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EQUIVARIANT QUANTUM COHOMOLOGY OF COTANGENT BUNDLE OF G/P CHANGJIANSU Abstract. Let G denote a complex semisimple linear algebraic group, P a parabolic subgroup of G and P = G/P. We identify the quantum multiplication by divisors in T∗P in terms of stable basis, which is 5 introduced in [9]. Using this and the restriction formulafor stable basis ([17]), we show that the G×C∗- 1 equivariantquantummultiplicationformulainT∗P isconjugatetotheformulaconjecturedbyBraverman. 0 2 r a M 1. Introduction 3 ThemaingoalofthispaperistostudytheequivariantquantumcohomologyofT∗P,whichisaspecialcase of symplectic resolutions. Recall from [7] that a smooth algebraic variety X with a holomorphic symplectic ] G form ω is called a symplectic resolution if the affinization map A X →X =SpecH0(X,O ) 0 X . h is projective and birational. Conjecturally all the symplectic resolutions of the form T∗M for a smooth t a algebraic variety M are of the form T∗P, see [7]. In [3], Fu proved that every symplectic resolution of m a normalization of a nilpotent orbit closure in a semisimple Lie algebra g is isomorphic to T∗P for some [ parabolic subgroup P in G. In[9],MaulikandOkounkovdefinedthestablebasisforawideclassofvarieties,whichincludesymplectic 2 resolutions. Other examples of symplectic resolutions include hypertoric varieties, resolutions of Slodowy v 3 slices, Hilbert schemes of points on C2, and, more generally, Nakajima varieties [11]. Their quantum coho- 1 mologies were studied in [10], [2], [14] and [9] respectively. The stable basis in the Springer resolutions are 5 justcharacteristiccyclesofVermamodulesuptoasign,see[5]andRemark3.5.3in[9],andtherestrictionof 7 stable basisto fixedpoints is obtainedin [17]. Inthe caseof Hilbertschemes ofpoints onC2, it corresponds 0 to Schur functions if we identify the equivariant cohomology ring of Hilbert schemes with the symmetric . 1 functions, while the fixedpoint basiscorrespondsto Jacksymmetric functions, see e.g. [9], [12], [13]. In this 0 case, Shenfeld obtained the transition matrix from the stable basis to fixed point basis in [15]. 5 To state our main Theorem, let us fix some notations. Let B be a Borel subgroup, R+ be the roots 1 : appearing in B, and R− = −R+. Let ∆ be the set of simple roots, I be a subset of ∆, and P = PI = v BwB be the parabolic subgroup containing B corresponding to I. It is well-known that every Xi pawr∈abWoIlic subgroup is conjugate to some parabolic subgroup containing the fixed Borel subgroup B, which S r is of the form PI for some subset I in ∆, and PI is not conjugate to PJ if the two subsets I and J are not a equal(see[16]). LetW thesubgroupoftheWeylgroupW generatedbythesimplereflectionsσ forα∈I, P α and R± be the roots in R± spanned by I. Let α∨ be the coroot corresponding to α. Let A be a maximal P torus of G contained in B, and C∗ scales the fiber of T∗P by a nontrivial character −~. Let T =A×C∗. Any weight λ that vanishes on all α∨ ∈I∨ determines a one-dimensional representation C of P. Define λ a line bundle L =G× C λ P λ on G/P. Pulling it back to T∗P, we get a line bundle on T∗P, which will still be denoted by L . Let λ D :=c (L ). Itiswell-knownthatthefixedpointset(T∗P)A isinone-to-onecorrespondencewithW/W . λ 1 λ P The stable envelope map stab will be defined in Section 2, and stab (y¯) is the image of the unit in H∗(y¯) + + T under the stable envelope map, where y¯ in H∗(y¯) is the fixed point in T∗P corresponding to yW . An T P element y ∈ W is called minimal if its length is minimal among the elements in the coset yW . As y runs P through the minimal elements, stab (y¯) form a basis in H∗(T∗P) after localization, which is called the + T stable basis. The result we are going to prove is: 1 Theorem 1.1. The quantum multiplication by D in H∗(T∗P) is given by: λ T D ∗stab (y¯)=y(λ)stab (y¯)−~ (λ,α∨)stab (yσ ) λ + + + α α∈R+X,yα∈R− qd(α) σ β −~ (λ,α∨) stab (yσ )+ α stab (y¯) , 1−qd(α)  + α β +  α∈RX+\R+P βY∈R+P   where y is a minimal representative in yW , and d(α) is defined by Equation 3.4. P Combining this and the restriction formula for stable basis ([17]), we get Theorem 1.2. Under the isomorphism HG∗×C∗(T∗P)≃(symt∗)WP[~], the operator of quantum multiplication by D is given by λ σ˜ (f (β−~)) σ β α α Dλ∗f =λf +~ (λ,α∨)1−qdq(αd)(α)  β∈QR(β+P −~) − β∈QR+P β f. α∈RX+\R+P  β∈R+ β∈R+   QP QP    This shows that the quantum multiplication formula is conjugate to the one (4.6) conjectured (through private communication) by Professor Braverman. The paper is organized as follows. In Section 2, we apply results in [9] to define the stable basis of T∗P. InSection3,weproveourmainTheorem1.1bycalculatingthe classicalmultiplicationandpurelyquantum multiplication separately. In the last section, we first show how to deduce the G×C∗-equivariant quantum multiplication in T∗(G/B) from Theorem 1.1, which is the main result of [2]. Then a similar calculation gives a proof to Theorem 1.2. Acknowledgments. I wish to express my deepest thanks to my advisor Professor Andrei Okounkov for suggesting this problem to me and his endless help, patience and invaluable guidance. I am grateful to Professor Alexander Braverman for suggesting the conjectured formula (4.6) to me. I also thank Chiu-Chu Liu,MichaelMcBreen,DaveshMaulik,AndreiNegut,AndreySmirnov,ZijunZhou,ZhengyuZongformany stimulatingconversationsandemails. AlotofthanksalsogotomyfriendPak-HinLeeforeditingaprevious version of the paper. 2. Stable basis for T∗P In this section, we apply the construction in [9] to T∗P. 2.1. Fixed point sets. It is well-known the A-fixed points of T∗P is in one-to-one correspondence with W/W . For any y ∈ W, let y¯ denote the coset yW and the corresponding fixed point in T∗P. Recall the P P Bruhat order ≤ on W/W is defined as follows: P y¯≤w¯ if ByP/P ⊆BwP/P. 2.2. Chamber decomposition. The cocharacters σ :C∗ →A form a lattice. Let aR =cochar(A)⊗ZR. Define the torus roots to be the A-weights occurring in the normal bundle to (T∗P)A. Then the root hyperplanes partition aR into finitely many chambers aR\ α⊥i = Ci. It is easy to see that in this case the torus roots[are justathe roots in G. Let + denote the chamber such that all root in R+ are positive on it, and − the opposite chamber. 