Quantum Cohomology of G/P

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

Last update: 10 December 2013

Lecture 1: February 7 1997

Course Outline:

Let G:semisimple algebraic group over BG:a Borel PB:a parabolic KG:maximal compact T=KB:maximal torus inK W=NK(T)/T:Weyl group Then G/B=K/T and W acts on the de Rham cohomology space H*(K/T). Moreover, since K/T maps to the classifying space BT, we have a morphism H*(BT)H*(K/T) of algebras. The map G/BG/P:gBgP gives inclusion H*(G/P) H*(G/B) = H*(K/T). G/P is a smooth projective variety.

The de Rham cohomology H*(G/P) can be used to answer the following question: suppose that three subvarieties x1,x2 and x3 of G/P are in general position, and that k=13dimxk=dimG/P. What is the number of points in the intersection x1x2x3?

The quantum cohomology qH*(G/P) answers a more general question: what is the number of holomorphic maps ϕ:1G/P with a fixed degree such that ϕ(0) X1 ϕ(1) X2 ϕ() X3? Some features of qH*(G/P):

So take equivariant cohomology HT(G/P), where T acts on G/P from the left by left translations.

Can define T-equivariant quantum cohomology qHT(G/P). Then

Geometrical models - the variety Y.

For each parabolic, have ypY and Yp± { yY:limt ?????=yp } Y = pYp± over and 𝒪(Yp+) qH*(G/P) over 𝒪(Yp-) H*(Ω(HP)) where Ω(KP) is the group of loops in KP. Moreover, Yp-n for some n, and Y=YG-= YB+ 𝒪(YG-) 𝒪(YG-YB+) 𝒪(YB+) || || || H*(ΩK) qH*(G/B)(q) qH*(G/B) Will express the Schubert basis elements as matrix entries of some representations. The variety Y lies in G/B, where G is the Langland dual of G.

Lecture 2: February 11, 1997

Kac-Moody root datum

Definition: A generalized Cartan matrix is a matrix A=(aij)i,jI with integer entries for some finite set I such that

(1) aii=2 iI
(2) aij0 ij
(3) aij=0aji=0

A Kac-Moody root datum consists of


?????ition: Simple roots: π={αi}iIh Simple coroots: π={αi}iIh Weight lattice: h Coweight lattice: h Root lattice: Q=iIαi Co-root lattice: Q=iIαi where Q h isom.of adjoint type Q h isom.of simply connected type.

Remark: In the classical case, root datum comes from connected reductive algebraic groups over .

Definition: We say that A=(Aij)i,jI is symmetrizable if A=(diagonal)·(symmetric).

Assumption: Will assume that A is symmetrizable.

The numbers mij: Define, for ij, i,jI mij= { 2 ifaijaji=0 3 ifaijaji=1 4 ifaijaji=2 6 ifaijaji=3 ifaijaji4 The Weyl group W is the group with generators ri, iI with relations ri2 = 1,iI, (rirj)mij = 1,i,jI,ij. The ri's are called the simple reflections.


?????tions of W on h*, h,Q, Q, S=S(h*)

The Nil-Hecke ring A

Definition: the Nil-Hecke ring A_ associated to the root datum ( A=(aij)i,jI, h*,h, {αi}iI, {αi}iI ) is the associated ring with 1 with generators λˆ,Ai, λˆh*, iI and relations: λˆ+μˆ = λ+μˆ, λˆμˆ = μˆλˆ, λ,μh*, Aiλˆ = riλAi+ λ,αi, λh*,iI AiAi = 0,iI, AiAjAi mij = AjAiAj mij , (ij,i,jI). The grading on A_ is defined to be degλˆ = 2, degAi = -2. For wW and for any w=ri1 ri2 rin[red], set Aw = Ai1Ai2 Ain (Aid = 1). Then it is clear that

(1) Aw is independent of the reduced expression
(2) AvAw= { Avw if(v)+ (w)=(vw), 0 otherwise.
Clearly SA as a subring.

Proposition: {Aw:wW} is an S-basis for A_.

(Does this need a proof?)

Proposition: The map ZW A_: ri 1-αiˆAi = Aiαiˆ-1,iI defines an injective ring homomorphism.


Only need to check ri2=1, iI and (rirj)mij=1 for ij. Injectivity is clear (?)

Proposition: The following defines an A-module structure on S: s·s = ss Ai·s = 1αi (s-ri·s)


The induced ri action on S is sri·s, as the usual one.

Remark: Suppose we need to check certain specified operators for sS and Ai on some space M is an action. We first check 1-αiˆAi=Aiαiˆ-1. Then this is how ri acts. If this gives a W-action, we are done.

Proposition: For sS, iI and wW ws = (w·s)w Ais = ri(s)Ai+ Ai·s Ais = sAi+(Ai·s) ri in A_.


