Quantum Cohomology of G/P

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last update: 19 December 2013

Lecture 16: April 22, 1997

Recall last time:

Today: Compare RP and qH*(G/P):

Theorem 1: We have an isomorphism SRP qH*(G/P) 1qτσP(w) qτσPw.


First the case of G/B: The map 1qτσB(w)qτσBw is bijective. Since both sides are generated by H2 and there is no torsion remains to check formula for multiplications by H2 on each side. We write these formulas down in Lectures 14 and 15. For any P, use the commutative diagram given at the end of Lecture 15.

Theorem 2: We have an isomorphism RPΛP HT(G/P) sσP(w)1 sσP(w).


For G/B, directly from the multiplication formula by H2. For G/P, take τ=0 and h=0 in the commutative diagram at the end of Lecture 15.

We have the commutative diagram RP qH*(G/P) HT(G/P) q=0 s=0 H*(G/P)

From now on, will denote RP = qHT(G/P), RP = qHT(G/P)(q) (quantum cohomology with the quantum parameter inverted).

The homomorphism ψP:

HT(ΩK) ψP qHT(G/P)(q) H*(ΩK) ψP qH(G/P)(q) When P=B, ψP is an isomorphism if we also invert the tranlational elements in H*(ΩK). Using ψP, we get structure constants for H*(ΩK) as Gromov-Witten invariants, which, since they are numbers of certain curves, are non-negative integers.

Results of Bott:

For certain hQ, construct K/TϕhΩK by ϕh(kT)(t) = thkt-h k-1

Geometrical Models

Will construct geometrical models for

Now we turn to the first model for 𝒰:

The first model 𝒰 of SpecHT(G/B):

Let e=iIeαi.

Lemma: For any hh_, the fixed points of the vector field Ve+h on G/B defined by e+h all lie in the cell ·B.

Theorem: 𝒰 = SpecHT(G/B) (G/B)e+h_ = { (e+h,x) (e+h_)× G/B:Ve+h (x)=0 } = { (e+h,u-·B): u--1·(e+h) b_ } where the second·is the adjoint action. But since U- stabilises e+b_-, when u--1·(e+h)b_, we have u--1· (e+h)b_ (e+b_-) =e+h_. ????? 𝒰 = { (e+h,u-·B): u--1· (e+h)e+h_ } = { (e+h,u-,e+h) :u--1· (e+h)=e+h } .

The groupoid structure on 𝒰:

As a model for SpecHT(G/B), we must have two W-actions on 𝒰 which gave wL and wR on HT(G/B). We identify these two actions in the next lecture.

Lecture 17: April 23, 1997

The following works for the general Kac-Moody case:

Set e = iIei n_+, fi(n) = finn! U(n_-). Then U(n_-) = fi(n)iI,n0 and using the action of 2ρ we can give U(n_-) a -grading with degfi=-2.

Define 𝒪(U-) = Hom (U(n_-),) (graded dual) and we use U- to denote the groupscheme defined by 𝒪(U-).

Lemma: For any wW, there exists a scheme morphism Uw: h_ U- s.t. Uw(h)· (e+h) = e+w·h. We have Uvw(h) = Uv(w·h) Uw(h) and Uri(h)= exp(αi,hfi) =yi(αi,h) where, recall, ϕi:SL(2,)G and for u (?) xi(u) = ϕi (1u01), yi(u) = ϕi (10u1).

The finite case:

In this case, a theorem of Kostant says that the element Uw(h)U- is unique for any given wW and hh_.

Example: For 𝔤=sl(3), h=diag(x1,x2,x3), have Ur1(h) = ( 100 x1-x210 001 ) Ur1r2r1 (h) = ( 100 x1-x310 (x1-x2)(x1-x3)x1-x31 ) ur2(h) = ( 100 010 0x2-x31 ) ur2r1(h) = ( 100 x1-x310 (x1-x2)(x1-x3)x1-x31 )

Fact: Uw0(tρ)=exp(tf), t, where {e,f,2ρ} is a TDS.

The affine case:

In this case, define Uri(h)=yi(αi,h) for iIaf and use Uvw(h) = Uv(w·h) Uw(h) to extend to any w. This is well-defined because of the braid relations: assume that 2<mij< and αj,αi=-1 for ij. Set a=αi,h, b=αj,h. Then mij=3: Uiji = Uri(rjri·h) Urj(ri·h) Uri(h) = yi(b) yj(a+b) yj(a) Ujij = yj(a) yi(a+b) yj(b) mij=4: Uijij = yi(a) yj(a+b) yi(a+2b) yj(b) Ujiji = yj(b) yi(a+2b) yj(a+b) yi(a) mij=6: Uijijij = yi(a) yj(3a+b) yi(2a+b) yj(3a+2b) yi(a+b) yj(b) Ujijiji = yj(b) yi(a+b) yj(3a+2b) yi(2a+b) yj(3a+b) yi(a) The fact that they are equal is due to Kostant's theorem (Uw(?????) is unique). These are called universal exponential solutions to the Yang-Baxter Equations by Fomin and Kirilov in their paper in Lett. Math. Phys (1996) 273-284.