2 2.3. Stable leaves. Let C be a chamber. Define the stable leaf of y¯by LeafC(y¯)= x∈T∗P limσ(z)·x=y¯ , z→0 n (cid:12) o where σ is any cocharacter in C; the limit is independen(cid:12)t of the choice of σ ∈C. In our case, (cid:12) Leaf (y¯)=T∗ P, + By¯P/P and Leaf (y¯)=T∗ P, − B−y¯P/P where B− is the opposite Borel subgroup. Define a partial order on W/W as follows: P w¯ (cid:22)C y¯ if LeafC(y¯)∩w¯ 6=∅. By the description of Leaf (y¯), the order (cid:22) is the same as the Bruhat order on W/W , and (cid:22) is the + + P − opposite order. Define the slope of a fixed point y¯by SlopeC(y¯)= LeafC(w¯). w¯[(cid:22)Cy¯ 2.4. Stable basis. For each y¯, define ǫ = eA(T∗P). Here, eA denotes the A-equivariant Euler class. Let y¯ y¯ N denote the normal bundle of T∗P at the fixed point y¯. The chamber C gives a decomposition of the y¯ normal bundle N =N ⊕N y¯ y¯,+ y¯,− into A-weights which are positive and negative on C respectively. The sign in ±e(N ) is determined by y¯,− the condition ±e(Ny¯,−)|H∗(pt) =ǫy¯. A The following theorem is the Theorem 3.3.4 in [9] applied to T∗P. Theorem 2.1 ([9]). There exists a unique map of H∗(pt)-modules T stabC :HT∗((T∗P)A)→HT∗(T∗P) such that for any y¯∈W/WP, Γ=stabC(y¯) satisfies: (1) suppΓ⊂Slope (y¯), C (2) Γ| =±e(N ), with sign according to ǫ , y¯ −,y¯ y¯ (3) Γ|w¯ is divisible by ~, for any w¯ ≺C y¯, where y¯ in stabC(y¯) denotes the unit in HT∗(y¯). Remark 2.2. (1) The map is defined by a Lagrangian correspondence between (T∗P)A ×T∗P, hence maps middle degree to middle degree. (2) From the characterization, the transition matrix from {stabC(y¯),y¯ ∈ W/WP} to the fixed point basis is a triangular matrix with nontrivial diagonal terms. Hence, after localization, {stabC(y¯),y¯∈ W/W } form a basis for the cohomology, which is the stable basis. P (3) Theorem4.4.1in[9]showsthat{stabC(y¯),y¯∈W/WP}and{(−1)mstab−C(y¯),y¯∈W/WP}aredual bases, where m=dimG/P. From now on, we let stab (y¯) denote the stable basis in H∗(T∗P), and let stab (y) denote the stable ± T ± basis in H∗(T∗B). We record two lemmas here, which will be important for the calculations. T Lemma 2.3 ([1]). Each coset W/W contains exactlyone element of minimal length, which is characterized P by the property that it maps I into R+. 3 Lemma 2.4 ([17]). Let y be a minimal representative of the coset yW . Then P ~ α (−1)l(y)+1 α∈R+ (mod ~2) if w¯ =yσ and yσ <y for some β ∈R+,  yβ Qyσ α β β β stab+(y¯)|w¯ ≡ α∈QR+P 0 (mod ~2) otherwise, and  ~ α (−1)l(y)+1 α∈R+ (mod ~2) if w¯ =yσ and yσ <y for some β ∈R+,  yβ Q yα β β stab−(w¯)|y¯ ≡ α∈QR+P 0 (mod ~2) otherwise, where < is the Bruhat order on the Weyl group W. 3. T-equivariant quantum cohomology of T∗P Now we turn to the study of equivariant quantum cohomology of T∗P. We denote T∗P by X in this section. Recall D := c (L ). We are going to determine the quantum multiplication by the divisor D in λ 1 λ λ terms of the stable basis. It is easy to see that yλ does not depend on the choice of representative in yW , P since W fix λ. P 3.1. Preliminaries on quantum cohomology. By definition, the operatorof quantum multiplication by α∈H (X) has the following matrix elements T (α∗γ ,γ )= qβhα,γ ,γ iX , 1 2 1 2 0,3,β β∈HX2(X,Z) where(·,·)denotesthestandardinnerproductoncohomologyandthequantityinanglebracketsisa3-point, genus 0, degree β equivariant Gromov–Witten invariant of X. If α is a divisor and β 6=0, we have hα,γ ,γ iX =(α,β)hγ ,γ iX . 