Note: All of this proof was crossed out in the scanned notes.

Just need to check that ri=1-αiˆ Ai=Aiαiˆ-1

The anti-automorphism * on A

*(S) = S *Aw = Aw-1 To check that this is an anti-automorphism, need to check only λˆAi=Ai riλˆ+ λ,αi 1λh*,iI. This is easy. Now since ri=1-αiˆAi =Aiαiˆ-1 we get *ri=αiˆAi -1=-ri. (Airi=-Ai,riAi=Ai) Consequently, *w=(-1)(w) w-1

Definitions of A_ on MSN and HomS(M,N)

Assume that M and N are A_-module and are thus ?????odules. Form MSN = MN/ {smn-msn} HomS(M,N) = { f:MN:f(sm) =sf(m) } want to define A_-module structures on MSN and HomS(M,N).

????? MSN: s·(mn) = smn Ai·(mn) = Ai·mn+ ri·mAi·n = mAi·n+Ai·m ri·n ????? check that this is an action, we first need to show that the above operators are well-defined. The s· operator is clearly OK. ????? sS, iI, we have, by definition Ai· ( smn-msn ) = (Ais)·mn+ (ris)·mAi ·n -Ai·msn- ri·m(Ais) ·n Using Ais = (ri·s)Ai+ Ai·s ris = (ri·s)ri ri = 1-αiˆAi ri·s = s-αiAi·s we get Ai· ( smn- msn ) = (ri·s)Ai ·mn+(Ai·s) mn +(ri·s)ri· mAi·n-Ai·m sn -ri·m(ri·s) Ai·n-ri·m (Ai·s)n = (ri·s)ri·m Ai·n-ri·m (ri·s)Ai·n +(s-αiˆAi·s) αi·mn-Ai·m sn +(Ai·s)mn- ( m-αiˆAi·m ) (Ai·s)n = (ri·s)ri·m Ai·n-ri·m (ri·s)Ai·n +sAi·mn-Ai msn - ( (Ai·s)αiˆAi ·mn-αiˆAi ·m(Ai·s)n ) +(Ai·s)mn- m(Ai·s)n Smn-m sn:m,nM,N . Hence Ai is well-defined.

Next, since ri=1-αiˆAi, we have Ai·mn+ri·m Ai·n = Ai·mn+ mAi·n- αiˆAi· mAi·n = Ai·mn- Ai·mαiˆ Ai·n+m Ai·n = Ai·mri·n +mAi·n = mαi·n+Ai· mri·n. This gives the 2nd expression for Ai·(mn).

Now for sS and iI, we need to show Ai·(s·(mn))= (ri·s)· (Ai·(mn))+ (Ai·s)·(mn) l.h.s. = (Ais)·mn+ (ris)·mAi ·n r.h.s. = (ri·s)Ai·m n+(ri·s)ri ·mAi·n+ (Ai·s)mn = (Ais)·mn+ (ris)·mαi·n = l.h.s. From this, we see that ri=1-αiˆAi=Aiαiˆ-1 acts by ri·(mn) = mn-αiˆAi ·mn-αiˆri ·mAi·n = ri·mn-ri ·mαiˆAi ·n = ri·mri·n This clearly induces an action of W on MSN. Thus we have proved that we indeed have an action of A_ on MSN.

On HomS(M,N), define: (s·f)(m) = sf(m) (Ai·f)(m) = f(Ai·m)+Ai ·f(ri·m) = Ai·f(m)-ri ·f(Ai·m) Need to check that this is indeed an action. Clearly s· is o????? First, since ri=1-αiˆAi and since f is S-linear, we have f(Ai·m)+Ai ·f(ri·m) = f(Ai·m)+Ai· (f(m)-αiˆf(Ai·m)) = Ai·f(m)+f(Ai·m) -(Aiαiˆ)· f(Ai·m) = Ai·f(m)-ri ·f(Ai·m) This shows that the two expressions for Ai·f are equal. Now we show that ?????(sm) = s(Aif)(m). ?????h.s. = f((Ais)·m)+ Ai·f((ris)·m) ?????h.s. = sf(Ai·m)+sAi ·f(ri·m) = f(sAi·m)+ ( Airi(s)+ Ai·s ) f(ri·m) = f(sAi·m)+ Ai·f(ri(s)ri·m) +f((Ai·s)ri·m) ????? ris = ri(s)ri Ais = sAi+(Ai·s)ri ????? see that l.h.s.=r.h.s.

????? shows that Ai·fHomS(M,N).

