## Quantum Cohomology of $G/P$

Last update: 19 December 2013

## Lecture 16: April 22, 1997

Recall last time:

• Defined $q{H}^{*}\left(G/P\right)$ from $Jn,τ: ⊗n-1H*(G/P) ⟶ H*(G/P).$
• $\sigma *\sigma \prime$ from ${J}_{3,\tau }\text{'s.}$
• Gave formula for ${\sigma }^{{r}_{i}}*$ in $q{H}^{2}\left(G/B\right)\text{.}$
• Comparison of $q{H}^{*}\left(G/B\right)$ and $q{H}^{*}\left(G/P\right)\text{.}$

Today: Compare ${R}_{P}$ and $q{H}^{*}\left(G/P\right)\text{:}$

Theorem 1: We have an isomorphism $ℤ⊗SRP ≅ qH*(G/P) 1⊗qτσP(w) ⟼ qτσPw.$

 Proof. First the case of $G/B\text{:}$ The map $1\otimes {q}_{\tau }{\sigma }_{B}^{\left(w\right)}\to {q}_{\tau }{\sigma }_{B}^{w}$ is bijective. Since both sides are generated by ${H}^{2}$ and there is no torsion remains to check formula for multiplications by ${H}^{2}$ 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. $\square$

Theorem 2: We have an isomorphism $RP⊗ΛPℤ ⟶∼ HT(G/P) sσP(w)⊗1 ⟼ sσP(w).$

 Proof. For $G/B,$ directly from the multiplication formula by ${H}^{2}\text{.}$ For $G/P,$ take $\tau =0$ and $h=0$ in the commutative diagram at the end of Lecture 15. $\square$

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 ${\psi }_{P}\text{:}$

$HT(ΩK) ⟶ψP qHT(G/P)(q) ↓ ↓ H*(ΩK) ⟶ψ‾P qH(G/P)(q)$ When $P=B,$ ${\stackrel{‾}{\psi }}_{P}$ is an isomorphism if we also invert the tranlational elements in ${H}_{*}\left(\Omega K\right)\text{.}$ Using ${\stackrel{‾}{\psi }}_{P},$ we get structure constants for ${H}_{*}\left(\Omega K\right)$ as Gromov-Witten invariants, which, since they are numbers of certain curves, are non-negative integers.

### Results of Bott:

For certain $h\in {Q}^{\vee },$ construct $K/T\stackrel{{\varphi }_{h}}{↪}\Omega K$ by $ϕh(kT)(t) = thkt-h k-1$

• $\exists$ $h$ s.t. $\text{Im} {{\varphi }_{h}}_{*}\left({H}_{•}\left(K/T\right)\right)$ generates ${H}_{*}\left(\Omega K\right)\text{.}$
• Can find $\text{Im} \left[\text{Prim} {H}^{*}\left(\Omega K\right)\right]$ in ${H}^{*}\left(K/T\right)\text{.}$
• Related to ${H}^{T}\left(K/T\right){\otimes }_{S}{H}_{T}\left(\Omega K\right)\to {H}_{T}\left(\Omega K\right)$ or $\text{Spec} {H}^{T}\left(K/T\right){×}_{\underset{_}{h}}\text{Spec} {H}_{T}\left(\Omega K\right)←\text{Spec} {H}_{T}\left(\Omega K\right)\text{.}$

