to show that this expression is equal to 0, combing the two sums and dividing by thefactor ( − k · ( n + m + k − n − m − gives k X s =0 ( n + m + 2 k − s )( s + n − s !( k − s )! (cid:20) s + n + m − s + m )( k + 1 − s )( n + m + k − − s − n ( s + 1)( n + k )( m + k − s ) (cid:21) . (37)We compute that the difference of the two fractions in the brackets in (37) simplifies to( m − s − k )( s + n − n + m + k + s )( s + m )( k + 1 − s )( k + m + n − s + 1)( n + k )( m + k − s ) . Thus, expression (37) being equal to zero is equivalent to the equation k X s =0 s !( k − s )! (cid:20) ( n + m + 2 k − s )(2 s − k )( n + m + k + s )( s + m )( k + 1 − s )( s + 1)( m + k − s ) (cid:21) = 0 . (38)Now letting f ( s, k ) := 1 s !( k − s )! (cid:20) ( n + m + 2 k − s )(2 s − k )( n + m + k + s )( s + m )( k + 1 − s )( s + 1)( m + k − s ) (cid:21) , we see that for 0 ≤ s ≤ k , we have f ( s, k ) = − f ( k − s, k ). This immediately implies (38), which inturn verifies (32), completing the proof.Now let ξ k := [ O X wkP ( − ∂X w k P )] denote the class of O X wkP ( − ∂X w k P ) in K ( ¯ X ). Then we have, in K ( ¯ X ), ξ n · ξ m = X k ≥ n + m b kn,m (1) ξ k , where b kn,m are defined as in (16).From Theorem 5.1 one may compute that d jn,m − d jn +1 ,m − d jn,m +1 + d jn +1 ,m +1 = ( − j · ( n + m + j − n ! m ! j ! · ( n + m + 2 j ) . (39)Then (39) and (17) together with an inductive argument give the following corollary: Corollary 5.2.
The structure constants in the basis ξ k in ordinary K -theory are given by b n + m + kn,m (1) = ( − k · ( n + m + k )! n ! m ! k ! . Structure constants for d SL in T -equivariant cohomology Let G min := d SL be the minimal affine Kac-Moody group associated to SL . Let P be the standardmaximal parabolic subgroup and let X := G min /P denote the standard affine Grassmannian.Let { ˆ ε i } ∞ i =0 denote the Schubert basis in T -equivariant cohomology of X . Here we use thenotation ˆ ε i := ˆ ε w i , where ˆ ε w i is defined as in [Ku, Theorem 11.3.9]. Let Z [ α , α ] denote the gradedring of polynomials with integral coefficients in the simple roots α , α . Further, let Z [ α , α ]( k )denote the k -th graded piece of Z [ α , α ].We define the T -equivariant cohomology structure constants c kn,m ∈ Z [ α , α ]( n + m − k ) (see[Ku, Corollary 11.3.17]) by ˆ ε n · ˆ ε m = n + m X k =max { n,m } c kn,m ˆ ε k . (40)By the Chevalley formula [Ku, Theorem 11.17 (i)], and using (18), we compute thatˆ ε · ˆ ε m = q m ˆ ε m + ( m + 1)ˆ ε m +1 , (41)where q m := l m m α + (cid:18)j m k + j m k(cid:19) α . (42)In particular, c m ,m = q m and c m +11 ,m = m + 1. Definition 6.1.
We let Q di,j denote the sum of all monomials of degree d in the j + 1 variables q i , q i +1 , . . . , q i + j .From (41) and induction, we have(ˆ ε ) n · ˆ ε m = n X i =0 ( m + i )! m ! Q n − im,i ˆ ε m + i . (43)Further, we compute that (ˆ ε ) n = n X k =1 k ! Q n − k ,k − ˆ ε k , (44)Solving for ˆ ε n in equation (43), we haveˆ ε n = 1 n ! " (ˆ ε ) n − n − X k =1 k ! Q n − k ,k − ˆ ε k . Assuming now that n ≤ m , a computation yieldsˆ ε n · ˆ ε m = n X i =0 n ! " ( m + i )! m ! Q n − im,i − n − X k =1 k ! Q n − k ,k − c m + ik,m ˆ ε m + i . Further, c jn,m = 0 whenever j < max { n, m } or j > n + m . Hence we have derived the followingrecursive formula for the structure constants: 15 roposition 6.2. For n ≤ m and ≤ i ≤ n , c m + in,m = 1 n ! ( m + i )! m ! Q n − im,i − n − X k =max { i, } k ! Q n − k ,k − c m + ik,m , (45) where Q di,j is defined as in Definition 6.1. Using the above formula (45), one may induct upwards on n to compute the structure constants.In addition, closed forms for the structure constants can be obtained inducting downwards on i .For example, it follows immediately, by letting i = n in (45), that c n + mn,m = (cid:18) n + mn (cid:19) . (46)Now to compute a closed form for c n + m − n,m , letting i = n − c n + m − n,m = ( n + m − n ! m ! n + m − X j =1 q j − m − X j =1 q j − n − X j =1 q j . (47)A closed form for P kj =1 q j is given by k X j =1 q j = ( k ( k + 1)( k + 2)( α + α ) k even ( k + 1)(3 + 2 k + k ) α + ( k − k + 1)( k + 3) α k odd . (48)Thus, one may verify that (47) gives c n + m − n,m = · ( n + m )!