????? need to check that Ai·(s·f)= (ri·s)· (Ai·f)+ (Ai·s)·f ????? m sf(Ai·m)+ Ais·f(ri·m) ????? m (ri·s)f(Ai·m) +(ri·s)Ai·f (ri·m)+(Ai·s) f(m) =(ri·s)f (Ai·m)+Ais· f(ri·m)-f ( (Ai·s)ri·m ) +f((Ai·s)m) ????? Ais=(ri·s) Ai+Ai·sand sAi=Ais-(Ai·s) ri ????? see l.h.s=r.h.s.

Finally, for ri=1-αiˆAi=Aiαiˆ-1, we see that (ri·f)(m) = f(m)-αiˆf (Ai·m)- αiˆAi·f (ri·m) = f(ri·m)- αiˆAi·f (ri·m) = ri·f(ri·m). Consequently, (w·f)(m)= w·f(w-1·m) This is certainly an action of W on HomS(M,N). Hence we have an action of A_ on HomS(M,N).

All these proofs seem to be longer than necessary.

But anyway, we have showed that s·(mn) = smn Ai·(mn) = Ai·mn+ri· mAi=mAi ·n+Ai·mri w·(mn) = w·mw·n (s·f)(m) = sf(m) (Ai·f)(m) = f(Ai·m)+Ai ·f(ri·m)= Ai·f(m)-ri ·f(Ai·m) (w·f)(m) = w·f(w-1·m) make MSN and HomS(M,N) A_-modules again.

?????ition Given A modules M, N and P (they are therefore also S-modules), the following canonical S-module maps are also A-module maps:

4. HomS(MSN,P) HomS(HomS(M,N),P)
5. MSHomS(N,P) HomS(HomS(M,N),P)
6. HomS(M,N)SP HomS(M,NSP)

Definition: For an A_-module P, set PA= { pP:Ai·p=0 iI } .

Proposition: For A_-modules M and N, HomA_(M,N) (HomS(M,N))A.

Example: Regard A_ as an A_ module by left multiplications. Then our previous constructions define an A_-module structure on A_SA_. Define: Δ: A_ A_SA_ by Δa = a·(11) Thus Δw = ww Δs = s1=1s ΔAi = Ai1+riAi =1Ai+Airi For any two A_ modules M and N, since we have a·(mn)= a(1)·m a(2)·n for aS or a=Ai, iI, where Δa=a(1)a(2), we have a·(mn)= a(1)·m a(2)·n aA.

Proposition: In the finite case, ΔAw0 = wWAw0w w0Aw = wWAw w0Aw0w


It is easy to show by induction on (w) that for any wW ΔAw=Aww+ v<wAvav for some avA. So ΔAw0=wW Awaw with aw0=w0. Now for any iI, AiAw0 = 0 0 = Δ(Ai) Δ(Aw0) 0 = ( Ai1+ riAi ) wW Awaw = wW ( AiAw aw+ri AwAiaw ) = wW ( AiAwaw+ (1-αiˆAi) AwAiaw ) = wWAi Awriaw- AwAiaw wWAi Awriaw = wWAw Aiaw Now l.h.s. = riw>w Ariw riaw ariw = -Aiawif riw<w aw0 = w0Aw0w. ?

Lecture 3: February 12, 1997

Recall that a Kac-Moody root datum consists of

The weight lattice is h
The coweight lattice is h
The root lattice is Q=defiIαi
The co-root lattice is Q=defiIαi
Say the datum is of the adjoint type if Qh:αiαi is an isomorphism.
Say the datum is of the simply connected type if Qh:αiαi is an isomorphism.

For sl(2,), use e,f,h for the standard generators such that [h,e] = 2e [h,f] = -2f [e,f] = h Given a Kac-Moody root datum (A,I,h,h,), set h_=h and regard it as a commutative Lie algebra. Set hi=αi

Theorem (see Kac?): For any Kac-Moody root datum, there exists a Lie algebra g_ over (of Kac-Moody type) and Lie algebra homomorphisms ϕ: h_ g_ ϕi: sl2() g_ iI such that

(1) ϕi(h) = ϕ(hi) [ ϕ(h), ϕi(e) ] = αi,h ϕi(e)hh_ [ ϕ(h), ϕi(f) ] = -αi,h ϕi(f)iI [ ϕi(e), ϕj(f) ] = 0(ij)
(2) For each iI, g_ as an sl2() module via ϕi (using the adj. rep) is a direct sum of finite-dimensional sl2()-module.
(3) If g_,ϕ, or ϕi are another such system, then there exists a unique ψ:gg such that ϕ=ψϕ and ϕi=ψϕi. Thus (g,ϕ,ϕi,iI) is unique.


(1) An sl2()-module V over is integrable if it is a direct sum of finite-dim. modules.
(2) An h_-module V over is integrable if V=μh* Vμ where Vμ= { vV:hv=μ(v)v for allhh_ }
(3) A g_-module V over is integrable if it is sl2()-integrable (via ϕi, for each iI) and h_-integrable via ϕ.
So the adjoint representation of g_ on g_ is integrable.


