Equivariant Quantum Cohomology of the Odd Symplectic Grassmannian
aa r X i v : . [ m a t h . AG ] J un EQUIVARIANT QUANTUM COHOMOLOGY OF THE ODDSYMPLECTIC GRASSMANNIAN
LEONARDO C. MIHALCEA AND RYAN M. SHIFLER
Abstract.
The odd symplectic Grassmannian IG : “ IG p k, n ` q parametrizes k di-mensional subspaces of C n ` which are isotropic with respect to a general (necessarilydegenerate) symplectic form. The odd symplectic group acts on IG with two orbits, and IGis itself a smooth Schubert variety in the submaximal isotropic Grassmannian IG p k, n ` q .We use the technique of curve neighborhoods to prove a Chevalley formula in the equi-variant quantum cohomology of IG, i.e. a formula to multiply a Schubert class by theSchubert divisor class. This generalizes a formula of Pech in the case k “
2, and it givesan algorithm to calculate any multiplication in the equivariant quantum cohomology ring. Introduction
Let E : “ C n ` be an odd-dimensional complex vector space and 1 ď k ď n `
1. An odd-symplectic form ω on E is a skew-symmetric bilinear form with kernel of dimension 1. The odd-symplectic Grassmannian IG : “ IG p k, E q parametrizes k -dimensional linear subspacesof E which are isotropic with respect to ω . One can find vector spaces F Ă E Ă r E such thatdim F “ n , dim r E “ n `
2, the restriction of ω to F is non-degenerate, and ω extends toa symplectic form (hence non-degenerate) on r E . Then the odd-symplectic Grassmannianis an intermediate space(1) IG p k ´ , F q Ă IG p k, E q Ă IG p k, r E q , sandwiched between two symplectic Grassmannians. This and the more general “odd-symplectic partial flag varieties” have been studied by Mihai [27] and Pech [37]. In par-ticular, Mihai showed that IG p k, E q is a smooth Schubert variety in IG p k, r E q , and that itadmits an action of Proctor’s odd-symplectic group Sp n ` (see [38]). If k ‰ n ` p k, E q with 2 orbits, and the closed orbit can be identifiedwith IG p k ´ , F q . If k “ p , E q “ P p E q and if k “ n ` p n ` , E q isisomorphic to the Lagrangian Grassmannian IG p n, F q .In this paper we are concerned with the study of the quantum cohomology ring QH ˚ T p IG q of the odd-symplectic Grassmannians, and its T -equivariant version, where T is the max-imal torus in Sp n ` . Since IG is a Schubert variety in the symplectic GrassmannianIG p k, r E q it follows that the (equivariant) fundamental classes of those Schubert varieties X p u q Ă IG p k, r E q included in IG form a basis for the (co)homology ring H ˚ p IG q ; we callthis the Schubert basis . This implies that the graded algebra QH ˚ T p IG q has a Schubert basis r X p u qs T over H ˚ T p pt qr q s , indexed by a particular subset of Schubert classes in the quantum Mathematics Subject Classification.
Primary 14N35; Secondary 14N15, 14M15.L.M. was supported in part by NSA Young Investigator Award 98320-16-1-0013 and a Simons Collabo-ration grant. cohomology of QH ˚ T p IG p k, r E qq . There are Schubert multiplication formulas in QH ˚ T p IG q , r X p u qs T ‹ r X p v qs T “ ÿ w,d c w,du,v q d r X p w qs T , where c w,du,v are the (equivariant) Gromov-Witten (GW) invariants for rational curves ofdegree d in IG, and q is the quantum parameter.Our main result is a combinatorial formula for the multiplication r X p u qs T ‹ r D s T of anySchubert class by the Schubert divisor class r D s T . This is called the (equivariant, quantum) Chevalley formula . Before stating this formula explicitly, we discuss its significance.It has been known at least since Knutson and Tao’s famous paper [20] that, despitethe fact that the Schubert divisor does not generate the cohomology ring, the equivariant
Chevalley formula gives a triangular system of equations calculating the Schubert structureconstants. Knutson and Tao worked in the geometric context of the (ordinary) equivariantcohomology of the Grassmannian, and they used previous work of Okounkov-Olshanski[34, 35] and Molev-Sagan [33] who studied a certain deformation of Schur polynomials,called factorial Schur functions. This system was extended to the equivariant quantumcohomology ring of flag manifolds in [29, 31] and very recently to quantum K theory [3],but in these cases it is no longer triangular. In this paper we further extend these resultsto the case of odd-symplectic Grassmannians: we use the Chevalley formula to obtain analgorithm for calculating any structure constant c w,du,v ; see theorem 12.2 below. Its corollary1.2, stated in the next section, makes precise the sense in which this formula determinesthe ring structure.The Chevalley formula technique was previously used to solve several cases of the Gi-ambelli problem : find a presentation of the (equivariant, quantum) cohomology ring bygenerators and relations, then identify the polynomials which represent Schubert classes;see e. g. [39] for more on the history of this problem. Such results were obtained for theequivariant quantum cohomology ring of the Grassmannian [32], of the orthogonal and La-grangian Grassmannians [18], and of the equivariant cohomology of non-maximal isotropicGrassmannians [40]. Although we do not pursue this application in this note, we believethat the Chevalley technique will be a key ingredient. A third application, for which wededicate the upcoming paper [24] joint with C. Li, is to verify Galkin, Golyshev and Iri-tani’s Conjecture O [9, 12] for the odd-symplectic Grassmannians. This uses the explicitcombinatorial formulation of the Chevalley formula.1.1. Statement of results.
To state the Chevalley formula, we need to introduce a variantof k -strict partitions of Buch, Kresch and Tamvakis [6]. This variant was used by Pech in[36] to study the ordinary cohomology ring of IG, and the quantum cohomology ring ofIG p , E q in [37].A partition p λ ě λ ě ¨ ¨ ¨ ě λ k q is p n ´ k q -strict if λ j ą n ´ k implies λ j ą λ j ` . LetΛ be the set of p n ´ k q -strict partitions p n ` ´ k ě λ ě ¨ ¨ ¨ ě λ k ě ´ q such that if λ k “ ´ λ “ n ` ´ k . For each λ P Λ there is a Schubert variety in X p λ q Ă IG ofcodimension | λ | : “ λ ` . . . ` λ k . If λ “ n ` ´ k and λ k ě λ ˚ “ p λ ě λ ě ¨ ¨ ¨ ě λ k ě q . If λ ă n ` ´ k or λ k “ ´ λ ˚ does not exist. If λ “ n ` ´ k and λ “ n ´ k then let λ ˚˚ “ p λ ě λ ě ¨ ¨ ¨ ě λ k ě ´ q . This can be explained by using that the localized equivariant cohomology is generated by divisors; see[3, § QUIVARIANT QUANTUM COHOMOLOGY OF THE ODD SYMPLECTIC GRASSMANNIAN 3 If λ ă n ´ k then λ ˚˚ does not exist. With this notation, the Schubert divisor is D “ X p q . Theorem 1.1 (Quantum Chevalley formula) . Let λ P Λ . Then the following holds in theequivariant quantum cohomology ring QH ˚ T p IG p k, C n ` q : r X p qs T ‹ r X p λ qs T “ Classical Part ` q r X p λ ˚ qs T ` q r X p λ ˚˚ qs T . (2) The terms involving λ ˚ or λ ˚˚ are omitted if the corresponding partitions do not exist. Theclassical part consists of terms which do not involve q , and it is combinatorially explicit; seeTheorem 11.7 below. This recovers Pech’s results in [37] for the non-equivariant ring QH ˚ p IG p , E qq and itverifies a conjecture for QH ˚ p IG p , E qq stated in [36]. It is an easy exercise to check thatfor k “ ˚ T p P n q and that for k “ n ` p n, F q from [7, 22].As we mentioned above, the Chevalley multiplication yields an algorithm to calculate any equivariant GW invariant c w,du,v , see §
12 below. Its immediate corollary is:
Corollary 1.2.
Let p A, ` , ˝q be a graded, commutative, H ˚ T p pt qr q s -algebra, with a H ˚ T p pt qr q s -basis t a λ u λ P Λ . Assume that the grading is the same as the one for the equivariant quantumring QH ˚ T p IG q , and assume that the Chevalley rule (2) holds in the basis t a λ u . Then theisomorphism of H ˚ T p pt q -modules A Ñ QH ˚ T p IG p k, n ` qq sending a λ ÞÑ r X p λ qs T is anisomorphism of algebras. The proof of Theorem 1.1 is based on the analysis of curve neighborhoods of Schubertvarieties [4, 8]. Let d P H p IG q be an effective degree and let M d : “ M , p IG , d q be theKontsevich moduli space of stable maps [11] equipped with evaluation maps ev , ev : M d Ñ IG. To a closed subvariety Ω Ă IG one can associate its
Gromov-Witten variety GW d p Ω q : “ ev ´ p Ω q and its curve neighborhood Γ d p Ω q : “ ev p GW d p Ω qq Ă IG;see § X p λ q Ă IG be a Schubert variety, and let Γ d p X p λ qq “ Γ Y Γ Y . . . Y Γ k bethe decomposition of the curve neighborhood into irreducible components. By the divisoraxiom, any component Γ i of “expected dimension” will contribute to the quantum product r X p qs T ‹ r X p λ qs T with d ¨ m i ¨ q d r Γ i s T , where m i is the degree of ev : GW d p X p u qq Ñ Γ d p X p λ qq over the given component. Therefore the main task is to find the components Γ i of expected dimension, and calculate the associated multiplicities m i . It was proved in [8]that for homogeneous spaces the curve neighborhoods of Schubert varieties are irreducibleand that all the multiplicities m i “
1. But IG is no longer homogeneous, and one easilyfinds reducible curve neighborhoods. Nevertheless, we proved that for any Schubert variety X p λ q , the only curve neighborhoods Γ d p X p λ qq of expected dimension are those when d “ X p λ q is included in the closed Sp n ` -orbit of IG. In this caseΓ p X p λ qq “ X p λ q Y X p λ q , and the components of the expected dimension correspond respectively to the partitions λ ˚ , λ ˚˚ from Theorem 1.1 above. (But it may happen, for instance, that X p λ q does nothave expected dimension, in which case λ ˚ does not exist.) We refer to Theorems 7.1 and9.2 for precise statements. Furthermore, the morphism ev : GW p X p λ qq Ñ Γ p X p λ qq isbirational over the relevant components, thus all multiplicities m i “
1. This is achievedin sections 8 and 9 by using a rather delicate analysis of the space of lines in IG. In thecourse of the proof we showed that each orbit of Sp n ` in IG contributes with at most LEONARDO C. MIHALCEA AND RYAN M. SHIFLER one component to the curve neighborhood. It is tempting to conjecture that this patternextends at least to the odd-symplectic partial flag varieties.Besides reducibility of curve neighborhoods of Schubert varieties, it is also worth point-ing out another difference between the (quantum) cohomology of odd-symplectic Grass-mannians and that of flag manifolds. Many arguments in the Gromov-Witten theory ofhomogeneous spaces rely on the Kleiman-Bertini transversality theorem, which makes theGW invariants enumerative. Variants of this theorem exist for varieties with a group act-ing with finitely many orbits (see e.g. [14]). But the lack of a transitive group actionimplies that occasionally cycles in IG cannot be translated to general position, and thatit is possible that certain Schubert multiplications might be non-effective. Indeed, Pechfound Pieri-type multiplications, both in ordinary and in quantum cohomology of IG p , E q ,which yield negative structure constants c w,du,v . For more such examples, equivariant or not,see the table for QH ˚ T p IG p , C qq in §
13, and the remark 11.9 below. We partially circum-vented the non-transversality problem by employing the aforementioned technique of curveneighborhoods.1.2.
Organization of the paper.
The sections 2-5 are dedicated to recalling the basicdefinitions and the relevant facts on (equivariant) quantum cohomology. In section 6 wediscuss curve neighborhoods of Schubert varieties. The main result is Theorem 6.6 aboutestimates on the dimension of curve neighborhoods. This is then used in section 7 to provethe vanishing of many GW invariants. In sections 8 we study the moduli space of lineson IG; our main result is Corollary 8.7 where we prove that if Ω is a Schubert varietyincluded in the closed orbit then the GW variety GW p Ω q , has two irreducible, genericallyreduced components. This result is used in Section 9 to obtain similar results about thecurve neighborhood Γ p Ω q . In section 10 we prove birationality results and use this tocalculate all the non-vanishing equivariant GW invariants which appear in the Chevalleyformula; see Theorem 10.1 and Corollary 10.2. In section 11 we re-interpret all results interms of k -strict partitions, and obtain the statement of Theorem 1.1 above. The algorithmto calculate the full multiplication table in QH ˚ T p IG q is presented in section 12. Section 13includes examples of products in QH ˚ T p IG p , C qq and QH ˚ T p IG p , C qq . Acknowledgements.
We would like to thank Dan Orr and Mark Shimozono for discussionsand valuable suggestions and to Pierre-Emmanuel Chaput, Changzheng Li, and NicolasPerrin for discussions and collaborations on related projects. Special thanks are due toAnders Buch for encouragement and interest in this project.2.
Preliminaries
The odd symplectic group.