1 2 0,3,β 1 2 0,2,β Since X has a everywhere-nondegenerate holomorphic symplectic form, it is well-known that the usual non-equivariant virtual fundamental class on M (X,β) vanishes for β 6= 0. However, we can modify g,n the standard obstruction theory so that the virtual dimension increases by 1 (see [2] or [14]). The virtual fundamental class [M (X,β)]vir has expected dimension 0,2 K ·β+dimX+2−3=dimX−1. X Hence the reduced virtual class has dimension dimX, and for any β 6=0, [M (X,β)]vir =−~·[M (X,β)]red, 0,2 0,2 where ~ is the weight of the symplectic form under the C∗−action. 3.2. Unbroken curves. Broken curves was introduced in [14]. Let f : C → X be an A-fixed point of M (X,β) such that the domain is a chain of rational curves 0,2 C =C ∪C ∪···∪C , 1 2 k with the marked points lying on C and C respectively. 1 k We say f is an unbroken chain if at every node f(C ∩C ) of C, the weights of the two branches are i i+1 opposite and nonzero. Note that all the nodes are fixed by A. Moregenerally,if(C,f) isanA-fixedpointofM (X,β), wesaythatf is anunbrokenmapifitsatisfies 0,2 one of the three conditions: (1) f arises from a map f :C →XA, (2) f is an unbroken chain, or 4 (3) the domain C is a chain of rational curves C =C ∪C ∪···C 0 1 k such that C is contracted by f, the marked points lie on C , and the remaining components form 0 0 an unbroken chain. Broken maps are A-fixed maps that do not satisfy any of these conditions. Okounkov and Pandharipande proved the following Theorem in Section 3.8.3 in [14]. Theorem 3.1 ([14]). Every map in a given connected component of M (X,β)A is either broken or unbro- 0,2 ken. Only unbroken components contribute to the A-equivariant localization of reduced virtual fundamental class. 3.3. Unbroken curves in X. Any α∈ R+\RP+ defines an SL2 subgroup Gα∨ of G and hence a rational curve Cα :=Gα∨ ·[P]⊂G/P ⊂X. ThisistheuniqueA-invariantrationalcurveconnectingthefixedpoints¯1andσ¯ ,becauseanysuchrational α curve has tangent weight at ¯1 in R−\R−, and uniqueness follows from the following lemma in Section 4 in P [4]. Lemma 3.2 ([4]). Let α,β be two roots in R+\R+. Then σ¯ =σ¯ if and only if α=β. P α β If C is an A-invariant rational curve in X, C must lie in G/P, and it connects two fixed points y¯ and w¯. Then its y−1-translate y−1C is still an A-invariant curve, which connects fixed points ¯1 and y−1w. So y−1C = C for a unique α ∈ R+ \R+, and y−1w = σ¯ . Hence the tangent weight of C at y¯ is −yα. In α P α conclusion, we have Lemma 3.3. There are two kinds of unbroken curves C in X: (1) C is a multiple cover of rational curve branched over two different fixed points, (2) C is a chain of two rational curve C =C ∪C , such that C is contracted to a fixed point, the two 0 1 0 marked points lie on C , and C is a multiple cover of rational curve branched over two different 0 1 fixed points. For any α∈∆\I, define τ(σ ):=Bσ P/P. Then α α {τ(σ )|α∈∆\I} α form a basis of H (X,Z). Let {ω |α ∈ ∆} be the fundamental weights of the root system. For any 2 α α∈R+\R+ , define degree d(α) of α by P (3.4) d(α)= (ω ,α∨)τ(σ ). β β β∈X∆\I Lemma 3.5 ([4]). The