Every ideal of g_ contained totally in n- or n+ is 0. b+ =def h_+n+=b (Borel subalgebra) b- =def h_+n-

Fact: g_ is the Lie algebra over with generators hh_,ei, fi,iI with relations [h,h] = 0 [h,ei] = αi,hei [h,fi] = -αi,hfi [ei,fj] = δijhi (adei)1-aij ej = 0(ij) (adfi)1-aij fj = 0(ij)

Warning: h_centralizer of h_ing_

The Q-grading of g_:

For βQ, the root lattice, set gβ= { xg:[h,x]= β,hx hh_ } . Then g_=βQ gβ and [gβ1,gβ2]gβ1+β2

?????ice that g0h_ gαi=ei g-αi=fi iI. + = {0,1,2,} Q+ = iI+αi Qsub-semigroup ????? β,νQ, say βν if β·νQ+.

????? n±= βQ± gβ Δ = { βQ:gβ0, β0 } ,set of roots Δ+ = ΔQ+,set of positive roots Π = {αi:iI}, set of simple roots

????? Δ- = -Δ+

????? Δ+ Δ- = Δ Δ+ Δ- = n± = βΔ± gβ

The principle -grading of g_:

Let ρQ be the unique(?) element such that αi,ρ 1iI. For βQ, the integer ht(β)=β,ρ is called the height of β. For n, set g_n= βQht(β)=n gβ Thus is a -grading for g_.

The set of real roots:

Need to define the Weyl group first. To define the Weyl group, need to define the Kac-Moody group.

Compact involution of g_:

This is the conjugation-linear automorphism of g_ such that ei -fi i????? h -h hh_ =det hh_.

Kac-Moody group

????? ():

For u, set t×, set x(u) = (1u01) y(u) = (10u1) h(t) = (t00t-1) SL2(). ?????l finite-dimensional representation of SL2() is said to be ?????ational if its matrix entries are regular functions on SL2(). ????? representation of SL2() on a vector space V over is said to be differentiable if it is a direct sum of finitely many finite dimensional rational representations.

?????t: Integrable representations of sl2()differentiable rep. of SL2(). (This is because SL2() is an algebraic group).

The complex torus H:

Define H=Hom(h,×). For hh and t×, define thH by th(λ)= tλ,h, λh. Thus, for each such hh, the map × H: t th is a homomorphism. Moreover th+h= th·th. A representation of H on V/ is said to be differentiable if it is a direct sum of 1-dimensional rational representations of H.

Fact: Differentiable representations of H integrable representations of h_.

Next: The Kac-Moody group G corresponding to the Kac-Moody root datum we started with at the beginning.

The Kac-Moody group G:

Given the Kac-Moody root datum, there is a group G with homomorphisms ϕ: H G ϕi: SL2() G iI ϕi(h(t)) = ϕ(thi) ϕ(th)ϕi (x(u))ϕ (t-h) = ϕi (x(tαi,hu)) ϕ(th)ϕi (y(u))ϕ (t-h) = ϕi (y(t-αi,hu)) ϕi(x(u)) ϕj(y(v)) = ϕj(y(v)) ϕi(x(u)), ij There exists a representation Ad of G on g_ such that under ϕ and ϕi, iI, the corresponding representations of H and SL2() on g_ differentiate to the representations of h_ and sl2() on g_ defined by ad.

If (G, ϕ and ϕi) is another system with above properties, then there exists a unique ψ:GG such that ϕ=ψϕ ϕi=ψϕi

The Weyl group W:

For each iI, u, set xi(u) = ϕi(x(u))= ϕi (1u01) G yi(u) = ϕi(y(u))= ϕi (10u1) G ni = yi(1)xi(-1) yi(1)=ϕi (0-110) G ????? ninjni mij = njninj mij ????? for w=ri1rin [red], let nw=ni1ni2 nin ????? nw·gβ= gw·β ????? in general nwnw-1 id. ????? N=ni,HiI G ????? the subgroup of G generated by {ni,iI} and H. ????? N/H W ni/H ri Warning: can have HZG(H).

The real roots:

Note that W·Δ=Δ so W permutes the root system.

Set Δre= iIW ·αi and call elements in Δre the real roots.

If β=w·αiΔre for some iI, then gβ=nw·gαi so dimgβ=1 and gmβ=0for |m|>1. Also, define rβ=wriw-1 W. Then

(1) rw1β= w1rβw1 for any w1W
(2) rβ·λ = λ- λ,β β λh* rβ·h = h-β,h βhh.

Lecture 4: February 19, 1997

I am moving the part on Bruhat decomposition of G/P to the end of lecture 3. The main part of this lecture is on

Equivariant Cohomology (due to Borel):

Example: BS1 = P ET = EK(becauseTK)

Definition: An L-space is a topological space X endowed with a continuous left L-action: L×X X: (,x) ·xx.