We recall next the definition and basic properties ofodd symplectic flag manifolds, following Mihai’s paper [27]; see also [28, 37]. Let E be acomplex vector space of dimension dim C E “ n `
1, and let ω be an odd-symplectic formon E , i.e. bilinear, skew-symmetric, with kernel of dimension 1. The odd symplectic group is the subgroup of GL p E q which preserves this symplectic form:Sp n ` p E q : “ t g P GL p E q : ω p g.u, g.v q “ ω p u, v q , @ u, v P E u . It will be convenient to extend the form ω to a nondegenerate symplectic form r ω on an evendimensional space r E Ą E , and to identify E Ă r E with a coordinate hyperplane C n ` Ă C n ` . For that, let t e , . . . , e n ` , e n ` , . . . , e ¯2 , e ¯1 u be the standard basis of r E : “ C n ` ,where ¯ i “ n ` ´ i . Set | i | “ min t i, ¯ i u , and consider r ω to be the nondegenerate symplecticform on r E defined by r ω p e i , e j q “ δ i, ¯ j for all 1 ď i ď j ď ¯1 . QUIVARIANT QUANTUM COHOMOLOGY OF THE ODD SYMPLECTIC GRASSMANNIAN 5
The form r ω restricts to the degenerate symplectic form ω on E : “ C n ` “ h e , e , ¨ ¨ ¨ , e n ` i such that the kernel ker ω is generated by e . Then ω p e i , e j q “ δ i, ¯ j for all 1 ď i ď j ď ¯2 . Let F Ă E denote the 2 n dimensional vector space with basis t e , e , ¨ ¨ ¨ , e n ` u . Since F X ker ω “ p q it follows that ω restricts to a nondegenerate form on F . Let Sp n p F q andSp n ` p r E q denote the symplectic groups which preserve respectively the symplectic form ω | F and r ω . Then with respect to the decomposition E “ F ‘ ker ω the elements of theodd-symplectic group Sp n ` p E q are matrices of the formSp n ` p E q “ "ˆ λ a S ˙ : λ P C ˚ , a P C n , S P Sp n p F q * . The symplectic group Sp n p F q embeds naturally into Sp n ` p E q by λ “ a “
0, butSp n ` p E q is not a subgroup of Sp n ` p r E q . Mihai showed in [27, Prop. 3.3] that there is asurjection P Ñ Sp n ` p E q where P Ă Sp n ` p r E q is the parabolic subgroup which preservesker ω , and the map is given by restricting g ÞÑ g | E . Then the Borel subgroup B n ` Ă Sp n ` p r E q of upper triangular matrices restricts to the (Borel) subgroup B Ă Sp n ` p E q .Similarly, the maximal torus T n ` : “ t diag p t , ¨ ¨ ¨ , t n ` , t ´ n ` , ¨ ¨ ¨ , t ´ q : t , ¨ ¨ ¨ , t n ` P C ˚ u Ă B n ` restricts to the maximal torus T “ t diag p t , ¨ ¨ ¨ , t n ` , t ´ n ` , ¨ ¨ ¨ , t ´ q : t , ¨ ¨ ¨ , t n ` P C ˚ u Ă B. The odd symplectic flag varieties.
Let 1 ď i ă . . . ă i r ď n `
1. The oddsymplectic flag variety IF p i , . . . , i r ; E q consists of flags of linear subspaces F i Ă . . . Ă F i k Ă E such that dim F i j “ i j and F i j is isotropic with respect to the symplectic form ω . The inclusion E Ă r E makes it a closed subvariety of the (even) symplectic flag varietyIF p i , . . . , i r ; r E q which consists of similar flags of subspaces, isotropic with respect to thesymplectic form r ω . The latter is a homogeneous space for Sp n ` p r E q . In fact, the inclusions F Ă E Ă r E realize the odd-symplectic flag variety as an intermediate variety between twoconsecutive symplectic flag varieties:IF p i ´ , . . . , i r ´ F q Ă IF p i , . . . , i r ; E q Ă IF p i , . . . , i r ; r E q , where the flags in IF p i ´ , . . . , i r ´ F q are isotropic with respect to ω | F . If r “ p i , . . . , i r ; E q which parametrizes flags of given dimensions in E . Itturns out that IF p i , . . . , i r ; E q is a smooth subvariety of Fl p i , . . . , i r ; E q of codimension i r p i r ´ q ; see [27, Prop. 4.1] for details. The odd-symplectic group acts on IF p i , . . . , i r ; E q ,but the action is no longer transitive. The next result, due to Mihai (see [27, Propositions5 and 6] describes the orbits of this action. Proposition 2.1.
The odd symplectic group Sp n ` p E q acts on IF p i , ¨ ¨ ¨ , i r ; E q with r ` orbits if i r ă n ` and r orbits if i r “ n ` . The orbits are: O j “ t V i Ă ¨ ¨ ¨ Ă V i r Ă E : e P V i j , e R V i j ´ u for all ď j ď r and O r ` “ t V i Ă ¨ ¨ ¨ Ă V i r Ă E : e R V i r u if i r ă n ` , However, Gelfand and Zelevinsky [13] defined another group Ă Sp n ` closely related to Sp n ` such thatSp n Ă Ă Sp n ` Ă Sp n ` . LEONARDO C. MIHALCEA AND RYAN M. SHIFLER where by convention V i “ p q . The only closed orbit is O , and it may be naturally identifiedto IF p i ´ , . . . , i r ´ F q .In particular, for ď k ď n the odd symplectic group Sp n ` p E q acts on the odd sym-plectic Grassmannian IG p k, E q with two orbits X c “ t V P IG p k, E q : e P V u the closed orbit X ˝ “ t V P IG p k, E q : e R V u the open orbit . The closed orbit X c is isomorphic to IG p k ´ , F q . If k “ n ` then IG p n ` , E q “ X c may be identified to the Lagrangian Grassmannian IG p n, F q . Mihai identifies the closures O i of the orbits and proves they are smooth. From nowon we will identify F Ă E Ă r E to C n Ă C n ` Ă C n ` with bases x e , . . . , e n ` y Ăx e , . . . , e n ` y Ă x e , . . . , e n ` y . The corresponding symplectic flag manifolds will bedenoted by IF p i ´ , . . . , i r ´
1; 2 n q Ă IF p i , . . . , i r ; 2 n ` q Ă IF p i , . . . , i r ; 2 n ` q . SimilarlySp n ` p E q will be denoted by Sp n ` etc.2.3. The Weyl group and odd-symplectic minimal representatives.
We recall nextthe indexing sets which we will use in the next section to define the Schubert varieties.Consider the root system of type C n ` with positive roots R ` “ t t i ˘ t j : 1 ď i ă j ď n ` u Y t t i : 1 ď i ď n ` u and the subset of simple roots ∆ “ t α i : “ t i ´ t i ` : 1 ď i ď n uYt α n ` : “ t n ` u . The associated Weyl group W is the hyperoctahedral group consistingof signed permutations , i.e. permutations w of the elements t , ¨ ¨ ¨ , n ` , n ` , ¨ ¨ ¨ , u satisfying w p i q “ w p i q for all w P W . For 1 ď i ď n denote by s i the simple reflectioncorresponding to the root t i ´ t i ` and s n ` the simple reflection of 2 t n ` . Each subset I : “ t i ă . . . ă i r u Ă t , . . . , n ` u determines a parabolic subgroup P : “ P I ď Sp n ` p r E q with Weyl group W P “ x s i : i ‰ i j y generated by reflections with indices not in I . Let∆ P : “ t α i s : i s R t i , . . . , i r uu and R ` P : “ Span Z ∆ P X R ` ; these are the positive rootsof P . Let ℓ : W Ñ N be the length function and denote by W P the set of minimallength representatives of the cosets in W { W P . The length function descends to W { W P by ℓ p uW P q “ ℓ p u q where u P W P is the minimal length representative for the coset uW P . Wehave a natural ordering 1 ă ă . . . ă n ` ă n ` ă . . . ă , which is consistent with our earlier notation i : “ n ` ´ i . Let P “ P k to be the maximalparabolic obtained by excluding the reflection s k . Then the minimal length representatives W P have the form p w p q ă w p q ă ¨ ¨ ¨ ă w p k q| w p k ` q ă . . . ă w p n ` q ď n ` q if k ă n ` p w p q ă w p q ă ¨ ¨ ¨ ă w p n ` qq if k “ n `
1. Since the last n ` ´ k are determined fromthe first, we will identify an element in W P k with the sequence p w p q ă w p q ă ¨ ¨ ¨ ă w p k qq . Example . The reflection s t ` t is given by the signed permutation s t ` t p q “ ¯2 , s t ` t p q “ ¯1 , and s t ` t p i q “ i for all 3 ď i ď n `
1. The minimal length representative of s t ` t W P k is p ă ă ¨ ¨ ¨ ă k ă ¯2 ă ¯1 q .The Weyl group W admits a partial ordering ď given by the Bruhat order. Its coveringrelations are given by w ă ws α where α P R ` is a root and ℓ p w q ă ℓ p ws α q . We will use the Hecke product on the Weyl group W . For a simple reflection s i the product is defined by w ¨ s i “ " ws i if ℓ p ws i q ą ℓ p w q ; w otherwise QUIVARIANT QUANTUM COHOMOLOGY OF THE ODD SYMPLECTIC GRASSMANNIAN 7
The Hecke product gives W a structure of an associative monoid; see e.g. [8, §
3] for moredetails. Given u, v P W , the product uv is called reduced if ℓ p uv q “ ℓ p u q ` ℓ p v q , or, equiva-lently, if uv “ u ¨ v . For any parabolic group P , the Hecke product determines a left action W ˆ W { W P ÝÑ W { W P defined by u ¨ p wW P q “ p u ¨ w q W P . We recall the following properties of the Hecke product (cf. e.g. [8]).
Lemma 2.3.
For any u, v P W there is an inequality ℓ p u ¨ vW P q ď ℓ p u q ` ℓ p vW P q . Ifthe equality holds then u ¨ vW P “ uvW P . If furthermore v P W P is a minimal lengthrepresentative, then the following are equivalent:(i) ℓ p u ¨ vW P q “ ℓ p u q ` ℓ p vW P q ;(ii) ℓ p u ¨ v q “ ℓ p u q ` ℓ p v q , and u ¨ v “ uv is a minimal length representative in W P .Proof. The first part of this lemma is explicitly stated in [8, § u ¨ v “ uv . For the converse, since v P W P , ℓ p u ¨ v q ě ℓ p u ¨ vW P q “ ℓ p u q ` ℓ p vW P q “ ℓ p u q ` ℓ p v q ě ℓ p u ¨ v q . Thus u ¨ v P W P and ℓ p u ¨ v q “ ℓ p u q ` ℓ p v q and this finishes the proof. (cid:3) Schubert Varieties in even and odd flag manifolds.
Let I : “ t i ă . . . ă i r u Ăt , . . . , n ` u and the associated parabolic subgroup P : “ P I . The even symplectic flagmanifold IF p i , . . . , i r ; 2 n ` q is a homogeneous space Sp n ` { P . For each w P W P let Y p w q ˝ : “ B n ` wB n ` { P be the Schubert cell . This is isomorphic to the space C ℓ p w q . Itsclosure Y p w q : “ Y p w q ˝ is the Schubert variety . We might occasionally use the notation Y p wW P q if we want to emphasize the corresponding coset, or if w is not necessarily a min-imal length representative. Recall that the Bruhat ordering can be equivalently describedby v ď w if and only if Y p v q Ă Y p w q . Set(3) w “ p , , . . . , n ` , q if k ă n ` p , , , . . . , n ` q if k “ n ` W . Let B : “ B n ` X Sp n ` be the odd-symplectic Borel subgroup.The following results were proved by Mihai [27, § Proposition 2.4. (a) The natural embedding ι : IF p i , ¨ ¨ ¨ , i r ; 2 n ` q ã Ñ IF p i , ¨ ¨ ¨ , i r ; 2 n ` q identifies IF p i , ¨ ¨ ¨ , i r ; 2 n ` q with the (smooth) Schubert subvariety Y p w W P q Ă IF p i , ¨ ¨ ¨ , i r ; 2 n ` q . (b) The Schubert cells (i.e. the B n ` -orbits) in Y p w q coincide with the B -orbits in IF p i , . . . , i r ; 2 n ` q . In particular, the B -orbits in IF p i , . . . , i r ; 2 n ` q are given by theSchubert cells Y p w q ˝ Ă IF p i , . . . , i r ; 2 n ` q such that w ď w . To emphasize that we discuss Schubert cells or varieties in the odd-symplectic case, foreach w ď w such that w P W P , we denote by X p w q ˝ , and X p w q , the Schubert cell,respectively the Schubert variety in IF p i , . . . , i r ; 2 n ` q . The same Schubert variety X p w q ,but regarded in the even flag manifold is denoted by Y p w q . For further use we note thatIG p k, n ` q has complex codimension k in IG p k, n ` q . Further, a Schubert variety X p w q in IG p k, n ` q is included in the closed orbit X c of if and only if it has a minimal lengthrepresentative w ď w such that w p q “ W odd : “ t w P W : w ď w u and call its elements odd-symplectic permuta-tions . This set consists of permutations w P W such that w p j q ‰ ¯1 for any 1 ď j ď n ` LEONARDO C. MIHALCEA AND RYAN M. SHIFLER
The following closure property of the Hecke product on odd-symplectic permutations willbe important later on.
Lemma 2.5.