degree of [C ] is d(α), and d(α)=d(wα) for any w∈W . α P 3.4. Classical part. We first calculate the classicalmultiplication by D in the stable basis. Let m denote λ the dimension of G/P. Since {stab (y¯)} and {(−1)mstab (y¯)} are dual bases, we only need to calculate + − D | ·stab (y¯)| ·(−1)mstab (w¯)| (3.6) (D ∪stab (y¯),(−1)mstab (w¯))= λ z¯ + z¯ − z¯. λ + − e(T X) z¯ w¯≤z¯≤y¯ X This will be zero if y¯<w¯. Assume y is a minimal representative. Note that the resulting expression lies in the nonlocalized coefficient ring due to the proof of Theorem 4.4.1 in [9], and a degree count shows that it is in H2(pt). There are two cases. T 3.4.1. Case y¯=w¯. There is only one term in the sum of the right hand side of Equation (3.6). Hence, D | ·stab (y¯)| ·(−1)mstab (y¯)| (D ∪stab (y¯),(−1)mstab (y¯))= λ y¯ + y¯ − y¯ =y(λ). λ + − e(T X) y¯ 5 3.4.2. Case y¯ 6= w¯. Notice that (D ∪stab (y¯),(−1)mstab (w¯)) ∈ H2(pt), and it is 0 if ~ = 0, because λ + − T every term in Equation (3.6) is divisible by ~. Hence, it is a constant multiple of ~. So in Equation (3.6), only z¯=y¯and z¯=w¯ have contribution since all other terms are divisible by ~2. Therefore, stab (w¯)| stab (y¯)| (D ∪stab (y¯),(−1)mstab (w¯))=y(λ) − y¯ +w(λ) + w¯ λ + − stab (y¯)| stab (w¯)| − y¯ + w¯ ~ part of stab (w¯)| ~ part of stab (y¯)| − y¯ + w¯ =y(λ) +w(λ) , yα wα α∈R+\R+ α∈R+\R+ Q P Q P where the first equality follows from stab (y¯)·stab (y¯))=(−1)me(T X). + − y¯ Lemma 2.4 shows this is zero if w¯ 6= yσ for any β ∈ R+ with yσ < y. However, if w¯ = yσ for such a β β β β, then since (−1)l(yσβ) =(−1)l(y)+1, we have (D ∪stab (y¯),(−1)mstab (w¯)) λ + − ~ α ~ α =y(λ)(−1)l(y)+1 α∈R+ +yσ (λ)(−1)l(y)+1 α∈R+ yβ Q yα β yβ Qyσ α β α∈R+ α∈R+ ~ ~ Q Q =− y(λ)+ yσ (λ) β yβ yβ =−~(λ,β∨). Notice that for any β ∈R+, yσ <y is equivalent to yβ ∈R−. To summarize, we get β Theorem 3.7. Let y be a minimal representative. Then the classical multiplication is given by D ∪stab (y¯)=y(λ)stab (y¯)−~ (λ,α∨)stab (yσ ). λ + + + α α∈R+X,yα∈R− 3.5. Quantum part. Let D ∗ denote the purely quantum multiplication. We want to calculate λ q (−1)m(D ∗ stab (y¯),stab (w¯))=− (−1)m~qβ(D ,β)(ev [M (X,β)]red,stab (y¯)⊗stab (w¯)). λ q + − λ ∗ 0,2 + − βeffective X whereev isthe evaluationmapfromM (X,β)to X×X. The−signappearsbecausethecotangentfibers 0,2 have weight −~ under the C∗−action. Since dim[M (X,β)]red =dimX, 0,2 and (ev [M (X,β)]red,stab (y¯)⊗stab (w¯)) ∗ 0,2 + − liesinthenonlocalizedcoefficientring(seeTheorem4.4.1in[9]),theproductisaconstantbyadegreecount. Thuswecanlet~=0,i.e.,wecancalculateitinA-equivariantchomology. Asintheclassicalmultiplication, there are two cases depending whether the two fixed points y¯and w¯ are the same or not. 3.5.1. Case y¯6=w¯. By virtual localization, Theorem 3.1 and Lemma 3.3, (ev [M (X,β)]red,stab (y¯)⊗stab (w¯)) ∗ 0,2 + − is nonzero if and only if w¯ = yσ for some α ∈ R+ \R+. Only the first kind of unbroken curves have α P contribution to (ev [M (X,β)]red,stab (y¯)⊗stab (yσ )), and only restriction to the fixed point (y¯,yσ ) ∗ 0,2 + − α α is nonzero in the localization of the product by the first and third properties of the stable basis. The A- invariantrationalcurvey[C ]connectsthetwofixedpointsy¯andyσ ,anditistheuniqueone. Forexample, α α if y[C ] is also such a curve, then yσ =yσ =w¯. Hence α=β by Lemma 3.2. Therefore, β α β (−1)m(D ∗ stab (y¯),stab (yσ ))=− (−1)m~qk·d(α)(D ,k·d(α)) λ q + − α λ k>0 X (ev [M (X,k·d(α))]red,stab (y¯)⊗stab (yσ )). ∗ 0,2 + − α 6 Let f be an unbroken map of degree k from C =P1 to y[C ]. Then α Aut(f)=Z/k. By virtual localization, e(T∗P)e(T∗ P)e′(H1(C,f∗TX)) k(ev [M (X,k·d(α))]red,stab (y¯)⊗stab (yσ ))= y¯ yσα . ∗ 0,2 + − α e′(H0(C,f∗TX)) Here e′ is the product of nonzero A-weights. We record Lemma 11.1.3 from [9]. Lemma 3.8 ([9]). Let A be a torus and let T be an A-equivariant bundle on C = P1 without zero weights in the fibers T and T . Then 0 ∞ e′(H0(T ⊕T∗)) =(−1)degT+rkT+ze(T ⊕T ) e′(H1(T ⊕T∗)) 0 ∞ where z =dimH1(T ⊕T∗)A, i.e., z counts the number of zero weights in H1(T ⊕T∗). Since f∗TX =T ⊕T∗ with T =f∗TP, Lemma 3.8 gives e(T∗P)e(T∗ P)e′(H1(C,f∗TX)) k(ev [M (X,k·d(α))]red,stab (y¯)⊗stab (yσ ))= y¯ yσ¯α ∗ 0,2 + − α e′(H0(C,f∗TX)) =(−1)degT+rkT+z. We now study the vector bundle T =f∗TP. First of all, rkT =dimP. By localization, (−yγ) (−yσ γ) α γ∈R+\R+ γ∈R+\R+ degT =k P−yPα + P Pyα      =k (γ,α∨)=k(2ρ−2ρ ,α∨) P γ∈RX+\R+P =2k (ω ,α∨) β β∈X∆\I is an even number, where ρ is the half sum of the positive roots, ρ is the half sum of the positive roots in P R+, and ω are the fundamental weights. P β The vector bundle T splits as a direct sum of line bundles on C T = L , i i M so L | = g , i 0 −yγ Mi γ∈RM+\R+P where g are the root subspaces of g. Suppose L | =g . Since yσ y−1 maps y to yσ , we have −yγ i 0 −yγ α α L | =g . i ∞ −yσαγ Hence there is only one zero weight in H1(T ⊕T∗), which occurs in H1(L ⊕L∗), where L | = g , i.e., i i i 0 −yα L is the tangent bundle of C. i Therefore z =1 and we have Lemma 3.9. qd(α) (−1)m(D ∗ stab (y¯),stab (yσ ))= ~qk·d(α)(D ,d(α))=−~ (λ,α∨). λ q + − α λ 1−qd(α) k>0 X 7 Proof. We only need to show (D ,d(α))=−(λ,α∨). λ By definition and localization, λ σ λ β (Dλ,d(α))= (ωβ,α∨) c1(Lλ)= (ωβ,α∨) + −β β β∈X∆\I Zτ(σβ) β∈X∆\I (cid:18) (cid:19) =− (ωβ,α∨)(λ,β∨)=− (ωβ,α∨)(λ,β∨) β∈X∆\I βX∈∆ =−(λ,α∨). (cid:3) 3.5.2. Case y¯ = w¯. In this case, only the second kind of unbroken curves have contribution to (D ∗ λ q stab (y¯),stab (y¯)). Let C = C ∪C be an unbroken curve of the second kind with C contracted to the + − 0 1 0 fixed point y¯, and C is a coverof the rationalcurve yC of degree k, where α∈R+\R+. Let p denote the 1 α P node of C, and let f be the map from C to X. Then the corresponding decoratedgraphΓ has two vertices, one of them has two marked tails, and there is an edge of degree k connecting the two vertices. Hence the automorphism group of the graph is trivial. The virtual normal bundle ([6]) is e′(H0(C,f∗TX))−yα/k e′(H0(C,f∗TX)) (3.10) e(Nvir)= =− , Γ e′(H1(C,f∗TX)) yα/k e′(H1(C,f∗TX)) where e′(H0(C,f∗TX)) denotes the nonzero A-weights in H0(C,f∗TX). Consider the normalization exact sequence resolving the node of C: 0→O →O ⊕O →O →0. C C0 C1 p Tensoring with f∗TX and taking cohomology yields: 0→H0(C,f∗TX)→H0(C ,f∗TX)⊕H0(C ,f∗TX)→T X 0 1 y¯ →H1(C,f∗TX)→H1(C ,f∗TX)⊕H1(C ,f∗TX)→0. 0 1 Since C is contracted to y¯, H0(C ,f∗TX)= T X and H1(C ,f∗TX)= 0. Therefore, as virtual represen- 0 0 y¯ 0 tations, we have H0(C,f∗TX)−H1(C,f∗TX)=H0(C ,f∗TX)−H1(C ,f∗TX). 