Definition: Given an L-space X, form the space EL×LX= (EL×X)/L where (e,x)·=(e-1,x) is a free left L-action. The L-equivariant cohomology of X is by definition the singular homology of EL×LX: HL(X)= H*(EL×LX)

Structures on HL(X):

  1. It is a graded ring, where the grading is nothing but the grading on H*(EL×LX). (And so is the ring structure ?????
  2. The fibration EL×LXEL/L=BL gives a graded ring homomorphism H*(BL) H*(EL×LX) i.e. HL(pt) HL(X) Thus HL(X) has a natural HL(pt)-module structure


Given an L-map of L-spaces f: X Y, form the map EL×LX EL×LY [e,x] [e,f(x)]. Have commutative diagram EL×LX EL×LY πx πy BL id BL [e,x] [e,f(x)] [e] = [e] have a graded ring homomorphism f*: HL(X) HL(Y) which is also a H*(BL)= HL(pt)-module map.

????? special case of Y=pt with f: X pt, ?????es πx: EL×LX EL×Lpt = BL, f*: HL(pt) HL(X) ????? just the one considered before.

L-equivariant homology

This is the space of HomHL(pt) (HL(X),HL(pt)). Than any L-space map f: X Y induces f*: HomHL(pt) (HL(X),HL(pt)) HomHL(pt) (HL(Y),HL(pt)).

The restriction homomorphism (or the evaluation at 0):

This is the map v0(X): HL(X) H*(X) induced by the map EL×X EL×LX. This is a graded -ring homomorphism.

For any L-space map f:XY, have commutative diagram HL(X) v0(X) H*(X) f* f* HL(Y) v0(Y) H*(Y)


1. L acts freely on X. Then HL(X)H*(X/L)


Have the following fibre bundle with contractible fibre EL. EL×LX EL X/L Thus HL(X)= H*(EL×LX) H*(X/L).

2. L acts trivially on X. Then HL(X) HL(pt) H*(X)


Have EL×LX BL×X.

Proposition: Have H*(BT)S(h*).


For λh*, define eλ: T ×: eλ(eh) = eλ,h,hh. If E is a principal T-bundle, form the complex line bundle -λ=E×T by [et,c]= [e,e-λ(t)c] tT,eE,c. Then λC1(E×T), the first Chern class gives a homomorphism S(h*) H*(E/T). In particular, take E=ET=EK. Then get S(h*) H*(ET/T) =HT(pt). One can then show that this is an isomorphism of graded rings if λh* is given deg=2.

The second S-module structure on HT(K/T)

Set Eu=ET=EK. The map Eu×T(K/T) Eu/T: [e,kT]ekT is another ring homomorphism, which we will denote by πR for reasons that will be clear next time; πR: S HT(K/T).


  1. πR, together with the map πL: S HT(K/T) incuded by K/Tpt, will be the source and target maps for the Hopf algebroid structure on HT(K/T) that will be discussed next lecture.
  2. Set Eu(2) = Eu×Eu/KEu= { (e1,e2) E×E:e1K=e2 K } Eu×Eu It is a (K×K)-inv. subset of Eu×Eu. Since K acts on Eu freely, we have the identification Eu(2) Eu×K: (e1,e2) (e1,k) ife2=e1. Under this identification, the T×T action on Eu(2) becomes the action (e,k)(t1,t2) (e1t1,t1-1kt2) of T×T on Eu×K. (easy to check this: (e,k) (e1,e1k) (t1,t2) (e1t1,e1kt2) (e1t1,e1t1t1-1kt2) (e1t1,t1-1kt2)). Thus we have Eu(2)/T×T Eu×TK/T The map Eu×T(K/T) Eu/T: [e,kT] ekT now is just the projection from Eu(2)/T×T to the 2nd factor Eu????? This will be used in the next lecture.

Proposition: For any T-space Y, we have HT(K×TY) HT(K/T)S HT(Y) where the S-module structure on HT(K/T) is via the second ring homomorphism πK:SHT(K/T). (The S-module structure on HT(Y) is the usual one).


Consider the following commutative square: (?????K×TY) p1 Eu×K(K×TY) Eu×TY p2 q1 (?????K×Tpt) q2 Eu×K(K×Tpt) | | ?????(K/T) Eu/T = [e,[k,y]]T p1 [e,[k,y]]K [ek,y] p2 q1 [e,[k,y]]T q2 [e,[k,p]]K [e,k] [ek] = [ek,pt] notice that q1*:SHT(Y) is the usual homo. (induced from Ypt). ????? q2*=πR:SHT(K/T) is the second homomorphism. Now since the square is commutative, ie. q2p2=q1p1, we get a ring homomorphism HT(K/T)SHT(Y) HT(K×TY). xy p2*(x)p1*(y) assuming even col????? To show that this is an isomorphism, we first notice that the fibration p1 has fibre K/T which is a CW-complex if only even dimensions. Thus Leray-Hitsch theorem tells us that HT(K×TY) is a free module over HT(Y) with basis coming from H*(K/T). The special case of Y=pt says that HT(K/T) is a free S=HT(pt)-module with basis coming from H*(K/T). Using a basis of H*(K/T), we see that the map HT(K/T)SHT(Y) HT(K×TY) is an isomorphism.