Let u, v P W be two odd-symplectic permutations, and assume that u p q “ .Then uv and u ¨ v are odd-symplectic permutations.Proof. We need to show that p uv qp j q ‰ ¯1 and p u ¨ v qp j q ‰ ¯1 for any 1 ď j ď n `
1. Inthe first situation, since u p q “
1, if p uv qp j q “ ¯1 for some 1 ď j ď n `
1, then v p j q “ ¯1,which contradicts that v is odd-symplectic. For the second, consider the signed permutation u : “ p u ¨ v q v ´ P W . By [8, Prop. 3.1] we have that u ď u and u v “ u ¨ v . The conditionthat u ď u implies that there is an inclusion of Schubert varieties X p u q Ă X p u q in the fullodd-symplectic flag manifold IF odd : “ IF p , , . . . , n `
1; 2 n ` q . Further, the hypothesisthat u p q “ X p u q is in the closed orbit of IF odd , thus X p u q is in the closedorbit as well. This implies that u p q “
1. Then u ¨ v “ u v and since u p q “ u v is again odd-symplectic, as claimed. (cid:3) Equivariant cohomology
Fix a parabolic subgroup P Ă Sp n ` containing the standard Borel subgroup B n ` . LetIF ev : “ IF p i , . . . , i r ; 2 n ` q be the corresponding symplectic flag variety. The Schubert cells Y p w q ˝ form a stratification of IF ev , when w varies in W P . This implies that the Schubertclasses r Y p w qs P H ℓ p w q p IF q form a basis of the (integral) homology of IF ev . Since IF ev issmooth, the Schubert classes determine cohomology classes r Y p w qs P H ´ ℓ p w q p IF ev q .The odd-symplectic flag manifold IF : “ IF p i , . . . , i r ; 2 n ` q is a smooth Schubert variety inIF ev , therefore its Schubert classes r X p w qs “ r Y p w qs P H ℓ p w q p IF q for w P W P X W odd forma Z -basis for both homology and cohomology H ˚ p IF q “ H ˚ p IF q . We will use that in theGrassmannian case, the inclusion ι : IG p k, n ` q Ñ IG p k, n ` q gives a group isomorphism H p IG p k, n ` qq » H p IG p k, n ` qq sending the Schubert curve r X p s k qs to itself. For X P t IF , IF ev u there is a nondegenerate Poincar´e pairing x¨ , ¨y : H ˚ p X q b H ˚ p X q Ñ H ˚ p pt q given by x γ , γ y “ ż X γ Y γ where the integral is the push-forward to the point, i.e. ş X γ : “ p ˚ p γ q and p : X Ñ pt isthe structure morphism. For a cohomology class γ P H ˚ p X q we denote by γ _ its Poincar´edual. Thus ş IF r X p u qs Y r X p v qs _ “ δ u,v . Remark . It is well known that the Poincar´e dual of a Schubert class in H ˚ p IF ev q is againa Schubert class; indeed this is true for any homogeneous space G { P [2]. This is no longertrue in the odd-symplectic case. Formulas for Poincar´e dual classes of Schubert classes inthe odd-symplectic Grassmannian IG p k, n ` q were calculated by Pech in [37, Prop. 3]for k “ k .We review some basic facts about the equivariant cohomology ring, following [1], andfocusing on H ˚ T p IF q . For any topological space Z with a left torus T action, its equivariantcohomology ring is the ordinary cohomology of the Borel mixed space Z T : “ p ET ˆ Z q{ T where ET Ñ BT is the universal T -bundle, and T acts on ET ˆ Z by t ¨p e, z q “ p et, t ´ z q . Inparticular, H ˚ T p pt q “ H ˚ p BT q is a polynomial ring Z r t , . . . , t s s where t i are an additive basisfor p Lie T q ˚ . The continuous map Z T Ñ BT gives a H ˚ T p pt q -algebra structure on H ˚ T p Z q .Let now Z “ IF with its natural T » p C ˚ q n ` action. The Schubert varieties X p u q Ă IFare T -stable, and the fundamental classes r X p u qs T P H T ℓ p u q p IF q give an H ˚ T p pt q -basis for theequivariant (co)homology H ˚ T p IF q “ H T ˚ p IF q . The inclusion ι : IF Ñ IF ev gives a natural QUIVARIANT QUANTUM COHOMOLOGY OF THE ODD SYMPLECTIC GRASSMANNIAN 9 restriction map H ˚ T n ` p IF ev q Ñ H ˚ T p IF q . The action of T n ` on IF factors through thatof T , therefore the natural morphism H ˚ T n ` p IF q Ñ H ˚ T p IF q is an algebra isomorphismover H ˚ T p pt q “ H ˚ T n ` p pt q “ Z r t , . . . t n ` s . Because of this, we will take T : “ T n ` fromnow on. We use the same conventions as in [8, §
8] for the geometric interpretation of thecharacters t i inside the equivariant cohomology ring. There is an equivariant version of thePoincar´e pairing x¨ , ¨y : H ˚ T p IF q b H ˚ T p IF q Ñ H ˚ T p pt q given by the (equivariant) push forwardmap to the point: x γ , γ y “ ż T IF odd γ Y γ : “ p T ˚ p γ Y γ q P H ˚ T p pt q . (Equivariant) Quantum cohomology In this section we recall some basic facts about equivariant Gromov-Witten (GW) in-variants and the equivariant quantum (EQ) cohomology rings, following [11, 31]. For thepurposes of this paper we specialize to the odd and even-symplectic Grassmannian case.4.1.
Equivariant Gromov-Witten invariants.
Set IG : “ IG p k, n ` q and IG ev : “ IG p k, n ` q , with ι : IG Ñ IG ev the natural embedding. Let X P t IG , IG ev u . Recallthat H p X q “ Z . A degree d in X is an effective homology class d P H p X q , and it can beidentified with a non-negative integer. Let M ,r p X , d q be the Kontsevich moduli of stablemaps to X of degree d to X with r marked points ( r ě M ,r p X , d q “ dim X ` ż r X p s k qs c p T X q ` r ´ T X denotes the tangent bundle of X . Lemma 4.1.
Let X p Div q and Y p Div q be the (unique) Schubert divisors in IG and IG ev .Then the following equalities hold:(a) ι ˚ r Y p Div qs “ r X p Div qs ;(b) c p T IG q “ p n ` ´ k qr X p Div qs and c p T IG ev q “ p n ` ´ k qr Y p Div qs .(c) ż r X p s k qs c p T X q “ n ` ´ k if X “ IG;2 n ` ´ k if X “ IG ev . Proof.
A more general version of the first identity was proved by Pech in her thesis [36, Prop.2.9]. The explicit calculation of the class of the tangent bundle in the even case can befound e.g. in [6]. In the odd case, the calculation is implicit Pech’s work (cf. [37, Prop.13], see also [36, Prop. 2.15]). Part (c) is a standard calculation based on the fact that ş IG ev r Y p Div qs X r Y p s k qs “ (cid:3) The points of the moduli space are (equivalence classes of) stable maps f : p C, pt , . . . , pt r q Ñ X of degree d , where C is a tree of P ’s and pt i P C are non-singular points. The modulispace M ,r p X , d q comes equipped with r evaluation maps ev i : M ,r p X , d q Ñ X sending p C, pt , . . . pt r ; f q to f p pt i q . For γ , . . . , γ r P H ˚ T p X q , the r -point, genus 0, (equivariant) GWinvariant is defined by x γ , . . . , γ r y d : “ ż T r M ,r p X ,d qs vir ev ˚ p γ q Y ev ˚ p γ q Y . . . Y ev ˚ r p γ r q P H ˚ T p pt q , where r M ,r p X , d qs virT P H T M ,r p X ,d q p M ,r p X , d qq is the virtual fundamental class . If X “ IG ev (or, more generally, X “ G { P ) then the moduli space M ,r p X , d q is an irreduciblealgebraic variety [19, 41], and the virtual fundamental class coincides to the fundamentalclass. In the (non-homogeneous) case X “ IG, and when d “
1, Pech used obstructiontheory to prove the following (cf. [36, Proposition 2.15]; for k “ Proposition 4.2.
Let r “ , , . Then the moduli space of stable maps M ,r p IG p k, n ` q , q is a smooth, irreducible, algebraic variety of complex dimension k p n ` ´ k q´ k p k ´ q `p n ` ´ k q ` r ´ . The GW invariants satisfy the “divisor axiom” property: if r D s T P H T p X q is a class incomplex codimension 1 then for any γ , . . . , γ r P H ˚ T p X q ,(4) xr D s T , γ , . . . , γ r y d “ pr D s X d qx γ , . . . , γ r y d . The (equivariant) quantum cohomology ring.
The quantum cohomology ringQH p IG q of IG : “ IG p k, n ` q is a graded Z p pt qr q s -algebra with a Z r q s -basis given bySchubert classes r X p u qs , where u P W P X W odd . The multiplication is given by r X p u qs ‹ r X p v qs “ ÿ d ě w P W P q d c w,du,v r X p w qs , where c w,du,v “ xr X p u qs , r X p v qs , r X p w qs _ y d is the GW invariant. The degree of q isdeg q “ ż r X p Div qs _ c p T IG p k, n ` q q “ n ` ´ k, by Lemma 4.1 above. The grading is equivalent to the requirement thatcodim X p u q ` codim X p v q “ codim X p w q ` d ¨ deg q. The quantum cohomology ring is a deformation of the ordinary cohomology ring, in thesense that if one makes q “ H ˚ T p IG q .As before there is an equivariant version of the quantum cohomology ring, denotedQH ˚ T p IG q , which deforms the multiplication in H ˚ T p IG q . This is a graded, free algebraover H ˚ T p pt qr q s with a basis given by equivariant Schubert classes r X p u qs T , where u variesin W P X W odd . The multiplication is defined as before, using the equivariant GW invariants.The structure constants c w,du,v P H ˚ T p pt q are homogeneous polynomials of polynomial degreedeg c w,du,v “ codim X p u q ` codim X p v q ´ codim X p w q ´ d ¨ deg q. If this degree equals 0, then one recovers the structure constant from the ordinary (non-equivariant) quantum cohomology ring.5.
The moment graph
Sometimes called the GKM graph, the moment graph of a variety X with a torus T actionhas a vertex for each T -fixed locus of X , and an edge for each 1-dimensional torus orbit.The description of the moment graphs for flag manifolds is well known, and it can be founde.g in [23, Ch. XII]. In this note we consider the moment graphs for IG : “ IG p k, n ` q Ă IG ev : “ IG p k, n ` q . Let P : “ P k Ă Sp n ` be the maximal parabolic for IG ev . Theminimal length representatives in w P W P are in one to one correspondence to sequences1 ď w p q ă . . . ă w p k q ď ¯1. Those corresponding to the odd-symplectic Grassmannian QUIVARIANT QUANTUM COHOMOLOGY OF THE ODD SYMPLECTIC GRASSMANNIAN 11 satisfy in addition that w p i q ď ¯2 for 1 ď i ď n `
1. The moment graph of IG ev has a vertexfor each w P W P , and an edge w Ñ ws α for each α P R ` z R ` P k “ t t i ´ t j : 1 ď i ď k ă j ď n ` u Y t t i ` t j , t i : 1 ď i ă j ď n ` , i ď k u . Geometrically, this edge corresponds to the unique torus-stable curve C α p w q joining w and ws α . This curve has degree d , where α _ ` ∆ _ P “ dα _ k ` ∆ _ P . The moment graph of IG is thefull subgraph of that of IG ev determined by the vertices w P W P X W odd . Notice that theorbits of T and T n ` coincide, therefore we do not distinguish between the moment graphsfor these tori. For later use, we list below the vertices adjacent to the identity element inthe moment graph of IG ev , together with the degrees of the corresponding curves. Recallthe convention ¯ s “ n ` ´ s . For now we let k ą p ă ă ¨ ¨ ¨ ă i ´ ă i ` ă ¨ ¨ ¨ ă k ă j q where k ă j ď n ` p ă ă ¨ ¨ ¨ ă i ´ ă i ` ă ¨ ¨ ¨ ă k ă ¯ j q where n ` ă ¯ j ď k ` p ă ă ¨ ¨ ¨ ă i ´ ă i ` ă ¨ ¨ ¨ ă j ´ ă j ` ă ¨ ¨ ¨ ă k ă ¯ j ă ¯ i q ;(iv) p ă ă ¨ ¨ ¨ ă i ´ ă i ` ă ¨ ¨ ¨ ă k ă i q .The edge in (i) corresponds to α “ t i ´ t j , those in (ii) and (iii) to α “ t i ` t j and thatin (iv) to α “ t i . In particular, only the edge in (iii) has degree 2, and the others havedegree 1. If k “
1, the case (iii) does not apply, and the remaining vertices in cases (i), (ii)and (iv) are respectively p j q , p ¯ j q and p ¯1 q . The figure below illustrates the moment graphsof IG p , q and IG p , q . 6. Curve Neighborhoods
Let X P t IG , IG ev u , let d P H p X q be an effective degree, and let Ω Ă X be a closedsubvariety. Consider the moduli space of stable maps M , p X , d q with evaluation mapsev , ev . The curve neighborhood of Ω is the subschemeΓ d p Ω q : “ ev p ev ´ Ω q Ă X endowed with the reduced scheme structure. This notion was introduced by Buch, Chaput,Mihalcea and Perrin [4] to help study the quantum K theory ring of cominuscule Grass-mannians. It was analyzed further for any homogeneous space by Buch and Mihalcea [8], inrelation to 2-point K-theoretic GW invariants, and to a new proof of the quantum Chevalleyformula. Often, estimates for the dimension of the curve neighborhoods provide vanishingconditions for certain GW invariants. In this paper we will use this technique to provevanishing of “Chevalley” GW invariants of degree d ě d p Ω q must be a (finite) union of Schubert varieties, stable under the same Borel subgroup.This follows because Ω is stable under the appropriate Borel subgroup, and ev , ev areproper, equivariant maps; thus Γ d p Ω q is closed and Borel stable. Further, it was provedin [4] that the curve neighborhood Γ d p Y p w qq of any Schubert variety is again a Schubertvariety. This Schubert variety was described in [8]: Γ d p Y p w qq “ Y p w ¨ z d W P q , where z d P W is defined by the condition that Γ d p .P q “ Y p z d W P q . We recall next a recursive formulafor z d . Recall also that q n ` denotes the quantum parameter for QH ˚ p IG ev q and it hasdegree 2 n ` ´ k . The maximal elements of the set t β P R ` z R ` P : β _ ` ∆ _ P ď d u arecalled maximal roots of d . The root β P R ` z R ` P is called P -cosmall if β is a maximal rootof β _ ` ∆ _ P P H p IG p k, n ` qq . In type C n ` , the P -cosmall roots are the roots 2 t i for1 ď i ď n `
1, and t i ´ t j for 1 ď i ă j ď n `
1. The following follows from [8, Corollary4.12, Theorem 6.2, Theorem 5.1, and Theorem 7.2].
Figure 1.
The figure is the moment graph of IG p , q without degree la-bels. The blue portion corresponds to vertices outside the Schubert varietyIG p , q . The red portion is inside the closed orbit IG p , q “ P . p ¯3 ă ¯2 qp ă ¯2 qp ă ¯3 q p ă ¯2 qp ă q p ă ¯3 qp ă qp ă qp ă ¯1 qp ă ¯1 q p ă ¯1 qp ă ¯1 q Proposition 6.1.