What about mij=? This is what is needed in the affine case?

Remark: In the affine case, the element Uw(h)U- is not necessarily unique for a given (w,h). For example, when t·h=h, have Ut(h)Z(e+h)U-.

The action of W on (e+h_)×Z and on S𝒪(Z)

Suppose that U- acts on a scheme Z. Then W acts on (e+h_)×Z by w((e+h,z)) = (e+wh,Uw(h)·z). Assume that Z is affine. Then W acts on 𝒪((e+h_)×Z) S𝒪(Z): (w·p1) (e=h,z) = p1(w-1·(e+h,z)) ,p1=sp S𝒪(Z).

Lemma: For spS𝒪(Z), ri·(sp) = n0 (ri·(αins)) fi(n)·p.


By definition, ri·(sp) (e+h,z) = (sp) (ri·(e+h,z)) = (sp) (e+ri·h,Uri(h)·z) = s(ri·h) p(Uri(h)·z) = (ri·s)(h) (Uri(h)-1·p)(z) = (ri·s)(h) ( exp(-αi,hfi·p) (z) ) = (ri·s)(h) ( n0 -αi,hnn! fin·p ) (z) = n0 ((-αi)nri·s) (h)(fi(n)·p)(z) = n0ri· (αins)(h) (fi(n)·p)(z) ri· (sp) = n0 (ri·(αins)) fi(n)·p.

Consequently, we get an integrable A_-module structure on S𝒪(Z) by Ai·(sp) = 1αi (1-ri)· (sp) = (Ai·s)p+ n1ri· (αin-1s) fi(n)·p. For each p𝒪(Z), this is a finite sum.

The groupoid scheme 𝒰=(e+h_)×U-:

Define PL=p1: 𝒰 e+h_: (e+h,u) e+h and PR: 𝒰 e+h_: (e+h,u) proj. ofu-1·(e+h) toe+h_in e+b_-=e+h_+n-. These are the source and target maps for the groupoid struture on 𝒰. Other structure maps: identitiesi: e+h_ 𝒰: e+h (e+h,1) multiplication:μ: 𝒰×e+h_𝒰 𝒰: (e+h,u)·(e+h,u) = (e+h,u) ifPR(e+h,u)=e+h=PL(e+h,u). inverse:ι: 𝒰 𝒰: (e+h,u) (PR(e+h,u),u-1). The idea now is to embed 𝒰 as a subgroupoid scheme of 𝒰. Here the groupoidscheme structure on 𝒰 is the one defined in Lecture 5. To this end, we use the integrable A_-module structure on 𝒰.

The groupoid morphism 𝒰𝒰:

Consider the WL action on (e+h_)×U-: wL· (e+h,u) = (e+wh,Uw(h)u). It satisfies PR·wL = PR by the definition of Uw. By the discussion in Lecture 17, we have an integrable A_L-module structure on 𝒪(𝒰). In other words we have a groupoid action ϕ: 𝒰×h_𝒰 𝒰 PRp2 PRp2 PR PR e+h_ e+h_ Also have 𝒰×h_𝒰×h_𝒰 id×μ 𝒰×h_𝒰 ϕ×id ϕ×id ϕ ϕ 𝒰×h_𝒰 μ 𝒰 where μ:𝒰×h𝒰𝒰 is the multiplication morphism for 𝒰. These imply that the following composition is a morphism of groupoid schemes over h_: 𝒰 = 𝒰×h_h_ id×i 𝒰×h_𝒰 ϕ 𝒰 where i:h_𝒰 is the identity morphism for 𝒰.

The groupoid isomoprhism 𝒰𝒰×e+b_-(e+h_)

Define PR: 𝒰 e+b_-: (e+h,u) u-1·(e+h) e+b_-. Form 𝒰×e+b_- (e+h_) 𝒰 using PR and e+h_e+b_- (the inclusion). We think of 𝒰×e+b_-(e+h_)=𝒰 as the subset of 𝒰: { (e+h,u): u-1· (e+h)e+h } . We claim that the morphism 𝒰𝒰 factors through 𝒰. To prove this, we look at 𝒪(𝒰) 𝒪(𝒰) HT(G/B). For each wW, recall that we have ψw:HT(G/B)S.