Definition: The morphism ε: HT(K/T) S induced by T/T K/T is called the co-unit map.

Definition: For any K-space X, the map ΔX: HT(X) HT(K/T)SHT(X) induced by the T-map μK: K×TX X: [k,x] kx, ie. Δx: HT(X) μK* HT(K×TX) HT(K/T)SHT(X) is called the co-module map.

Proposition: For any K-space X, we have (εid)ΔX= id|HT(X) and (ΔK/Tid)ΔX = (idΔX)ΔX: HT(X) HT(K/T)SHT(K/T)SHT(X).

Definition: A groupoid scheme (𝒴,𝒮) consists of two schemes 𝒴 and 𝒮 and five morphisms: PL,PR: 𝒴 𝒮 : 𝒮 𝒴 i: 𝒴 𝒴 μ: 𝒴×S𝒴 𝒴 (fibre produ????? -×S refers to PL, and ×S- refers to PR) They must satisfy: PL=id𝒴 =PR PLi=PR PRi=PL PLμ=p2p1 PRμ=PR p2 μ(id𝒴,PR) =id𝒴 μ(PL,id𝒴) =id𝒴 μ(id𝒴,i)= iPRμ (i,id𝒴)= PR μ(id𝒴×μ)= μ(μ×id𝒴) These imply ii=id𝒴.

If 𝒴=SpecR and 𝒮=SpecS, then 𝒴×S𝒴=Spec (R×SR).

Lecture 5: February 25, 1997

Recall the concept of a groupoid:

A groupoid is a small category with every morphism invertible

Example: Let G be a group acting on a space X. Then we can form a groupoid (𝒴,𝒮), where 𝒮=X, 𝒴= { (x,g,y): x,yX,x=g·y } Multiplication is given by (x,g,y) (x,g,y)= (x,gg,y) ify=x Source map: 𝒴S: (x,g,y)y Target map: 𝒴S: (x,g,y)x Inverse map: 𝒴𝒴: (x,g,y) (y,g-1,x) Units: S𝒴: x(x,e,x).

An action ϕ:𝒴×SXX of a groupoid scheme (𝒴,S) on a scheme X ????? S with structure morphism PX:XS is one such that

(1) ϕ(μ×idX)= ϕ(id𝒴×ϕ)
(2) PXϕ= PLP1 where P1:𝒴×X𝒴, (y,x)?????
(3) ϕ((ePX)×idX)=idX.

The groupoid scheme 𝒰=SpecHT(K/T)

Let Eu be the principal K (and thus also T)-bundle. For n1, let Eun = Eu××Eu ntimes Kn = K××K ntimes Tn = T××T ntimes Set Eu(n)= { (e1,,en) Eun:e1L= =enK } Eun. As a subset of Eun, the set Eu(n) is invariant under the Kn-action, so Eu(n) is a principal Kn-bundle.

Set B(n)= Eu(n)/Tn Then it is easy to check that B(2) is a groupoid over B(1)=E/T=BT with the following structure maps: (This is a subquotient of the coarse groupoid E×E over E):

We now pull back all the above structure maps on cohomology:

First note that Eu(2)Eu×K by (e1,e2) (e1,k) ife2=e1k Under this identification, the T2 action on Eu(2) becomes (e1,k) (e1,e1k) (e1t1,e1kt2) (e1t1,e1t1t1-1kt2) (e1t1,t1-1kt2) Thus we get an induced identification Eu(2)/T2 Eu×TK/T [e1,e2] [e1,kT] ife2=e1k Similarly, we have Eu(3) Eu×K×K : (e1,e1k1,e1k1k2) (e1,k1,k2) and (e1t1,e1k1t2,e1k1k2t3) = ( e1t1, e1t1t1-1 k1t2, e1k1t2 t2-1k2t3 ) ( e1t1, t1-1k1 t2,t2-1 k2t3 ) so Eu(3)/T3 (Eu×K×K)/T3 where the T3 action on Eu×K×K is (e1,k1,k2)· (t1,t2,t3)= ( e1t1, t1-1k1t2, t2-1k2t3 ) But (Eu×K×K)/T3 Eu×T (K×TK/T) so we have the identifications B(2) Eu×T K/T B(3) Eu×T (K×TK/T) Hence H(B(2)) HT(K/T) H(B(3)) HT(K×TK/T) HT(K/T)S HT(K/T) (from last time) where the last identification is due to the general fact we proved last time that for any K-space Y, HT(K×TY) HT(K/T)S HT(Y). We also have H(B(1))= H(E/T)=S Therefore, the pull-backs on cohomology of all the structure maps for the groupoid B(2) over B(1) give the groupoid structure on 𝒰=SpecHT(K/T).