Let d P H p IG p k, n ` qq be an effective degree and w P W P . Then thefollowing hold: (1) If α P R ` ´ R ` P is a maximal root of d , then s α ¨ z d ´ α _ W P “ z d W P ; (2) dim Γ d p Y p w qq ď ℓ p w q ` ℓ p z d W P q ď ℓ p w q ` d ¨ deg q n ` ´ . Furthermore, if thesecond equality occurs then d “ α _ ` Z ∆ _ P and α is a P -cosmall root. Corollary 6.2. (a) If k ą then there is an equality z W P “ s t W P and the minimallength representative of z W P is p ă ă ¨ ¨ ¨ ă k ă q .(b) There is an inequality ℓ p z d W P q ď d deg q n ` ´ with equality if and only if d “ .(c) If k ą and d “ then z W P “ s t ` t W P and ℓ p z W P q “ q n ` ´ .(d) If k “ then z W P “ z W P and ℓ p z W P q “ n ` ă q n ` ´ .Proof. The first part follows directly from the part (1) of the proposition. The equality in(b) follows by direct calculation of ℓ p s t W P q , using its minimal length representative. Acalculation of degrees of roots shows that no degree d ě QUIVARIANT QUANTUM COHOMOLOGY OF THE ODD SYMPLECTIC GRASSMANNIAN 13
For part (c), notice that 2 t is a maximal root of d “
1, therefore z W P “ s t W P . Bythe recursion in Proposition 6.1 we obtain z W P “ s t ¨ s t W P . Now observe the following: s t ¨ s t “ p s . . . s n ` . . . s q ¨ p s . . . s n ` . . . s q “ s ¨ s t ¨ s ¨ s t ¨ s “ s ¨ s t ` t ¨ s “ s t ` t ¨ s ¨ s “ s t ` t ¨ s . Since s P W P , the above shows that z W P “ s t ` t W P as claimed. The equality ℓ p z W P q “ q n ` ´ s t ` t W P . Finally, part p d q follows from the observation that if k “
1, then IG p , n ` q “ P n ` , and then Γ p id q “ P n ` , thus ℓ p z W P q “ dim IG p , n ` q . (cid:3) In what follows we give estimates for the dimension of the curve neighborhoods of Schu-bert varieties X p w q Ă IG, using known estimates for the dimension in the even case. Wewill need the following lemma.
Lemma 6.3.
Let w “ p w p q ă w p q ă ¨ ¨ ¨ ă w p k qq P W P . Then ℓ p w ¨ z W P q “ ℓ p w q ` ℓ p z W P q if and only if w p q “ . In particular, dim Γ p Y p w qq “ ℓ p w q ` deg q n ` ´ if andonly if Y p w q Ă X c is a Schubert variety in the closed orbit of IG .Proof. Let z : “ p ă ă ¨ ¨ ¨ ă k ă q P W P be the minimal length representative of z W P .By Lemma 2.3, ℓ p w ¨ z W P q “ ℓ p w q ` ℓ p z W P q if and only if the product wz is reduced andit is a minimal length representative. We calculate wz “ p w p q , w p q , . . . , w p k q , w p q , w p k ` q , . . . , w p n ` qq . If w p q “ wz P W P , and one checks ℓ p wz q “ ℓ p w q ` ℓ p z q .Conversely, if w ¨ z “ wz P W P , then one uses the bijection between W P and the strictpartitions described in § ℓ p wz q ´ ℓ p w q “ n ` ´ w p q ´ k ` t j : w p q ` w p j q ą n ` u . The length condition forces w p q “
1. (For a similar proof see Proposition 11.4 below). (cid:3)
Curve neighborhoods for IG p k, n ` q . Let w P W P X W odd and let d P H p IG q be an effective degree. As mentioned above, the curve neighborhood Γ d p X p w qq of X p w q isa closed, B -stable subvariety of IG, therefore it must be an union of Schubert varieties:Γ d p X p w qq “ X p w q Y ¨ ¨ ¨ Y X p w r q where w i P W P X W odd . As noticed in [8, § w i canbe determined combinatorially from the moment graph. Proposition 6.4.
Let w P W P X W odd . In the moment graph of IG p k, n ` q , let t v , ¨ ¨ ¨ , v s u be the maximal vertices in the moment graph which can be reached from any u ď w using a path of degree d or less. Then Γ d p X p w qq “ X p v q Y ¨ ¨ ¨ Y X p v s q .Proof. Let Z w,d “ X p v q Y ¨ ¨ ¨ Y X p v s q . Let v : “ v i P Z w,d be one of the maximal T -fixedpoints. By the definition of v i ’s and the moment graph there exists a chain of T -stablerational curves of degree less than or equal to d joining u ď w to v . It follows that v P Γ d p X p w qq , thus X p v q Ă Γ d p X p w qq , whence Z w,d Ă Γ d p X p w qq .For the converse inclusion, let v P Γ d p X p w qq be a T -fixed point. By [25, Lemma 5.3] thereexists a T -stable curve joining a fixed point u P X p w q to v . This curve corresponds to apath in the moment graph of IG p k, n ` q , thus v P Z w,d . Since Bruhat order is compatiblewith inclusion of Schubert varieties, this completes the proof. (cid:3) In what follows we will obtain estimates for the dimension of the curve neighborhoodΓ d p X p w qq , using estimates obtained in the even case. We start with the observation thatthe “odd” curve neighborhoods are proper subvarieties of the “even” ones. Lemma 6.5.
Let w P W P X W odd and d ě an effective degree. Then there is a strictinclusion Γ d p X p w qq Ĺ Γ d p Y p w qq .Proof. Consider the identity 1 .P P Γ d p X p w qq . There is a T -stable degree 1 curve (i.e.a line) in IG p k, n ` q that contains the T -fixed points 1 .P and p ă ă ¨ ¨ ¨ ă ¯1 q P IG p k, n ` qz IG p k, n ` q . (cid:3) The next result is the key technical requirement needed for the vanishing of certainChevalley GW invariants.
Theorem 6.6.
Let w P W P X W odd . Then the following inequalities hold: dim Γ p X p w qq ´ dim X p w q ď deg q ´ d p X p w qq ´ dim X p w q ă d deg q ´ for all d ě Further, if the Schubert variety X p w q is not contained in the closed orbit X c of IG then dim Γ p X p w qq ´ dim X p w q ă deg q ´ . Proof.
Recall that deg q n ` “ deg q `
1. If d “
1, by Lemma 6.5 and Proposition 6.1dim Γ p X p w qq ` ´ dim X p w q ď dim Γ p Y p w qq ´ dim Y p w q ď deg q n ` ´ p X p w qq ´ dim X p w q ď deg q n ` ´ “ deg q ´
1. Let now d “
2. If k ą p X p w qq´ dim X p w q ď dim Γ p Y p w qq´ ´ ℓ p w q ď ℓ p z W P q´ ď q n ` ´ ă q ´ . For arbitrary d , let Γ d p X p w qq “ X p v q Y ¨ ¨ ¨ Y X p v s q . Then each v i is joined to some u i ď w in the moment graph of IG p k, n ` q by j edges of degrees d i P t , u , where ř ji “ d i ď d .By applying repeatedly the estimates for d “ , X p v i q ´ dim X p w q ď dim X p v i q ´ dim X p u i q ď j ÿ i “ p d i deg q ´ q ď d deg q ´ j. If j ě j “ d P t , u , a case treated before.This proves the first two inequalities. For the last inequality, we notice that the hypothesisimplies that w is determined by a sequence p w p q ă ¨ ¨ ¨ ă w p k qq such that w p q ą
1. Thenby Lemma 6.3 combined with Proposition 6.1 we obtaindim Γ p X p w qq ´ dim X p w q ď dim Γ p Y p w qq ´ ´ ℓ p w q ă deg q n ` ´ “ deg q ´ . This finishes the proof. (cid:3) Vanishing of Chevalley Gromov-Witten invariants
The main result of this section is the following.
Theorem 7.1.
Let d ě be a degree in H p IG p k, n ` qq . Let X p v q , X p w q Ă IG p k, n ` q betwo Schubert varieties and X p Div q the Schubert divisor. If dim Γ d p X p v qq ă ℓ p v q` d deg q ´ then the equivariant GW invariant xr X p Div qs T , r X p v qs T , r X p w qs _ T y d “ . In particular, the equivariant Gromov-Witten invariant above vanishes if either d ě or if d “ and X p v q is not included in the closed orbit X c Ă IG p k, n ` q . QUIVARIANT QUANTUM COHOMOLOGY OF THE ODD SYMPLECTIC GRASSMANNIAN 15
Proof.
By the divisor axiom xr X p Div qs T , r X p v qs T , r X p w qs _ T y d “ d xr X p v qs T , r X p w qs _ T y d . By definition, xr X p v qs T , r X p w qs _ T y d “ ż T r M , p IG p k, n ` q ,d q vir s T ev ˚ r X p v qs T Y ev ˚ r X p w qs _ T “ ż T IG p k, n ` q r X p w qs _ T X p ev q ˚ p ev ˚ r X p v qs T X r M , p IG p k, n ` q , d q vir s T q . The cycle p ev qp ev ´ r X p v qsq is supported on the curve neighborhood Γ d p X p v qq , and thepush-forward p ev q ˚ p ev ˚ r X p v qs T X r M , p IG p k, n ` q , d q vir s T q is non-zero only if the curveneighborhood has components of dimensionexpdim M , p IG p k, n ` qq ´ codim X p v q “ deg q d ´ ` ℓ p v q . However, the hypothesis implies that dim Γ d p X p v qq is strictly less than this quantity. Thelast statement follows from Theorem 6.6. (cid:3) Lines in IG p k, n ` q As before, we set IG : “ IG p k, n ` q . If k ‰ n ` n ` acts with two orbits: X o (the open orbit) and X c (the closed orbit). If k “ n ` n ` , and IG “ X c is isomorphic to the Lagrangian Grassmannian IG p n, n q . Allstatements remain true in this case after making X o “ H , with almost identical proofs.According to Theorem 7.1, the only equivariant GW invariants xr X p Div qs T , r X p v qs T , r X p w qs _ T y d which maybe non-zero are those when d “ X p v q is included inthe closed orbit X c » IG p k ´ , n q . To calculate these invariants, we will analyze thegeometry of the moduli spaces of stable maps M ,r p IG , q Ñ IG where r “ ,
2, and thegeometry of the
Gromov-Witten varieties GW p w q : “ ev ´ p X p w qq Ă M , p IG , q . For X p w q Ă X c , we will show that GW p w q is a scheme which has 2 irreducible, genericallyreduced, components. One component parametrizes lines in IG contained in the closedorbit X c , and the other those lines which intersect the open orbit X ˝ . The restrictionof the evaluation maps to each of these components will be a surjective map, which iseither birational, or it has general fiber of positive dimension. We will deduce from thisthat the curve neighborhood Γ p X p w qq has two components, and that if non-zero, the GWinvariant is equal to 1 precisely in the cases when r X p w qs _ is Poincar´e dual to one of thesecomponents.From now on, a line in X will mean an irreducible, reduced, curve of degree 1. Recallthat there is a sequence of embeddingsIG p k, n ` q Ă IG p k, n ` q Ă Gr p k, n ` q Ă P p k ľ C n ` q where the last is the Pl¨ucker embedding. The image of a line in IG under the compositionof these embeddings is a projective line. Indeed, a calculation in coordinates shows that theimage of the Schubert curve in IG is the Schubert curve in Gr p k, n ` q , and the image ofthis Schubert curve is a projective line. Let r “ , ,
3. We recall from Prop. 4.2 that M ,r p IG , q is a non-singular, irreduciblescheme of dimensiondim M ,r p IG , q “ dim IG ` deg q ` r ´ “ k p n ` ´ k q ´ k p k ´ q ` n ` ´ k ` r ´ . There is a natural isomorphism M , p Gr p k, n ` q , q » Fl p k ´ , k, k `
1; 2 n ` q (a 3-step flagvariety) such that the evaluation map ev is the projection π k : Fl p k ´ , k, k `
1; 2 n ` q Ñ Gr p k, n ` q . To see the isomorphism explicitly, one can use e.g. the kernel-span techniqueof Buch [5] to observe that to any line L Ă Gr one can associate its kernel K : “ Ş V P L V and its span S : “ Span t V : V P L u , which have dimension k ´
1, respectively k ` p p P L q is sent to p ker L, p,
Span L q . Although logically not neededin what follows, we remark that one can identify M , p IG , q to a subvariety of the three-step flag variety, by noticing that if V k ´ P IG p k ´ , n ` q is a kernel of a line, then atriple p V k ´ Ă V k Ă V k ` q corresponds to a line in IG if and only if V k ´ is isotropic and V k ` Ă V K k ´ . Therefore M , p IG , q can be identified set theoretically with M , p IG , q » tp V k ´ Ă V k Ă V k ` q : V k ´ P IG p k ´ , n ` q , V k ` Ă V K k ´ u . Under this identification, ev corresponds to the projection to the component V k .8.1. Lines intersecting the open orbit X ˝ . Consider the open subvariety M ˝ Ă M , p IG , q parametrizing 1-pointed lines intersecting the open orbit X ˝ Ă IG: M ˝ : “ tp p, L q : L X X ˝ ‰ Hu . Since the kernel of a line L intersecting X ˝ cannot contain e (which spans the kernel of theodd symplectic form), the variety M ˝ can be realized as the flag bundle F ℓ p , S K k ´ { S k ´ q over the open orbit IG p k ´ , n ` q ˝ , where S k ´ denotes the tautological subbundle. Inthis case rank p S K k ´ q “ n ` ´ p k ´ q . Let π : M ˝ Ñ X denote the natural projectionmap. Key to the calculation of the GW invariants is the following result, analyzing thegeometry of the fibres of π . Theorem 8.1. (a) The natural projection map π : M ˝ Ñ IG p k, n ` q is surjective, andall its fibers are irreducible, generically smooth, of dimension dim M ˝ ´ dim IG p k, n ` q .(b) The inverse image π ´ p X c q is isomorphic to an Sp n ` orbit in IF p k ´ , k, k `
1; 2 n ` q . In particular, it is smooth and irreducible. Before proving the theorem, we recall the description of the Sp n ` -orbits of the odd-symplectic 3-step partial flag variety IF p k ´ , k, k `
1; 2 n ` q : K “ t V k ´ Ă V k Ă V k ` P IF p k ´ , k, k `
1; 2 n ` q : e P V k ´ u K “ t V k ´ Ă V k Ă V k ` P IF p k ´ , k, k `
1; 2 n ` q : e P V k , e R V k ´ u K “ t V k ´ Ă V k Ă V k ` P IF p k ´ , k, k `
1; 2 n ` q : e P V k ` , e R V k u K “ t V k ´ Ă V k Ă V k ` P IF p k ´ , k, k `
1; 2 n ` q : e R V k ` u . We also need the following lemma:
Lemma 8.2.
Let L be a line such that L X X c ‰ H and L X X ˝ ‰ H . Then Span L is anisotropic subspace in C n ` .Proof. Let x P L X X c and y P L X X ˝ . Since x P X c and y R X c we can choose a basis t e , x , ¨ ¨ ¨ , x k ´ u for x such that t x , ¨ ¨ ¨ , x k ´ u is a basis for x X y and choose a basis t x , ¨ ¨ ¨ , x k ´ , f u for y . Then t e , x , ¨ ¨ ¨ , x k ´ , f u is a basis for Span L “ x x, y y . Clearly x x i , f y “ e P ker ω it follows that x e , f y “
0. This finishes the proof. (cid:3)
QUIVARIANT QUANTUM COHOMOLOGY OF THE ODD SYMPLECTIC GRASSMANNIAN 17
We note that this is the best result possible. For instance let n “ k “ T -fixed points p ă q and p ă q (this is a line included inthe closed orbit X c » IG p , q » P ). Then Span L “ h e , e , e i is not isotropic, because ω p e , e q “
1. Similarly, the line joining p ă q to p ă ¯2 q (a line in the open orbit) hasagain non-isotropic span; see figure 2 below for more examples.We will need to calculate dim K . For that, observe that to construct a triple in K onefirst chooses V k P IG p k, n ` q c » IG p k ´ , n q , then V k ´ in an open set in Gr p k ´ , V k q ,and then finally an open set of V k ` P Gr p , V K k ´ { V k q . (The spaces V k ` obtained this wayare automatically isotropic, because e P V k .) This yields(5)dim K “ dim IG p k ´ , n q`p k ´ q`p n ´ k ` q “ p k ´ qp n ´ k ` q´ p k ´ qp k ´ q ` n ´ k. Proof of Theorem 8.1.
The definition of M ˝ implies that π is surjective over the open orbit X ˝ . By [4, Prop. 2.3] this is a locally trivial fibration, and because both M ˝ and X ˝ aresmooth and irreducible, it follows that the fibers over X ˝ are also smooth and irreducible.Notice that the same result implies that π ´ p X c q is a locally trivial fibration over X c . Toprove (a) it remains to show that the fibre π ´ p .P q is nonempty, irreducible and genericallysmooth.As explained in §
5, there is a line joining 1 .P to x e , e , . . . , e k , e k ` y P X ˝ . Thus π ´ p .P q ‰ H . We prove next that the reduced support p π ´ p X c qq red is irreducible, whichimplies that π ´ p X c q is again irreducible. Then we will use a local calculation to find anopen dense set of π ´ p .P q where it is smooth. In the process we will simultaneously proveboth (a) and (b).To start, there is a bijective morphism K Ñ p π ´ p X c qq red defined as follows: to eachpointed line p p P L q in IG such that p P L X X c and L X X ˝ ‰ H one associates theelement p ker L, p,
Span L q P IF odd p k ´ , k, k `
1; 2 n ` q . (The fact that the span of L isisotropic follows from Lemma 8.2.) Conversely, to each element p V k ´ Ă V k Ă V k ` q P K one associates the line L : “ P p V k ` { V k ´ q and the point V k P X c . Since V k ` is isotropic itfollows that L is a line in IG; the condition e R V k ´ implies that L cannot be included in theclosed orbit, so L X X ˝ ‰ H . The fact that this is an algebraic morphism follows e.g. because K is an orbit of Sp n ` . This proves that π ´ p X c q is irreducible. Since π ´ p X c q Ñ X c isa locally trivial fibration, it follows that π ´ p .P q is irreducible, and that it has dimensiondim π ´ p .P q “ dim K ´ dim X c “ n ´ k “ dim M ˝ ´ dim IG . Turning to smoothness, we will show that there exist open sets U Ă IG and U Ă M ˝ suchthat 1 .P P U , U Ă π ´ p U q , U i ’s are isomorphic to open sets in some affine spaces A N i , i “ , N i ), and such that the induced map U Ñ U is smooth. Usingthe coordinate charts in Gr p k, n ` q one defines the open set U around 1 .P to be givenby the column space of the matrix ˆ I k A ˙ where A “ p a i,j q is a p n ` ´ k q ˆ k matrix.The isotropy constraints on the coordinates can be arranged in a triangular system withequations of the form a i,j ` quadratic terms “
0, where 1 ď j ď k ´ k ď i ď j ` U is isomorphic to an affine space A dim IG .To define U , observe that an open set in the dual projective space of codimension 1 sub-spaces V k ´ Ă V k “ x v , . . . , v k y P U where e R V k ´ is given by h v i ` c i v : 2 ď i ď k, c i P C i .Then an open set U around triples containing such V k is given by the column span of the matrix C : “ p C | C | . . . | C k | C k ` q where C i are column vectors in C n ` , defined as follows: C k “ e ` ¯2 ÿ j “ k ` a j, e j ; C k ` “ e k ` ` ¯2 ÿ j “ k ` d j e j , and C i “ c i ` e ` e i ` ` ¯2 ÿ j “ k ` p a j,i ` ` c i ` a j, q e j ; 1 ď i ď k ´ . By definition, the span Σ k ´ : “ span p C , . . . , C k ´ q of the first k ´ C k and C k ` are perpendicular to Σ k ´ ; the projectionto IG p k ´ , n ` q ˝ sends the matrix C to Σ k ´ . The isotropy conditions translate intolinear constraints which determine the coordinates d ¯2 , . . . , d ¯ k and the coordinates a i,j , where1 ď j ď k ´ k ď i ď j ` U ).There are p n ` ´ k qp k ´ q ` n ´ k ` k p k ´ q ` k ´ U » A dim M ˝ . In these coordinates themap π U : U Ñ U becomes the linear map given by c i ÞÑ d i ÞÑ
0. In particular, thismap is smooth, and the fiber π ´ p .P q X U is smooth. This finishes the proof. (cid:3) Example . We illustrate the local calculation for k “ n “
3. The open sets U Ă IG p , q and U Ă M ˝ » F ℓ p , S K { S q (a flag bundle over IG p , q ˝ ) are given by: U “ ¨˚˚˚˚˚˚˚˚˝ c c a , ` c a , a , ` c a , a , a ¯4 , ` c a ¯4 , a ¯4 , ` c a ¯4 , a ¯4 , d ¯4 a ‚ ¯3 , ` c a ‚ ¯3 , a ¯3 , ` c a ‚ ¯3 , a ‚ ¯3 , d ‚ ¯3 a ¯2 , ` c a ‚ ¯2 , a ¯2 , ` c a ‚ ¯2 , a ‚ ¯2 , d ‚ ¯2 ˛‹‹‹‹‹‹‹‹‚ π U ÝÑ U “ ¨˚˚˚˚˚˚˚˚˝ a , a , a , a ¯4 , a ¯4 , a ¯4 , a ‚ ¯3 , a ‚ ¯3 , a ¯3 , a ‚ ¯2 , a ¯2 , a ¯2 , ˛‹‹‹‹‹‹‹‹‚ The coordinates with ‚ are determined from linear equations, using the isotropy contraints.For instance, a ‚ ¯2 , in U is determined by imposing that the first and second column arepependicular, i.e. a ‚ ¯2 , ¨ ` a ¯4 , ¨ a , ´ a , ¨ a ¯4 , “ . The third and fourth column vectors from U are each perpendicular to the first two columnvectors. The dimension of U is 17 (coordinates) - 5 (linear constraints) “
12, which equalsdim M , p IG p , q , q , as claimed.8.2. Lines in the closed orbit.
Set M : “ M , p IG , q and consider the closed subvariety M c : “ M z M ˝ “ tp p, L q P M , p IG , q : L Ă X c u , which consists of lines included in the closed orbit. In terms of triples of flags this consistsof triples p V k ´ Ă V k Ă V k ` q such that V k ´ belongs to the closed orbit in IG p k ´ , n ` q (i.e e P V k ´ ), and V k ` Ă V K k ´ . Since e spans the kernel of the odd-symplectic form ω , this is a smooth subvariety of Fl p k ´ , k, k `
1; 2 n ` q , and the universal property forthe moduli space of stable maps gives a bijective morphism M c Ñ M , p X c , q . It followsthat M c is isomorphic to the moduli space M , p X c , q . Recall that X c is isomorphic to thehomogeneous space IG p k ´ , n q , thus the moduli space M , p X c , q is a smooth, irreduciblevariety of dimensiondim M c “ dim IG p k ´ , n q ` n ` ´ p k ´ q ´ “ dim IG p k ´ , n q ` n ´ k. QUIVARIANT QUANTUM COHOMOLOGY OF THE ODD SYMPLECTIC GRASSMANNIAN 19 (Note the coincidence dim M c “ dim K .) We recall the following result, proved in Thm.2.5 and Cor. 3.3 from [4]: Lemma 8.4.
For every V P X c , the fibre ev ´ p V q of the restricted map ev : M , p X c , q Ñ X c is an irreducible, normal variety of dimension dim M c ´ dim X c . We combine the previous lemma to Theorem 8.1 to obtain the main result of this section.
Theorem 8.5.
Consider the evaluation map ev : M , p IG , q Ñ IG . Then the followinghold:(a) For any V P IG , the fibre ev ´ p V q is pure dimensional of dimension dim M , p IG , q´ dim IG , and each of its components is generically smooth. In particular, ev is flat.(b) For any Schubert variety X p w q Ă X c , the preimage ev ´ p X p w qq has two irreduciblecomponents: ev ´ p X p w qq : “ A Y A , where A is the closure of the subvariety of pointed lines p p, L q such that L X X ˝ ‰ H ,and A is the closed subscheme corresponding to p p, L q such that L is included in the closedorbit X c . Further, each irreducible component is generically smooth of expected dimension dim M , p IG , q ´ codim IG X p w q .Proof. Since Sp n ` acts transitively on the open orbit X ˝ , the morphism ev is flat, andthe fibres have the stated dimension. Transitivity implies that all fibers over the closed orbitare isomorphic, thus it suffices to take V “ .P . Let F : “ ev ´ p .P q be the fibre. Recallthe notation M ˝ and M c . Clearly F can be written as the disjoint union F “ F ˝ Y F c where F ˝ : “ F X M ˝ is open in F and F c : “ F z F ˝ is closed in M c . It follows from Theorem8.1 that F ˝ is irreducible, generically reduced, and of the stated dimension. On the otherside, Lemma 8.4 implies that F c is irreducible, reduced, of dimensiondim F c “ dim M c ´ dim X c “ dim M ´ dim IG;(the last equality is a simple calculation). Therefore F c cannot in the closure of F ˝ , andthe statements about F hold. The flatness follows from [26, Theorem 23.1], taking intoaccount that both source and target of ev are smooth varieties, and that all fibers havethe same dimension. Flatness implies that the GW variety GW p w q from part (b) is puredimensional of expected dimension. Further, using transitivity and applying [4, Prop. 2.3]to each irreducible component of ev ´ p X c q implies that the map ev : ev ´ p X c q Ñ X c is alocally trivial fibration with fibre F . Then the restriction to ev ´ p X p w qq is a locally trivialfibration over X p w q with fibre F , and the statement in (b) follows. (cid:3) Lines with two marked points.
Define ξ : M , p IG , q ÝÑ M , p IG , q to be themap forgetting the second marked point. Proposition 8.6.
The forgetful map ξ : M , p IG , q ÝÑ M , p IG , q is a locally trivial P -fibration.Proof. Consider the embedding IG Ă Gr : “ Gr p k, n ` q . We first prove the statementwith IG replaced by Gr. Recall that the moduli space M , p Gr , q may be identified to thepartial flag manifold Fl p k ´ , k, k `
1; 2 n ` q . It follows in particular that M , p Gr , q admits a transitive action of SL : “ SL n ` . Then by [4, Prop. 2.3] the forgetful map Another way to see that F c Ĺ F ˝ is to notice that every line in F ˝ has isotropic span, therefore anyline in the closure must satisfy the same property. But we have seen that there exist lines in X c with nonisotropic span. M , p Gr , q Ñ M , p Gr , q is an SL-equivariant locally trivial fibration with fibres isomor-phic to P . Consider the commutative diagram: M , p IG , q FP M , p IG , q M , p Gr , q M , p Gr , q ψ π ξ j π ξ j where F P denotes the fibre product and j , j are the closed embeddings determined by theembedding IG Ă Gr. The map ψ is determined by the universal property for fibre products.It is easy to check that ψ is bijective. Since both M , p IG , q and F P are smooth varieties ψ is in fact an isomorphism, by Zariski’s Main Theorem. Since the right vertical arrow isa P -fibration, so is the left vertical arrow F P » M , p IG , q Ñ M , p IG , q . This provesthe statement. (cid:3) Combining Proposition 8.6 and Theorem 8.5 imply the main result of this section. Re-call that GW p w q denotes the Gromov-Witten variety ev ´ p X p w qq . Obviously ev is thecomposition of the forgetful map ξ with the evaluation map from M , p IG , q . Corollary 8.7.