( n − m − α + α ) n, m even14 · ( n + m )! n ! m ! ((1 + nm ) α + ( − nm ) α ) n, m odd14 · ( n + m − n − m ! (cid:0) ( − nm + m ) α + (1 + nm + m ) α (cid:1) n even , m odd14 · ( n + m − n !( m − (cid:0) ( − nm + n ) α + (1 + nm + n ) α (cid:1) n odd , m even . In general, to obtain a closed form for c n + m − dn,m given closed forms for c n + mn,m , . . . , c n + m − d +1 n,m , oneneeds closed forms for Q i,j , . . . , Q di,j . It is easy to see that the following recurrence relation holds: Q di,j = Q d − i,j q i + j + Q di,j − . (49)16y (48) we obtain Q i,j = (3 i + 2 j + 6 ij + 3 i j + 3 j + 3 ij + j ) α + (6 i + 3 i + 2 j + 6 ij + 3 i j + 3 j + 3 ij + j ) α i, j even (1 + j )(3 i + 3 i + 2 j + 3 ij + j )( α + α ) i, j odd (1 + j )(3 + 3 i + 3 i + 2 j + 3 ij + j ) α + (1 + j )( − i + 3 i + 2 j + 3 ij + j ) α i even , j odd (3 + 6 i + 3 i + 5 j + 6 ij + 3 i j + 3 j + 3 ij + j ) α + ( − i − j + 6 ij + 3 i j + 3 j + 3 ij + j ) α i odd , j even . Now using the above and (49) we obtain a closed form for Q i,j , although we do not provide ithere for the sake of space. Then, using (45) we derive the following closed form for the structureconstants c n + m − n,m : c n + m − n,m = a n + m − n,m (cid:0) b n + m − n,m (0) α + b n + m − n,m (1) α α + b n + m − n,m (2) α (cid:1) , where a n + m − n,m = · ( n + m )!( n − m − · n + m − n, m even18 · ( n + m )! n ! m ! · ( n − m − n + m − n, m odd18 · ( n + m − n !( m − · ( n − n even , m odd18 · ( n + m − n − m ! · ( m − n odd , m even .b n + m − n,m (0) = (cid:0) nm + n m − n − m − nm + 4 (cid:1) n, m even14 (cid:0) nm + n m − nm − (cid:1) n, m odd14 (cid:0) nm + n m − n − nm + 2 n + 3 (cid:1) n even , m odd14 (cid:0) nm + n m − m − nm + 2 m + 3 (cid:1) n odd , m even . n + m − n,m (1) = (cid:0) nm + n m − n − m − nm + 2 n + 2 m − (cid:1) n, m even12 (cid:0) nm + n m − nm − (cid:1) n, m odd12 (cid:0) nm + n m − n − nm − (cid:1) n even , m odd12 (cid:0) nm + n m − m − nm − (cid:1) n odd , m even .b n + m − n,m (2) = (cid:0) nm + n m − n − m − nm + 4 n + 4 m − (cid:1) n, m even14 (cid:0) nm + n m − n − nm + 3 (cid:1) n, m odd14 (cid:0) nm + n m − n − nm − n − (cid:1) n even , m odd14 (cid:0) nm + n m − m − nm − m − (cid:1) n odd , m even . Lastly, from [Ku, Lemma 11.1.10 and Proposition 11.1.11 (1) and (3)] one may verify usinginduction that for n ≤ m , a closed form for c mn,m is given by c mn,m = (cid:18) m + n n (cid:19) n − Y i =0 (cid:18)(cid:18) m − n i (cid:19) α + (cid:18) m − n i (cid:19) α (cid:19) n, m even (cid:18) m + n n (cid:19) n − Y i =0 (cid:18)(cid:18) m − n i (cid:19) α + (cid:18) m − n i (cid:19) α (cid:19) n, m odd (cid:18) m + n − n (cid:19) n − Y i =0 (cid:18)(cid:18) m − n + 12 + i (cid:19) α + (cid:18) m − n + 12 + 1 + i (cid:19) α (cid:19) n odd , m even (cid:18) m + n − n (cid:19) n − Y i =0 (cid:18)(cid:18) m − n + 12 + 1 + i (cid:19) α + (cid:18) m − n + 12 + i (cid:19) α (cid:19) n even , m odd . References [G] W. Graham. Equivariant K-theory and Schubert varieties. Preprint.[He] X. He. Minimal length elements in some double cosets of Coxeter groups.
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