Set R=HT(K/T), S=HT(pt)=H(BT)=H(B(1)). Then from: p1: B(2) B(1): [e1,e2] [e1] p2: B(2) B(1): [e1,e2] [e2] d: B(1) B(2): [e] [e,e] t: B(2) B(2): [e1,e2] [e2,e1] μ: B(3) B(2): [e1,e2,e3] [e1,e3] we get: πL = p1*: S R πR = p2*: S R ε = d*: R S c = t*: R R Δ = μ*: R RSR

Theorem: The above maps πL, πR, ε, c and Δ make (𝒰=SpecR,h_=SpecS) into a groupoid scheme. Moreover, if X is any K-space, the map ΔX = { K×TXX: [k,x]kx } * : HT(X) HT(K/T)HT(X) is the composition of an action of (𝒰,h_) on SpecHT(X), (assuming that H*(X) is even)

Characteristic operators

Definition: A characteristic operator for (K,T) is a rule that assigns to each K-space X an HK(X)-linear endomorphism ϕX:HT(X)HT(X) such that if F:XY is a K-map then F*ϕY=ϕXF*.

Remark: When K=T, any HT(X)-linear endomorphism of HT(X) must be a multiplication operator by characters. This is why the name characteristic operators.

Fact: The set Aˆ_ of all characteristic operators is an S-algebra.

Definition: We say that a characteristic operator is of compact support if there exists a compact subset K0K which is T-stable such that given any K-space X, a T-stable subset X0 of X and an element zHT(X) vanishing in HT(K0X0), the element ϕX(z0) must vanish in HT(X0).

Remark: In the finite case, can take K0=K and every characteristic operator is compact.

Definition-Notation: Aˆ_c = theS-subalgebra ofAˆ_ of all characteristic operators of compact support.

Proposition: For any characteristic operator a and any K-space X, we have ΔXa = (aid)ΔX: HT(X) HT(K/T)SHT(X)

Corollary 1: For a characteristic operator a, we have a=0 a=0on HT(K/T) εa=0 HomS(HT(K/T),S).


If εa=0:HT(K/T)S, then for any K-space X, aonHT(X) = (εid) ΔXa (because (εid) ΔX= idHT(X)) = (εid) (aid)ΔX (by Proposition) = (εaid) ΔX = 0.

Corollary 2: Aˆ_ has no S-torsion.


If sS and aAˆ_ are such that sa=0anda0 then for any zHT(K/T) 0=(εsa)(z) = ε(s(a·z)) = sε(a·z) But since a0, we know by Corollary 1 that εa????? so z0 s.t. ε(a·z)0S. Since S is a polynomial algebra, it has no S-torsion. Thus S=0. This shows that Aˆ_ has no S-torsion.

Corollary 3 (added by me) (of Corollary 1): The action of aAˆ_ on HT(X) is expressed using ΔX: HT(X) HT(K/T)SHT(X) and the map εa:HT(K/T)S by aonHT(X) = (εaid) ΔX.

Remark: Should think of Aˆ_ as the dual of HT(K/T) by aεaHom(HT(K/T),S).

Integration over the fibre

Assume that P:EB is a fibration over a pathwise connected base B with b0B. Let F=P-1(b0). Assume that this fibration is orientable. This means that the holonomy around b0 acts trivially on H*(F). Since B is pathwise connected, the weak homotopy type of F is independent of the choice of b0. Then we have, assuming Hγ(F)=0 for γ>n Hom(Hn(F),) ( HomH*(B) (H*(E),H*(B)) degree-n ) denoted by τ τ obtained as follows by using the Serre spectral sequence: Hm+n(E) Em,n E2m,n Hm(B,Hn(F)) τ Hm(B,).


(1) This is just the identity map when B=pt.
(2) It is functorial over pullbacks.
(3) It preserves certain Mayer-Vietoris sequences
(4) Can do this for relative cohomology as well.

The A_-action on H*(E/T) for any principal K-bundle E

If E is a principal T-bundle, then we have a ring homomorphism ch: S Heven(E/T) : λ c1(-λ=E×Te-λ) H2(E/T). We call it the characteristic homomorphism. Using the characteristic homomorphism, we get an S-module structure on H*(E/T): s·z = ch(s)z.