Consider a Schubert variety X p w q Ă X c . Then the Gromov-Witten variety GW p w q has two irreducible components GW p w q “ GW p q p w q Y GW p q p w q , where GW p q p w q “ ξ ´ p A q is the closure of the subvariety corresponding to lines L suchthat L X X ˝ ‰ H , and GW p q p w q “ ξ ´ p A q is the closed subscheme corresponding to lines L included in the closed orbit X c . Further, each irreducible component is generically smoothand it has dimension dim M , p X, q ´ codim X X p w q . Line neighborhoods
In this section we analyze the curve neighborhoods Γ p w q : “ Γ p X p w qq (i.e. the lineneighborhoods ) in the case when X p w q Ă X c . By Theorem 7.1 these are the only oneswhich may contribute to non-zero GW invariants. By Corollary 8.7, the Gromov-Wittenvariety GW p w q has two components, each of expected dimension. It follows that the curveneighborhood Γ p w q has at most two components, and we have an equalityΓ p w q “ Γ p q p w q Y Γ p q p w q , where ev : GW p i q p w q Ñ Γ p i q p w q : “ ev p GW p i q p w qq ( i “ , p q p w qX X ˝ ‰H , Γ p q p w q Ă X c , and each of Γ p i q p w q is irreducible and stable under the standard Borelsubgroup; therefore each must be a Schubert variety. Further, since the second component GW p q p w q is the GW variety of lines in the closed orbit X c - isomorphic to the homogeneous QUIVARIANT QUANTUM COHOMOLOGY OF THE ODD SYMPLECTIC GRASSMANNIAN 21 space IG p k ´ , n q - it follows from Corollary 6.2 that Γ p q p w q “ X p w ¨ O W P q where X p O q “ Γ X c p id q is the line neighborhood of the Schubert point in X c . Next we willidentify the components Γ p i q p w q . Proposition 9.1.
Consider the minimal length representatives O “ p ă ă ¨ ¨ ¨ ă k ă k ` q and O “ p ă ă ă ¨ ¨ ¨ ă k ă q . Then the line neighborhood of the Schubertpoint in IG p k, n ` q is Γ p id q “ X p O q Y X p O q and ℓ p O q “ ℓ p O q “ n ` ´ k . (Observethat this equals deg q ´ .) Before the proof, we contrast the result above to that for curve neighborhoods in ahomogeneous space. For the latter, it was proved in [4] and [8] that any curve neighborhoodof a Schubert variety is a single Schubert variety. For the quasi-homogeneous space IG, thisalready fails for Γ p id q , but we observe that the components correspond naturally to theorbits of Sp n ` on IG. Proof.
The properties of O follow from Corollary 6.2. We observed in § O to the identity. The fact that ℓ p O q “ n ` ´ k follows immediatelyfrom the equation (6) below, where we describe O in terms of partitions. Finally, since ℓ p O q “ deg q ´
1, Theorem 6.6 implies that X p O q is a component of Γ p w q . (cid:3) Theorem 9.2.
Let w “ p w p q ă ¨ ¨ ¨ ă w p k qq P W P X W odd be an odd-symplectic minimallength representative such that X p w q Ă X c (i.e. w p q “ ). Then the cosets w ¨ O W P and w ¨ O W P have representatives in W odd and Γ p X p w qq “ X p w ¨ O W P q Y X p w ¨ O W P q . Proof.
The existence of representatives in W odd follows from Lemma 2.5. To prove theequality, since both sides are B -stable, it suffices to check they have the same T -fixedpoints. Using the action of Sp n ` , a line passing through the Schubert point X p id q canbe translated so it contains any point in the closed orbit. In particular, a T -stable lineguaranteed by Proposition 9.1, joining X p id q to O i W P ( i “ ,
2) is translated to one joiningany T -fixed point v P X p w q to vO i W P . Since the minimal length representatives satisfy v ď w it follows that vO i ď v ¨ O i ď w ¨ O i , therefore Γ p X p w qq Ă X p w ¨ O W P qY X p w ¨ O W P q .For the converse inclusion we will consider only lines L which intersect both X p w q and theopen orbit X ˝ (those included in the closed orbit are already accounted by the equalityΓ p q p w q “ X p w ¨ O W P q ). Let v “ p w ¨ O q O ´ , where the products are performed in W .Then v ď w by [8, Prop.3.1] and vO “ w ¨ O in W . If L is the line joining X p id q to O in IG then v.L joins v P X p w q to vO W P “ p w ¨ O q W P P X p w ¨ O W P q . This proves therequired inclusion. (cid:3) We record an immediate consequence of Lemma 2.3, which gives necessary and sufficientconditions for the components of the curve neighborhood to have the expected dimension.
Lemma 9.3.
Let w P W P X W odd such that X p w q Ă X c , and let z P t O , O u . Then dim X p w ¨ zW P q ´ dim X p w q ď deg q ´ . Furthermore, the following are equivalent:(i) dim X p w ¨ zW P q ´ dim X p w q “ deg q ´ ;(ii) ℓ p w ¨ z q “ ℓ p w q ` ℓ p z q , w ¨ z “ wz and w ¨ z is a minimal length representative in W odd . Gromov-Witten invariants of lines
The main result of this section is the following:
Figure 2.
The moment graph of IG p , q . The thick edges have degree 2and the rest have degree 1. Each red vertex is in X c . The blue edges leavingupwards from each red vertex w are connected to w ¨ O or w ¨ O . p ¯3 ă ¯2 qp ă ¯2 qp ă ¯3 q p ă ¯2 qp ă q p ă ¯3 qp ă qp ă q Theorem 10.1.
Let X p w q Ă X c be a Schubert variety in the closed orbit of X , and let z P t O , O u . Consider the restricted evaluation map ev : GW p i q p w q Ñ X p w ¨ O i q p i “ , q . Then dim GW p i q p w q ě dim X p w ¨ O i W P q with equality if and only if the restricted map ev is birational. In particular, the following holds: p ev q ˚ r GW p i q p w qs T “ r X p w ¨ O i qs T if w ¨ O i P W P X W odd and ℓ p w ¨ O i q “ ℓ p w q ` ℓ p O i q ;0 otherwise . Proof.
By definition X p w ¨ O i W P q “ ev p GW p i q p w qq , therefore the inequality on dimensionsis immediate. In the case of equality it remains to prove the birationality statement. Firstobserve that in this case dim X p w ¨ O i W P q “ dim X p wW P q ` deg q ´
1, and by Lemma 9.3 w ¨ O i is a minimal length representative satisfying ℓ p w ¨ O i q “ ℓ p w q ` ℓ p O i q . Given this, wewill drop W P from the notation.Recall from Corollary 8.7 that GW p i q p w q is irreducible and generically smooth. Sincethe evaluation map ev is B -equivariant, [4, Prop. 2.3] implies that ev is a locally trivialfibration over the open cell X p w ¨ O i q ˝ . The preimage ev ´ p X p w ¨ O i q ˝ q , being open anddense, intersects the smooth locus of GW p i q p w q . Therefore all fibres over the open cell, whichby hypothesis are discrete, must be reduced. To prove birationality it suffices to show thatfor some x P X p w ¨ O i q ˝ there exists a unique line L such that x P L and L X X p w q ‰ H . If i “ i “ QUIVARIANT QUANTUM COHOMOLOGY OF THE ODD SYMPLECTIC GRASSMANNIAN 23
We consider the fibre over x “ wO “ w ¨ O . This fibre contains the line L w , obtainedby w -translating the unique, T -stable, line joining X p id q and O . If X p v q Ă X p w q suchthat v ‰ w thendim Γ p X p w qq ´ dim X p v q “ ℓ p w ¨ O q ´ ℓ p v q “ n ` ´ k ` p ℓ p w q ´ ℓ p v qq ą n ` ´ k. Then theorems 6.6 and 9.2 imply that there is no line joining X p v q to the open cell X p wO q ˝ .We deduce that any line passing through wO and X p w q cannot intersect the boundary X p w qz X p w q ˝ of X p w q . Let L be any line such that wO P L and y P L X X p w q ˝ . If y “ w then L “ L w is T -stable, so assume y ‰ w ; in particular L is not T -stable. Weshow that existence of this L leads to a contradiction. Consider a general C ˚ Ă T suchthat the T and C ˚ fixed points in IG coincide. (Pick the C ˚ to be a regular 1-parametersubgroup as in [17, Ch. 24].) A line t.L in the (infinite) family of lines t t.L : t P C ˚ u contains w ¨ O and it passes through t.y P X p w q ˝ . The limits at 0 and exist by theproperness of the appropriate Hilbert scheme [16, Prop. 3.9.8], and they correspond to twolines passing through two (distinct) T -fixed points lim t Ñ t.y, lim t Ñ8 t.y P X p w q . The twolines are necessarily T -stable, and this contradicts the uniqueness of L w . (cid:3) As a corollary, we can calculate the Chevalley GW invariants not covered by Theorem7.1. Recall that X p Div q denotes the Schubert divisor in IG. Corollary 10.2.
Let u, w P W P X W odd such that X p w q Ă X c . Then the Gromov-Witteninvariant xr X p Div qs T , r X p w qs T , r X p u qs _ T y “ if u “ wO i and ℓ p u q “ ℓ p w q ` ℓ p O i q , and itis equal to otherwise.Proof. As in the proof of Theorem 7.1 we obtain xr X p Div qs T , r X p w qs T , r X p u qs _ T y “ ż T IG p k, n ` q p ev q ˚ p ev ˚ r X p w qs T q X r X p u qs _ T . (We omitted the virtual class, since for d “ is flat. Then by Corollary 8.7,ev ˚ r X p w qs T “ r ev ´ p X p w qqs T “ r GW p q p w qs T ` r GW p q p w qs T . Then the result follows from Theorem 10.1 and Poincar´e duality. (cid:3)
The previous corollary together with Theorem 7.1 give the quantum terms in the equi-variant quantum Chevalley formula for X . Recall that the Chevalley formula is given by r X p Div qs T ‹ r X p w qs T “ ÿ d ě u P W n ` c u,dDiv,w q d r X p u qs T , where c u,dDiv,w is a homogeneous polynomial of degree codim X p w q` ´p codim X p u q` d deg q q .The terms when d “ d ą ˚ p IG p , n ` qq [37] and they were conjectured in few casesfor QH ˚ p IG p , n ` qq [36]. Theorem 10.3.
Let u, w P W P X W odd and d ą . The equivariant quantum Chevalleycoefficients c u,dDiv,w “ for d ě or if w p q ‰ (i.e. X p w q Ę X c ). If d “ and w p q “ then c u, Div,w “ if u “ wO i and ℓ p u q “ ℓ p w q ` ℓ p O i q for i “ , otherwise . In the next section we will rewrite this formula in terms of partitions.
Equivariant Quantum Chevalley Rule with p n ´ k q -strict partitions The goal of this section is to give an explicit formulation of the equivariant quantumChevalley formula using partitions.11.1.
A dictionary permutations - partitions.
In this section we introduce a variantof Buch, Kresch and Tamvakis k -strict partitions [6]. This variant, due to Pech [37], isconvenient to describe the cohomology of the odd-symplectic Grassmannian X “ IG p k, n ` q . Recall that if P k is the maximal parabolic subgroup of Sp n ` determined by the simpleroot α k , then the minimal length representatives W P k have the form p w p q ă w p q ă ¨ ¨ ¨ ă w p k qq . Consider the set of partitions λ “ p n ` ´ k ě λ ě ¨ ¨ ¨ ě λ k ě q which are p n ` ´ k q -strict , i.e. λ j ą λ j ` whenever λ j ą n ` ´ k . We denote this set by Λ n ` k .There is a bijection between Λ n ` k and the set W P k of minimal length representatives givenby: λ ÞÑ w is defined by w p j q “ n ` ´ k ´ λ j ` t i ă j : λ i ` λ j ď p n ` ´ k q ` j ´ i u ,w ÞÑ λ is defined by λ j “ n ` ´ k ´ w p j q ` t i ă j : w p i q ` w p j q ą n ` u . See [6, Proposition 4.3]. Recall that the minimal length representative of the element w defined in (3) indexes IG as a Schubert variety inside IG p k, n ` q . Under the bijectionabove, the coset of w W P k corresponds to the p n ` ´ k q -strict partition 1 k : “ p , , . . . , q if k ă n ` p k, , . . . , q if k “ n `
1. The minimal length representatives for oddsymplectic permutations w P W odd are in bijection with the subset of Λ n ` k consisting ofthose p n ` ´ k q -strict partitions satisfying the additional condition that if λ k “ λ “ n ` ´ k ; in other words, if the first column is not full, then the first row must befull. Pech introduced an equivalent indexing set, which is more convenient in the contextof the odd-symplectic Grassmannians:Λ : “ t λ “ p n ` ´ k ě λ ě ¨ ¨ ¨ ě λ k ě ´ q : λ is n ´ k -strict , if λ k “ ´ λ “ n ` ´ k u . Pictorially, the partitions in Λ are obtained by removing the full first column 1 k from thepartitions in Λ n ` k , regardless of whether a part equal to 0 is present. Example . Let k “ n “
7, and w “ p ă ă ¯8 ă ¯7 ă ¯2 | ă ă q P W odd . Then λ “ p λ ě λ ě λ ě λ ě λ q P Λ n ` is given by λ “ p , , , , q and the correspondingpartition in Λ is p , , , , ´ q . Pictorially, ´ “ Example . Let k “ n ` “
5, so IG p , q » IG p , q is the Lagrangian Grassmannian.Then the codimension 0 class is the ´ λ “ p , ´ , ´ , ´ , ´ q “ .For λ P Λ define | λ | “ λ ` . . . ` λ k . If w corresponds to λ then ℓ p w q “ k p n ` ´ k q ´ k p k ´ q ´ | λ | , i.e. the codimension of the Schubert variety X p w q in X equals | λ | ; see[6, Proposition 4.4] and [37, Section 1.1.1]. The partitions associated to the elements O and O from Proposition 9.1 are: λ p O q “ p n ´ k ě n ´ k ´ ě n ´ k ´ ě ... ě n ´ k ` ě q ; λ p O q “ p n ´ k ` ě n ´ k ´ ě ... ě n ´ k ` ě ´ q . (6) One word of caution: the Bruhat order does not translate into partition inclusion. For example, p n ` ´ k, , . . . , q ď p , , . . . q in the Bruhat order for k ă n ` QUIVARIANT QUANTUM COHOMOLOGY OF THE ODD SYMPLECTIC GRASSMANNIAN 25
A Schubert variety X p w q is included in the closed orbit X c Ă IG if its partition λ P Λsatisfies λ “ n ` ´ k . In order to translate the conditions from Lemma 9.3 in terms ofpartitions we need the following definition. Definition 11.3.