The map 𝒪(𝒰) 𝒪(𝒰) HT(G/B) ψw S corresponds to the scheme morphism h_ 𝒰: h (e+h,Uw-1(h)-1). Since (Uw-1(h)-1)-1 ·(e+h)=Uw-1(h)· (e+h)=e+w-1·h we see that (e+h,Uw-1(h)-1) 𝒰. Since {ψw:wW} is a basis for HomS(HT(K/T),S), we conclude that the morphism 𝒰𝒰 factors through 𝒰 to give 𝒰 𝒰.

Theorem: 𝒰 𝒰 as groupoid schemes over h_.

Lecture 18: April 29, 1997

Last time we had morphisms of groupoid schemes over h_ SpecHT(K/T)=𝒰 (e+h_)×𝒰-𝒰 [(e+h_)×𝒰-] ×e+b_- (e+h_)=𝒰

Consider the corresponding ring homomorphism 𝒪(𝒰) = S𝒪(𝒰-) 𝒪(U) = HT(K/T). (*)

Definition: wW is called G-abelian if the following equivalent conditions hold.

(1) rirjri, where aij=-1, does not occur as a consecutive subexpression for any reduced expression of w.
(2) 𝒰-wBw-1 is commutative.

Lifting of σG/B(w) for G-abelian w to 𝒪(𝒰-)

Consider the quotient of 𝒰(n_-) by the 2-sided ideal generated by {fi(m)|iI,m2}. The resulting ring with identity 𝒰(n_-)/fi(2);iI is given by generators {fi|iI} and relations: fifi=0, fifjfi=0 ifaij=-1, andfifj=fjfi ifaij=0. For G-abelian w with reduced expression ri1riN, put fw = fi1 fiN. These fw define a basis of 𝒰(n_-)/fi(2);iI. The dual basis gives us elements in 𝒪(𝒰-): fw*Hom (𝒰(n_-)/fi(2);iI,) Hom(𝒰(n_-),) =𝒪(𝒰-).

Claim: Under the homomorphism (*) S𝒪(𝒰-) HT(G/B), 1fw* σG/B(w).


Write fw* for 1fw*. The statement is clear for the identity elements: f1*σG/B(1). Suppose riww. Then riw is again G-abelian, and we have (ri·fw*) (h,u)= fw*(uri(h)u) =αi(h)friw* (u)+fw*(u). Therefore rifw* = αifriw* +fw*, Ai·fw* = -friw*. Similarly, Aj·fw*=0 if wrjw. Define xHT(G/B) by fw* σG/B(w)+x. Then Ai·fw* AiσG/B(w)+Aix. We can assume by induction that friwσG/B(riw) whenever riww. Therefore Ai·x=0 in this case. Also Aj·x=0 for rjww, by the above. So x=0.

Miniscule representations

Definition: A representation is miniscule if the following equivalent conditions hold.

(1) All weights lie in the same W-orbit.
(2) The representation has highest weight λ such that 0λ,α1 for all αϕ+.

Let V=V(λ) be a miniscule representation of G with highest weight v+V(λ). The stabilizer of the λ weight space is the parabolic subgroup P=Pλ=BWλB (where Wλ is the stabilizer of λ in W). The weights of V(λ) are precisely {w·λ|wWP}.

Lemma: All wWP, for P as above, are G-abelian, and {vw=fw·v+|wWP} gives a basis of V(λ).


WP is characterised as WP = { wW|α ϕ+,w,α <0 λ,α =1 } . Therefore 𝒰-wBw-1 (for wWP) is generated by 1-parameter subgroups 𝒰-α=exp𝔤-α for which λ,α=1. Any two such subgroups 𝒰-α,𝒰-β commute, since λ,α+β=2 and thus α+β is not a root of 𝔤 (by condition (2) for miniscule λ). So w is G-abelian.

That fw·v+Vw·λ is proved inductively. Let w=riw with (w)=(w)+1. Then wWP and fw·v+= fifw· v+Vw·λ-αi. On the other hand riw·λ=w·λ-αi,w·λαi. We have αi,w·λ=(w)-1αi,λ=1, since riw=w lies in WP and takes the positive weight (w)-1αi to -αi. Thus w·λ=w·λ-αi and fw·v+Vw·λ (and fw·v+ is nonzero).

Corollary: All matrix coefficients in 𝒪(𝒰-) of the miniscule representation V(λ) go to Schubert basis elements in HT(G/B) under the homomorphism (*) (matrix coefficients with respect to {vw}, that is).


This follows since fi(2) acts on V(λ) by 0.