Now assume that E is also a principal K-bundle, so thus also a T-bundle. Then we can use the K-action to define the following W-action on H*(E/T): for wW, w·z = w*z where w:E/TE/T: w·eT=ewT. Because of the following basic property of the characteristic map, w*c1(-λ) = c1(w*-λ)= c1(-w·λ) ie.w*ch(λ) = ch(w·λ) we have, for any wW and sS ws = (w·s)w as operators on H*(E/T). Therefore we have an action of the smashed product algebra WS on H*(E/T).

Now for each iI, consider the fibre bundle E/T πi E/Ki which has fibre Ki/TPi/BP1 so it has a preferred orientation σiHom(H*(Ki/T,)) namely the fundamental cycle. Integration over the fibre gives H*(E/T) H*-2(E/Ki) : z σiz Now define Ai: H*(E/T) H*-2(E/T) : Ai·z = πi*σiz.

Proposition: For any zH*(E/T), αi·(Ai·z) = z-ri·z (*)


We will check this over (Why?). The fibration πi:E/TE/Ki gives a H*(E/Ki)-module structure on H*(E/T). Since the fibre is P1, this is in fact a free H*(E/Ki)-module, a basis of which is given by 1 and 12ch(αi)H*(E/T). For z0H*(E/T) we use the same letter to denote the pull back πi*?????H*(E/T). We will check (*) for z=z0 and z=12ch(αi)z0. Clearly Ai·z0=0 and ri·z0=z0. Thus (*) holds for z=z0. Now for z=12ch(αi)z0, αi·(Ai·z) = αi· ( Ai·(ch(αi)2) z0 ) .

Lemma: Ai·ch(αi)=2. (a calculation over P1)

Assume Lemma. Then αi·(Ai·z)= αi·z0=ch (αi)z0. On the other hand, z-ri·z0 = 12ch(αi)z0- ri·(12ch(αi)z0) = 12ch(αi)z0- ri·(12ch(αi)) ri·z = 12ch(αi)z0+ 12ch(αi)z0 = ch(αi)z0. Hence (*) holds for z=12ch(αi)z0.

It is strange to carry the 12 around. Why necessary?

Therefore we have

Theorem: For any principal K-bundle E, the following define an A_-action on H*(E/T): s·z = ch(s)z w·z = w*z Ai·z = πi* σiz Moreover, the characteristic morphism ch: S Heven(E/T): λ c1(-λ) is an A_-map, where Ai acts on sS by Ai·s = s-ri·s αi as before (see Lecture 2).

Example: E=K with right action of K by right multiplications. Then the Ai's on H*(K/T) are the BGG-operators.

Example: If E1fE2 is a K-map, then f*:H*(E2/T)H*(E1/T) is clearly an A_-map.

A_-action on HT(X) for K-space X:

Example: Let X be a K-space and let Eu=EK be the universal principal bundle of K. Let E = Eu×X with the K-action given by (e,x)·k = (ek-1,kx). Then E/T = Eu×TX so get an action of A_ on HT(X). If f:XY is a K-map, then Eu×X Eu×Y : (e,x) (e,f(x)) is a K-map, so f*: HT(Y) HT(X) is an A_-map. Finally, the A_-action on HT(X) is clearly HK(X)-linear. Thus we can think of elements of A_ as characteristic operators.

Property: For any K-space X, the morphism S HT(X) (=(Xpt)*) is an A_-map.


This is the same as the characteristic morphism. ?????

Proposition: For any K-space X, the multiplication map HT(X)SHT(X) HT(X) is an A_-map.

T-equivariant homology

Example: Suppose Y is a T-space such that Hr(Y)=0 for r>n. Then integration over the fibre for Y Eu×TY Eu/T gives a map Hom(Hn(Y),) HomS(HT(Y),S) τ τ For each iI, we have a map Hom(Hn(Y),) Hom(Hn+2(Ki×TY),) : τ σi*τ where σi*τHom(Hn+2(Ki×TY),) is the composition Hn+2(Ki×TY) τ H2(Ki/T) σi using integration over the fibre first for the bundle Y Ki×TY Ki/T . Consequently we have a map Hom(Hn(Y),) Hom(Hn+2(Ki×TY),Z) HomS(HT(Ki×TY),S) τ σi*τ σi*τ. Now suppose that X is a K-space with K-action μ: K×X X. Assume that F:YX is a T-equivariant map. Then for τHom(Hn(Y),Z), we have τHomS(HT(Y),S), so F*τ HomS(HT(X),S) and thus Ai·F*τ HomS(HT(X),S). On the other hand, we have Ki×TY Fi Ki×TX μ X [ki,y] [ki,f(y)] ki·f(y) and σi*τ HomS(HT(Ki×TY),S).

Fact: Ai·F*τ = μ*Fi* σi*τ HomS (HT(X),S).



This fact will be used in the next lecture for Y=XwP, a Schubert variety in ?????

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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