Let λ “ p λ , . . . , λ k q be a partition in Λ such that λ “ n ` ´ k .(a) If λ k ě then let λ ˚ “ p λ ě λ ě ¨ ¨ ¨ ě λ k ě q . If λ k “ ´ then λ ˚ does not exist.(b) If λ “ n ´ k then let λ ˚˚ “ p λ ě λ ě ¨ ¨ ¨ ě λ k ě ´ q . If λ ă n ´ k then λ ˚˚ does not exist. In both situations notice that | λ ˚ | “ | λ ˚˚ | “ | λ | ´ p n ` ´ k q . As an example, if ρ “ p n ´ k ` , n ´ k, . . . , n ´ k ` q is the partition indexing the Schubert point,then λ p O q “ ρ ˚ and λ p O q “ ρ ˚˚ . It is easy to produce examples when only one of λ ˚ or λ ˚˚ exist. For instance, if k “ , n “
4, and λ “ p , , ´ q then λ ˚ does not exist, but λ ˚˚ “ p , ´ , ´ q ; if λ “ p , , q then λ ˚ “ p , , q and λ ˚˚ does not exist. Proposition 11.4.
Let w P W P k X W odd such that w p q “ and let w ÞÑ λ “ p n ` ´ k, λ , . . . , λ k q be the partition in Λ corresponding to w . The following hold:(a) The partition λ ˚ exists if and only if wO is a minimal length representative in W odd . If any of these conditions is satisfied then wO ÞÑ λ ˚ , thus in particular ℓ p wO q “ ℓ p w q ` ℓ p O q .(b) The partition λ ˚˚ exists if and only if wO is a minimal length representative in W odd and ℓ p wO q “ ℓ p w q ` ℓ p O q . In this case wO ÞÑ λ ˚˚ .Proof. By Lemma 2.5 wO i P W odd so one only needs to check the claims about minimallength representatives. Let w “ p ă ă . . . ă j ă w p j ` q ă . . . ă w p k q| w p k ` q ă . . . ă w p n ` qq where j ě w p j ` q ą j ` w p n ` q ď n `
1, since w is a minimal lengthrepresentative; this last condition is omitted if k “ n `
1. Notice that either w p k q “ j ` j ` ď w p k ` q , or w p k q ď j ` w p k ` q “ j `
1. By the definition of O and O , we have wO “ p w p q , w p q , . . . , w p k q , w p k ` q| , w p k ` q , . . . , w p n ` qq ; wO “ p , w p q , ¨ ¨ ¨ , w p k q , w p q| w p k ` q , ¨ ¨ ¨ w p n ` qq , as elements in W . Therefore wO is not a minimal length representative if and only if w p k q ą w p k ` q , i.e. w p k q “ j ` j ` ď w p k ` q . Similarly, wO R W P k if and onlyif w p k q ą w p q .We now proceed to prove the statement (a). If λ ˚ exists but w p k ` q ‰ j `
1, then thepreceding considerations imply that w p k q “ j `
1. Using the bijection W P k X W odd Ñ Λ,we calculate λ k “ n ` ´ k ´ j ` ` t i ă k : w p i q ` j ` ą n ` u“ j ´ k ` t i ă k : w p i q ą j ` u“ j ´ k ` p k ´ ´ j q“ ´ . This contradicts that λ k ě
0. Therefore w p k q ď j ` w p k ` q “ j `
1, which meansthat wO P W P k . Conversely, if wO is a minimal length representative, let wO ÞÑ µ underthe bijection W P k X W odd Ñ Λ. Then for 1 ď s ď k ´ µ s “ n ` ´ k ´ wO p s q ` t i ă s : wO p i q ` wO p s q ą n ` u“ n ` ´ k ´ w p s ` q ` t i ă s : w p i ` q ` w p s ` q ą n ` u . Since w p q “
1, t i ă s : w p i ` q ` w p s ` q ą n ` u “ t i ă s ` w p i q ` w p s ` q ą n ` u , thus µ s “ λ s ` . We calculate µ k separately: µ k “ n ` ´ k ´ wO p k q ` t i ă k : wO p i q ` wO p k q ą n ` u“ n ` ´ j ` ` t i ă k : w p i ` q ` j ` ą n ` u“ j ´ k ` t i ă k : w p i ` q ą j ` u“ j ´ k ` p k ´ j q“ . Then µ “ λ ˚ , and in particular the length condition is satisfied.We now prove (b). We first observe that λ “ n ` ´ k ´ w p q . Then λ ˚˚ exists ifand only if λ “ n ´ k , i.e. w p q “
2. Then clearly w p k q ă w p q , therefore wO P W P k .Let wO ÞÑ µ . As before we calculate µ “ n ` ´ k , µ s “ λ s ` for 2 ď s ď k ´
1, andthat µ k “ w p q ´ “ ´
1. This proves one implication. For the converse, we notice thatonce wO P W P k , same calculations show that µ “ n ` ´ k and that µ s “ λ s ` for2 ď s ď k ´ w p q “ ℓ p wO q implies that | λ | ´ | µ | “ n ` ´ k , which forces µ k “ ´
1, thus µ “ λ ˚˚ and the propositionis proved. (cid:3) The equivariant quantum Chevalley formula.
To formulate the equivariantquantum Chevalley formula we will first recall the (non-quantum) equivariant Chevalley for-mula for IG. This is due to Pech [36], but for the convenience of the reader we briefly recallthe main steps. (Pech works in the non-equivariant setting, and a minor argument is neededfor the equivariant extension.) In a nutshell, Pech uses the embedding ι : IG Ñ IG p k, n ` q to reduce the calculation to the Chevalley formula in H ˚ p IG p k, n ` qq . Since IG p k, n ` q is homogeneous, the classical work of Chevalley [10], and its equivariant generalization (seee.g. [21]), give this formula with Schubert classes indexed by Weyl group representatives.Buch, Kresch and Tamvakis [6] proved a more general (non-equivariant) Pieri rule, and inthe process re-stated the formulas in terms of strict partitions.In this section we will use the notation X p λ q to denote the Schubert variety in IG and Y p λ ` q to denote the same Schubert variety, but now regarded in IG p k, n ` q “ Sp n ` { P .The notation is consistent with the fact that the partitions in Λ n ` k are obtained from the“odd-symplectic partitions” λ P Λ by adding one box to each row. Set X p q respectively Y p q to be the Schubert divisors in IG and in IG p k, n ` q . Pech proved that in H ˚ p IG q thereis an equality ι ˚ r Y p qs “ r X p qs . Therefore in the equivariant cohomology ι ˚ r Y p qs T “r X p qs T ` C p t q , where C p t q P H T p pt q is a homogeneous linear form. After localization atthe point w W P (the torus-fixed point in the open Schubert cell in IG), and using that w W P R X p q we obtain that C p t q “ ι ˚ w r Y p qs T , where ι ˚ w is the localization map. For thenext result, let w denote the longest element in W . Lemma 11.5.
Let w P W be a signed permutation. Then the localization coefficient ι ˚ w r Y p qs T “ w p ω k q ´ w p ω k q . In particular, C p t q “ ι ˚ w r Y p qs T equals C p t q “ t k ` ´ t if k ă n ` ´ t if k “ n ` . Proof.
Let ϕ w : IG p k, n ` q Ñ IG p k, n ` q be the left multiplication by w . This is anautomorphism of IG p k, n ` q which is equivariant with respect to the map T Ñ T given QUIVARIANT QUANTUM COHOMOLOGY OF THE ODD SYMPLECTIC GRASSMANNIAN 27 by t ÞÑ w tw ´ . There is a commutative diagramIG p k, n ` q ϕ w / / IG p k, n ` qt w u ι w O O ϕ w / / t w w u ι w w O O For v P W P let Y p v q denote the Schubert variety which is stable under B ´ n ` , the oppositeBorel subgroup B n ` ; then Y p v q has codimension ℓ p v q . The morphism ϕ w induces a ringisomorphism ϕ ˚ w P Aut p H ˚ T p IG p k, n ` qqq which satisfies ϕ ˚ w r Y p w vW P qs T “ r Y p vW P qs T ,and it acts on H ˚ T p pt q by twisting by w . Since in our situation r Y p qs T “ ϕ ˚ w r Y p s k qs T wededuce that ι ˚ w r Y p qs T “ ι ˚ w ϕ ˚ w r Y p s k qs T “ ϕ ˚ w ι ˚ w w r Y p s k qs T “ ϕ ˚ w p ω k ´ w w p ω k qq “ w p ω k q ´ w p ω k q . The third equality follows from localization formulas of Schubert classes for opposite Borelsubgroups, see e.g. [21]. The claim on C p t q follows from taking into account the expressionfor w from (3), that ω k “ t ` . . . ` t k , and that w “ p ¯1 , . . . , n ` q . (cid:3) Consider the expansions r X p qs T Y r X p λ qs T “ ÿ µ P Λ c µ p q ,λ r X p µ qs T P H ˚ T p X q ; r Y p qs T Y r Y p λ ` qs T “ ÿ µ P Λ ˜ c µ p q ,λ r Y p µ ` qs T P H ˚ T n ` p IG p k, n ` qq , (7)where c µ p q ,λ , ˜ c µ p q ,λ P H ˚ T p pt q . Notice that ι ˚ r X p λ qs T “ r Y p λ ` qs T therefore the product r Y p qs T Y r Y p λ ` qs T will only contain cohomology classes supported on IG. We apply ι ˚ to both sides of (7) and the projection formula to obtain ι ˚ pr X p qs T Y r X p λ qs T q “ ι ˚ pp ι ˚ r Y p qs T ´ C p t qq Y r X p λ qs T q“pr Y p qs T ´ C p t qq Y r Y p λ ` qs T “r Y p qs T Y r Y p λ ` qs T ´ C p t qr Y p λ ` qs T It follows from this and Lemma 11.5 that(8) c µ p q ,λ “ ˜ c µ p q ,λ if λ ‰ µ ;˜ c λ p q ,λ ´ C p t q “ w p ω k q ´ w λ p ω k q if λ “ µ, where w λ P W is any permutation such that wW P corresponds to λ . Notice in particularthat if λ ‰ µ , the coefficients ˜ c µ p q ,λ are non-negative integers. We recall next the formulafor these integers obtained in [6]. Definition 11.6.
Represent λ P Λ n ` k as a Young diagram. The box in row r and column c of λ is p n ` ´ k q - related to the box in row r and column c if | c ´ n ` k ´ | ` r “ | c ´ n ` k ´ | ` r . Given λ, µ P Λ n ` k with λ Ă µ , the skew diagram µ { λ is called a horizontal strip (resp.vertical) strip if it does not contain two boxes in the same column (resp. row).Following [6, Definition 1.3] we say λ Ñ µ for any n ` ´ k -strict partitions λ, µ if µ canbe obtained by removing a vertical strip from the first n ` ´ k columns of λ and adding ahorizontal strip to the result, so that (1) if one of the first n ` ´ k columns of µ has the same number of boxes as the samecolumn of λ , then the bottom box of this column is n ` ´ k -related to at most onebox of µ z λ ; and (2) if a column of µ has fewer boxes than the same column of λ , the removed boxes andthe bottom box of µ in this column must each be n ` ´ k -related to exactly one boxof µ z λ , and these boxes of µ z λ must all lie in the same row.If λ Ñ µ , we let A be the set of boxes of µ z λ in columns n ` ´ k through n ` ´ k which are not mentioned in (1) or (2). Then define N p λ, µ q to be the number of connectedcomponents of A which do not have a box in column n ` ´ k . Here two boxes are connectedif they share at least a vertex. We refer to [6] for examples of these coefficients. Combining theorem 10.3, proposi-tion 11.4, and equation (8) above, together with the formulation of the Chevalley rule for H ˚ p IG p k, n ` qq obtained in [6, Theorem 1.1] yields the equivariant quantum Chevalleyformula. To shorten notation we set A p λ, µ q : “ N p λ ` , µ ` q . Theorem 11.7.
Let λ P Λ be an n ´ k strict partition. Then the following equality holdsin the equivariant quantum cohomology ring QH ˚ T p IG p k, n ` qq : r X p qs T ‹ r X p λ qs T “ ´ÿ A p λ,µ q r X p µ qs T ¯ ` p w p ω k q ´ w λ p ω k qq r X p λ qs T ` q r X p λ ˚ qs T ` q r X p λ ˚˚ qs T , (9) where the first sum is over partitions µ P Λ such that λ ` Ñ µ ` and | µ | “ | λ | ` , andwhere w λ p j q “ n ` ´ k ´ λ j ` t i ă j : λ i ` λ j ď p n ´ k q ` j ´ i u . When λ ˚ or λ ˚˚ donot exist then the corresponding quantum term is omitted.Example . Consider the Schubert class indexed by r X p , , qs T P QH ˚ T p IG p , qq . Thepermutation corresponding to λ “ p , , q is w “ p , , , ¯4 , ¯3 | q . Then r X p qs T ‹ r X p , , qs T “ ´p t ` t ` t ` t qr X p , , qs T ` r X p , , qs T ` q r X p , qs T . More examples can be found in section 13.