1 1 Due to Equation (3.10) and the analysis in the last case, we get e′(H0(C ,f∗TX)) e(Nvir)=− 1 Γ e′(H1(C ,f∗TX)) 1 =(−1)me(T P)e(T P). y¯ yσα 8 Then by virtual localization formula, we have e(T∗P)2 (−1)m(D ∗ stab (y¯),stab (y¯))=−~ (D ,d(α))qk·d(α) y¯ λ q + − λ e(T P)e(T P) α∈R+X\R+P,k>0 y¯ yσα yβ =~ (λ,α∨) qd(α) β∈RQ+\R+P 1−qd(α) yσ β α α∈RX+\R+P β∈R+\R+ Q P yβ yσαβ =~ (λ,α∨)1−qdq(αd)(α) β∈QR+yσ β β∈QR+P yβ α α∈RX+\R+P β∈R+ β∈R+ Q QP σ β α =−~·y (λ,α∨) qd(α) β∈QR+P . 1−qd(α) β α∈RX+\R+P β∈R+   QP    Here we have used yβ =(−1)l(y) β, and (−1)l(yσα) =(−1)l(y)+l(σα) =(−1)l(y)+1. βY∈R+ βY∈R+ Notice that for any root γ ∈R+, σ preserves R+\R+. For any α∈R+\R+, d(σ (α)) =d(α), (λ,α∨)= P γ P P γ (λ,σ (α)∨) and σ β =− β. Hence, γ γ β∈R+ β∈R+ QP QP qd(α) σ (λ,α∨) σ β γ 1−qd(α) α  α∈RX+\R+P βY∈R+P   qd(α) = (λ,α∨) σ σ β 1−qd(α) σγα γ α∈RX+\R+P βY∈R+P qd(α) =− (λ,α∨) σ β. 1−qd(α) α α∈RX+\R+P βY∈R+P Therefore (λ,α∨) qd(α) σ β is divisible by β. But they have the same degree, so 1−qd(α) α α∈R+\R+ β∈R+ β∈R+ P P QP QP σ β α qd(α) β∈R+ (3.11) (λ,α∨) QP 1−qd(α) β α∈RX+\R+P β∈R+ QP is a scalar. To summarize, we get Theorem 3.12. The purely quantum multiplication by D in H∗(T∗P) is given by: λ T σ β α Dλ∗qstab+(y¯)=−~ (λ,α∨)1−qdq(αd)(α) stab+(yσα)−~ (λ,α∨)1−qdq(αd)(α)β∈QR+P β stab+(y¯). α∈RX+\R+P α∈RX+\R+P β∈R+ QP Remark 3.13. 9 (1) The scalar σ β α −~ (λ,α∨) qd(α) β∈QR+P 1−qd(α) β α∈RX+\R+P β∈R+ QP can also be determined by the condition D ∗ 1=0. λ q (2) The element y is not necessarily a minimal representative. (3) The Theorem is also true if we replace all the stab by stab . + − 3.6. Quantum multiplications. Combining Theorem 3.7 and Theorem 3.12, we get our main Theorem 1.1. Taking I =∅, we get the quantum multiplication by D in H∗(T∗B). λ T Theorem 3.14. The quantum multiplication by D in H∗(T∗B) is given by: λ T D ∗stab (y)=y(λ)stab (y)−~ (λ,α∨)stab (yσ ) λ + + + α α∈R+X,yα∈−R+ qα∨ −~ (λ,α∨) (stab (yσ )+stab (y)). 1−qα∨ + α + αX∈R+ 3.7. Calculation of the scalar in type A. We can define an equivalence relation on R+\R+ as follows P α∼β if d(α)=d(β). Then w(α)∼α for any w ∈W . We have P σ β α qd(α) β∈R+ (λ,α∨) QP 1−qd(α) β α∈RX+\R+P β∈R+ QP σα′β qd(α) β∈R+ = (λ,α∨) QP . 1−qd(α) β α∈(R+X\R+P)/∼ αX′∼α β∈R+ QP It is easy to see that σα′β β∈R+ QP β α′∼α X β∈R+ QP is a constant, which will be denoted by C (α). P In this section, we will determine the constant C (α) when G is of type A. We will first calculate this P number in T∗Gr(k,n) case, and the general case will follow easily. Now let G=SL(n,C) and let x be the i function on the Lie algebra of the diagonal torus defined by x (t ,··· ,t )=x . i 1 n i 3.7.1. T∗Gr(k,n) case. Let P be a parabolic subgroup containing the upper triangular matrices such that T∗(G/P) is T∗Gr(k,n). Then R+ ={x −x |1≤i<j ≤k, or k <i<j ≤n}, R\R+ ={x −x |1≤i≤k <j ≤n} P i j P i j and all the roots in R\R+ are equivalent. The number C (α) will be denoted by C . By definition, P P P (rs) (x −x ) (x −x ) i j p q 1≤r≤k<s≤n 1≤i<j≤k 1+k≤p<q≤n ! (3.15) C = , P P Q Q (x −x ) (x −x ) i j p q 1≤i<j≤k 1+k≤p<q≤n Q Q where (rs) means the transposition of x and x . r s 10