Example: Consider the standard representation V(ρ1) of SL3. It is clearly miniscule. The homomorphism 𝒪(𝒰-)HT(G/B) gives rise to the "tautological" element u= ( 1 σG/B(r1) 1 σG/B(r2r1) σG/B(r2) 1 ) 𝒰-(HT(G/B)). Similarly the structure maps πL and πR:𝒪(h_)HT(G/B) correspond to hL= ( πL(ρ1) πL(ρ2-ρ1) πL(-ρ2) ) , hR= ( πR(ρ1) πR(ρ2-ρ1) πR(-ρ2) ) in h_(HT(G/B)).

Then the following relation holds. ( 1 σG/B(r1) 1 σG/B(r2r1) σG/B(r2) 1 ) ·(e+hL) = e+hR. This implies the factorization 𝒪((e+h_)×𝒰-) HT(KT) 𝒪(𝒰) from before expliticly.

Remarks: The map S𝒪(𝒰-)HT(G/B) gives rise to (after applying S and dualizing) a map H*(G/B)𝒰(n_-). So to any representation V with highest weight v+ one can define a subspace of V by applying the image of H*(G/B) in 𝒰(n_-) to v+. If v+ is of weight λ then the map H*(G/B)v+V factors through H*(G/B)H*(G/Pλ). It seems natural to ask whether the resulting map H*(G/Pλ)V is injective. If λ is miniscule then this map is in fact bijective.

There is also a similar construction for H*(ΩK). It will be shown later that H*(ΩK)𝒰(n_+e). Therefore one can apply it to the lowest weight vector v- of a representation V to obtain a subspace of that representation. If V is miniscule we again recover all of V (in types ADE). This is seen as follows.

Let s_+ be the centralizer in n_+af of e+e0=e+te-θ. Any representation of Gaf with miniscule highest weight ρi is isomorphic to V(ρ0af), the representation with highest weight ρ0 (since there is an admissible graph automorphism of the extended Dynkin diagram taking the vertex i to the 0 vertex). We have the following commutative diagram V*(ρi) V*(ρ2af) V*(ρ0af) ·v- 𝒰(n_+e) ev0 𝒰(s_+) By a theorem in the Kac-Moody case, the map 𝒰(s_+)V*(ρ0af) on the right hand side is bijective. Hence the composition is surjective and so is 𝒰(n_+e)V*(ρi).

From now on let us assume that G is finite-dimensional and 𝔽 a field.

Lemma: We have the following inclusion of 𝔽-valued points (not schematically) ZG(e)B.


Suppose gZG(e). Then, by the Bruhat decomposition, g=b1b2 for b1,b2B and nNG(T). We have b1b2·e=e, hence nb2·e = b1-1·e. Let wW be the Weyl group element represented by n. Then the left hand side of the above equation lies in the sum of weight spaces αw·Δ+gα, while the right hand side has nonzero components in all the 𝔤αi, for αiΠ. Thus Πw·Δ+, which implies that w=id.

Consider the morphism ϕ: (e+h_)×𝒰- e+b_- (e+h,u) u-1·(e+h). Let X(e+h_)×𝒰- and Ye+b_-. Then 𝒪(X) and 𝒪(Y) are graded polynomial rings over in N=#Δ+ generators, where the grading is given as follows. For 𝒪(X)=S𝒪(𝒰-) let S be graded as usual by degh_*=2, and 𝒪(𝒰-) by deg𝔤-α*=2ht(α). The grading on 𝒪(Y)=𝒪(b_-) is given by deg𝔤-α*=2(ht(α)+1). Then we get that ϕ*: 𝒪(Y) 𝒪(X) is a homomorphism of graded polynomial rings. Choose homogeneous generators of 𝒪(Y) and 𝒪(X). So 𝒪(Y)=[y1,,yN] and 𝒪(X)=[x1,,xN].

Lemma: ϕ*(y1),,ϕ*(yN) form a regular sequence in 𝒪(X)𝔽.


Let 𝒮=ϕ*(y1),,ϕ*(yN). Since the ϕ*(yi) are homogeneous elements in a graded ring it suffices to show that the depth of 𝒮 (or equivalently 𝒮) equals N. The following claim will imply that 𝒮=x1,,xN and hence this lemma.

Claim: Let hh_(𝔽) and u𝒰-(𝔽), then u-1·(e+h)=e u=1.


Consider the semisimple part of u·e=e+h. Since the semisimple part of e is zero it must be zero. On the other hand it must be conjugate to h. Hence h=0* and u·e=e. So uZG(e)(𝔽) which by a previous lemma is contained in B(𝔽). Therefore u=1.

We aim to prove the following

Theorem: The map 𝒪(((e+h_)×𝒰-)×e+b_(e+h_)) HT(G/B) is an isomorphism.

Notes and references

This is a typed version of Lecture Notes for the course Quantum Cohomology of G/P by Dale Peterson. The course was taught at MIT in the Spring of 1997.

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