Remark . By Kleiman-Bertini theorem, the GW invariants for homogeneous spacesare enumerative; cf. [11]. There is an equivariant version of positivity [15, 30] which statesthat (quantum) equivariant multiplication of B ´ stable Schubert classes yields structureconstants which are polynomials in positive simple roots with (weakly) negative coefficients.Both the ordinary and equivariant positivity statements hold for the coefficients of theChevalley formula (9). Since IG is not a homogeneous space, one expects that in generalpositivity will fail. Based on theorem 12.2 below we calculated that in QH ˚ T p IG p , qq , r X p , ´ qs T ‹ r X p , ´ qs T “ p t ´ t qr X p , ´ qs T ` p t ` t qr X p qs T ` r X p , qs T ´ q thus c p q , p , ´ q , p , ´ q “ ´ c p q , p , ´ q , p , ´ q “ t ` t . The last coefficient fails the expected equivariant positivity. The full multiplicationtable in QH ˚ T p IG p , q , containing more such examples, can be found in section 13.1.12. Application: an algorithm for the structure constants of QH ˚ T p IG p k, n ` qq One of the main applications of the equivariant quantum Chevalley formula is a recursivealgorithm calculating the structure constants in the equivariant quantum cohomology ringQH ˚ T p IG q . This is possible despite the fact that the divisor class does not generate the QUIVARIANT QUANTUM COHOMOLOGY OF THE ODD SYMPLECTIC GRASSMANNIAN 29 ring. The key is that the extra equivariant parameters introduce sufficient rigidity to allowfor a recursive formula. Similar algorithms, in various levels of generality, were obtained in[20, 33, 35] in relation to equivariant cohomology of Grassmannians. These were generalizedfor equivariant quantum cohomology and equivariant quantum K theory of flag manifolds in[3, 29, 31]. Although the odd-symplectic Grassmannian is not homogeneous, the shape of theequivariant quantum Chevalley formula is almost identical to the one for the Grassmannian.In particular, the non-quantum terms are governed by the Bruhat order, there are no “mixedterms” (i.e. no terms which contain both equivariant and quantum coefficients), and thereare two quantum terms with coefficient 1 (in the Grassmannian case, there is just one suchterm). Therefore it should not be a surprise that almost the same algorithm as the onefrom [29] extends to this case, with essentially the same proof. We present next the preciseresults, while indicating the salient points in their proofs, but we shall leave it to the readerto check the details.We need to introduce few additional notations. For a partition λ P Λ there is at mostone partition λ ` P Λ such that p λ ` q ˚ “ λ . Similarly, there exists at most one partition λ `` such that p λ `` q ˚˚ “ λ . Proposition 12.1.
Let λ, µ, ν P Λ and d P H p X q a non-negative degree. The structureconstant c ν,dλ,µ satisfy the following equation: p w ν p ω k q ´ w λ p ω k qq c ν,dλ,µ “ ÿ η A p λ,η q c ν,dη,µ ´ ÿ ξ A p ξ,ν q c ξ,dλ,µ ` p c ν,d ´ λ ˚ ,µ ´ c ν ` ,d ´ λ,µ q ` p c ν,d ´ λ ˚˚ ,µ ´ c ν `` ,d ´ λ,µ q , (10) where w λ P W is the partition corresponding to λ , the first sum is over η P Λ such that λ ` Ñ η ` and | η | “ | λ | ` , and the second sum is over ξ P Λ such that ξ ` Ñ ν ` and | ξ | “ | ν | ´ ; the terms involving λ ˚ , λ ˚˚ , ν ` , ν `` are omitted if the corresponding partitiondoes not exist.Proof. This is an immediate calculation obtained by collecting the coefficient of q d r X p ν qs T in both sides of the associativity equation r X p qs T ‹ pr X p λ qs T ‹ r X p µ qs T q “ pr X p qs T ‹r X p λ qs T q ‹ r X p µ qs T . (cid:3) The system of equations (10) gives a recursive procedure to calculate any structure con-stant c ν,dλ,µ . We briefly recall the main ideas, following [29], where a similar equation appearedin the study of the equivariant quantum cohomology of Grassmannians; see [3, 31] for moregeneral algorithms. The procedure can be summarized as follows: given λ, µ, ν P Λ and d adegree, the first sum contains coefficients c ν,dη,µ where η is smaller in Bruhat order than λ ; thesecond sum contains coefficients c ξ,dλ,µ where ξ is larger than ν in Bruhat order; the remainingterms involve degree d ´ ă d , known inductively. Given this, the recursion can be runwhenever w ν p ω k q ´ w λ p ω k q ‰
0, which is equivalent to asking that λ ‰ ν . If λ “ ν oneruns the recursion for the coefficient c ν,dµ,λ “ c ν,dλ,µ , using commutativity of the quantum ring.If λ “ µ “ ν one uses the system of equations (10) to write down a linear equation in theunknown coefficient c λ,dλ,λ where all other terms in this equation will be known recursively;see [29, Prop. 6.2], and also [31, Prop. 7.4] or [3, Prop.5.4] for similar statements. The exis-tence of the linear equation in c λ,dλ,λ requires that the linear form F λ,ν : “ w λ p ω k q´ w ν p ω k q is anonzero, positive, combination of simple roots whenever w λ ă w ν in Bruhat ordering. This But the Schubert divisor generates the ring QH ˚ T p IG q localized at the equivariant parameters. We referto [3, §
5] for details. follows easily by induction on the length ℓ p w ν q ´ ℓ p w λ q . Alternatively, F λ,ν “ ˜ c νν, p q ´ ˜ c λλ, p q ,and the required positivity follows from [31, Appendix], applied to G { P “ IG p k, n ` q .This proves the following: Theorem 12.2.
The EQ coefficients are determined (algorithmically) by the following for-mulas: (1) c p q ,d p q , p q “ d ą d “ (commutativity) c µ,dλ,ζ “ c µ,dζ,λ for al partitions λ, ζ, and µ ; (3) (EQ Chevalley) The coefficients c µ,d p q ,λ from theorem 11.7, for all partitions λ and µ ,and all degrees d ; (4) The system of equations (10) for all partitions λ, ζ, µ such that λ ‰ ν . The theorem immediately implies Corollary 1.2, stated in the introduction.13.
Examples
In this section we present several examples. All multiplications are in the equivariantquantum cohomology ring, but we will ignore the subscripts T . The Chevalley formula forQH ˚ T p IG p , qq is: λ r X p qs ‹ r X p λ qsp q ´p t qr X p qs ` r X p , ´ , ´ qs ` r X p qsp q ´p t qr X p qs ` r X p , , ´ qs ` r X p , qs ` r X p qsp , , ´ q ´p t ` t qr X p , , ´ qs ` r X p , , ´ qs ` r X p qsp , q ´ p t ` t qr X p , qs ` r X p , , ´ qs ` r X p , qsp q ´p t qr X p qs ` r X p , qs ` r X p qsp , , ´ q ´p t ` t ` t qr X p , , ´ qs ` r X p , , ´ qs ` r X p , qsp , q ´ p t ` t qr X p , qs ` r X p , qs ` r X p , qsp q ´p t ` t qr X p qs ` r X p , qs ` q p , , ´ q ´p t ` t ` t qr X p , , ´ qs ` r X p , qs ` r X p , , ´ qsp , q ´ p t ` t qr X p , qs ` r X p , , qs ` r X p , qsp , q ´p t ` t ` t qr X p , qs ` r X p , , ´ qs ` r X p , qs ` q r X p qsp , , ´ q ´p t ` t ` t qr X p , , ´ qs ` r X p , qs ` q r X p , ´ , ´ qsp , , q ´ p t ` t ` t qr X p , , qs ` r X p , , qsp , q ´p t ` t ` t qr X p , qs ` r X p , , qs ` r X p , qs ` q r X p qsp , , q ´p t ` t ` t ` t qr X p , , qs ` r X p , , qs ` q r X p , qsp , q ´p t ` t ` t qr X p , qs ` r X p , , qs ` q r X p , , ´ qs ` q r X p qsp , , q ´p t ` t ` t ` t qr X p , , qs ` r X p , , qs ` q r X p , , ´ qs ` q r X p , qsp , , q ´p t ` t ` t ` t qr X p , , qs ` q r X p , , ´ qs ` q r X p , qs QUIVARIANT QUANTUM COHOMOLOGY OF THE ODD SYMPLECTIC GRASSMANNIAN 31
Multiplication table for QH ˚ T p IG p , qq . r X p qs ‹ r X p qs “ ´ t r X p qs ` r X p , ´ qs ` r X p qsr X p qs ‹ r X p qs “ ´ t r X p qs ` r X p , qs ` r X p qsr X p qs ‹ r X p , ´ qs “ ´p t ` t qr X p , ´ qs ` r X p qsr X p qs ‹ r X p , qs “ ´ p t ` t qr X p , qs ` r X p , qsr X p qs ‹ r X p qs “ ´p t ` t qr X p qs ` r X p , qs ` q r X p qs ‹ r X p , qs “ ´p t ` t ` t qr X p , qs ` r X p , qs ` q r X p qsr X p qs ‹ r X p , qs “ ´p t ` t ` t qr X p , qs ` q r X p qs ` q r X p , ´ qsr X p qs ‹ r X p qs “ t p t ´ t qr X p qs ´ t r X p , qs ´ p t ´ t qr X p qs ` r X p , qsr X p qs ‹ r X p , ´ qs “ ´p t ` t qr X p qs ` q r X p qs ‹ r X p , qs “ t p t ` t qr X p , qs ´ p t ` t qr X p , qs ` r X p , qsr X p qs ‹ r X p qs “ p t ` t qp t ´ t qr X p qs ´ p t ` t qr X p , qs ´ p t ´ t q q ` q r X p qsr X p qs ‹ r X p , qs “ p t ` t qp t ` t qr X p , qs ´ p t ` t qr X p , qs ´ p t ` t q q r X p qs` q r X p , ´ qs ` q r X p qsr X p qs ‹ r X p , qs “ t p t ` t qr X p , qs ´ p t ` t q q r X p , ´ qs ´ t q r X p qs ` q r X p qsr X p , ´ qs ‹ r X p , ´ qs “ p t ´ t qr X p , ´ qs ` p t ` t qr X p qs ` r X p , qs ´ q r X p , ´ qs ‹ r X p , qs “ ´p t ` t qr X p , qs ´ r X p , qs ` q r X p qsr X p , ´ qs ‹ r X p qs “ p t ´ t qr X p qs ` r X p , qs ´ p t ´ t q q r X p , ´ qs ‹ r X p , qs “ p t ´ t qr X p , qs ´ p t ` t qr X p , qs ´ p t ´ t q q r X p qs ` q r X p qsr X p , ´ qs ‹ r X p , qs “ p t ´ t qr X p , qs ´ p t ´ t q q r X p qs ` q r X p , qsr X p , qs ‹ r X p , qs “ ´ t t p t ` t qr X p , qs ` t p t ` t qr X p , qs ´ p t ` t qr X p , qs` q r X p , ´ qsr X p , qs ‹ r X p qs “ p t ` t qp t ` t qr X p , qs ´ p t ´ t qr X p , qs ´ p t ` t q q r X p qs ` q r X p qsr X p , qs ‹ r X p , qs “ ´ t p t ` t qp t ` t qr X p , qs ` t p t ` t qr X p , qs´ p t ` t q q r X p , ´ qs ` t p t ` t q q r X p qs ´ p t ` t q q r X p qs ` q r X p qsr X p , qs ‹ r X p , qs “ ´ t p t ` t qp t ` t qr X p , qs ` p t ` t qp t ` t q q r X p , ´ qs` t p t ` t q q r X p qs ´ p t ` t ` t q q r X p qs ` q r X p qs ‹ r X p qs “ ´p t ` t qp t t ´ t t ´ t ` t t qr X p qs ` p t ´ t qr X p , qs ´ t r X p , qs` p t t ´ t t ´ t ` t t q q ´ p t ´ t q q r X p qs ` q r X p , ´ qs ` q r X p qsr X p qs ‹ r X p , qs “ ´p t t ` t t ´ t ´ t t qr X p , qs ` p t ` t ` t t qr X p , qs´ p t ` t q q r X p , ´ qs ` p t t ` t t ´ t ´ t t q q r X p qs ´ p t ` t q q r X p qs` q r X p , qs ` q r X p qsr X p qs ‹ r X p , qs “ ´ t p t ´ t qr X p , qs ` p t ´ t q q r X p , ´ qs` t p t ´ t q q r X p qs ´ t q r X p , qs ´ p t ´ t q q r X p qs ` q r X p , qsr X p , qs ‹ r X p , qs “ t p t ´ t qp t ` t qr X p , qs ´ p t t ` t t ` t t ` t t qr X p , qs` p t ` t t ` t q q r X p , ´ qs ´ t p t t ` t t ´ t ´ t t q q r X p qs` t p t ` t q q r X p qs ´ p t ` t q q r X p , qs ´ p t ` t ` t q q r X p qs ` q r X p , qs ` q r X p , qs ‹ r X p , qs “ t p t t ` t t ´ t t ´ t qr X p , qs ´ p t ` t qp t ´ t q q r X p , ´ qs´ t p t t ` t t ´ t t ´ t q q r X p qs ` t p t ` t q q r X p , qs` p t ` t ` t qp t ´ t q q r X p qs ´ p t ` t ` t q q r X p , qs ´ p t ´ t q q ` q r X p qs r X p , qs ‹ r X p , qs “ ´ t p t t ´ t t ´ t t ` t qr X p , qs ` p t t ´ t t ´ t t ` t q q r X p , ´ qs` t p t t ´ t t ´ t t ` t q q r X p qs ´ t p t ´ t q q r X p , qs´ p t ` t ` t qp t t ´ t t ´ t t ` t q q r X p qs ` p t ` t t ` t ´ t q q r X p , qs` p t t ´ t t ´ t t ` t q q ´ p t ` t q q r X p qs ` q r X p qs References [1] Dave Anderson,
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