### Geometrical Models

Will construct geometrical models for

• the groupoid scheme $𝒰=\text{Spec} {H}^{T}\left(G/B\right)$ (finite $G\text{);}$
• scheme ${𝒰}_{G/P}=\text{Spec} {H}^{T}\left(G/P\right)$ with a groupoid $𝒰\text{-action;}$
• group schemes: $𝒜ˆ = Spec HT(ΩK) 𝒜 = Spec HT(Ω0K)$ (do not assume $G$ is simply connected);
• $q𝒰 = Spec qHT(G/B), q𝒰G/P = Spec qHT(G/P) (q𝒰G/P)q = Spec qHT(G/P)q.$ All equipped with groupoid $𝒰\text{-actions;}$
• The variety $𝒴$ (used to be denoted by $Y\text{)}$
• It is a projective scheme over $\underset{_}{h}$ "with pieces $q{𝒰}_{G/P}\text{."}$
• It has an "open piece" ${𝒴}_{\left(q\right)}$ where the $\ell$ distinguished line bundles have nonvanishing sections (?) (will explain later).
• Has $ℤ\text{-points}$ ${Y}_{P}\in 𝒴\left(ℤ\right)$ for each parabolic $P\text{.}$
• ${𝔾}_{m}=\text{Spec} ℤ\left[t,{t}^{-1}\right]$ acts on all and gives gradings;
• Have homomorphism $\stackrel{ˆ}{𝒜}\to 𝒜$ as group schemes;
• $𝒜,$ as a groupoid scheme, acts on $𝒴,$ and can identify the $𝒜\text{-orbit}$ through ${Y}_{G}$ with ${𝒴}_{\left(q\right)}\text{;}$
• Have natural morphisms $𝒰G/P ⟶ q𝒰G/P (corresponding to qHT(G/P)⟶q=0HT(G/P)) q𝒰G/P ⟶ 𝒴 imbeddings (q𝒰G/P)(q) ⟶ 𝒴(q)$
• $q𝒰G/P∩q𝒰G/P′ = ∅if P≠P′$ but $q𝒰G/P∩𝒜 = (q𝒰G/P)(q).$ (In the Peterson lingo, "the quantum cohomology res do not see each other, but they all see the homology of $\Omega K\text{").}$

Now we turn to the first model for $𝒰\text{:}$

### The first model $𝒰$ of $\text{Spec} {H}^{T}\left(G/B\right)\text{:}$

Let $e=\sum _{i\in I}{e}_{{\alpha }_{i}}\text{.}$

Lemma: For any $h\in \underset{_}{h},$ the fixed points of the vector field ${V}_{e+h}$ on $G/B$ defined by $e+h$ all lie in the cell $·B\text{.}$

Theorem: $𝒰 = Spec HT(G/B) ≃(G/B)e+h_ = { (e+h,x)∈ (e+h_)× G/B:Ve+h (x)=0 } = { (e+h,u-·B): u--1·(e+h) ∈b_ } where the second · is the adjoint action.$ But since ${U}_{-}$ stabilises $e+{\underset{_}{b}}_{-},$ when ${u}_{-}^{-1}·\left(e+h\right)\in \underset{_}{b},$ 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 $𝒰\text{:}$

• $𝒰\underset{t}{\overset{s}{⇉}}\underset{_}{h}:$ $\left(e+h,{u}_{-},e+h\prime \right)\genfrac{}{}{0}{}{\stackrel{s}{↦}}{\underset{t}{↦}}\genfrac{}{}{0}{}{e+h}{e+h\prime }$
• $𝒰{×}_{h}𝒰\to 𝒰:$ $\left(e+h,{u}_{-},e+h\prime \right)·\left(e+h\prime ,u\prime ,e+h″\right)=\left(e+h,\text{?????}\right)$
• $\underset{_}{h}\to 𝒰:$ $h↦\left(e+h,1,e+h\right)$ (identities)
• inverse: $𝒰\to 𝒰:$ $\left(e+h,{u}_{-},e+h\prime \right)↦\left(e+h\prime ,{u}_{-}^{-1},e+h\right)$

As a model for $\text{Spec} {H}^{T}\left(G/B\right),$ we must have two $W\text{-actions}$ on $𝒰$ which gave ${w}_{L}$ and ${w}_{R}$ on ${H}^{T}\left(G/B\right)\text{.}$ We identify these two actions in the next lecture.

## Lecture 17: April 23, 1997

The following works for the general Kac-Moody case:

Set $e = ∑i∈Iei∈ n_+, fi(n) = finn! ∈U(n_-).$ Then $U(n_-)ℤ = ⟨fi(n)⟩i∈I,n≥0$ and using the action of $2{\rho }^{\vee }$ we can give $U{\left({\underset{_}{n}}_{-}\right)}_{ℤ}$ a $ℤ\text{-grading}$ with $\text{deg} {f}_{i}=-2\text{.}$

Define $𝒪(U-) = Homℤ (U(n_-)ℤ,ℤ) (graded dual)$ and we use ${U}_{-}$ to denote the groupscheme defined by $𝒪\left({U}_{-}\right)\text{.}$

Lemma: For any $w\in W,$ 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 $Uri(h)= exp(⟨αi,h⟩fi) =yi(⟨αi,h⟩)$ where, recall, ${\varphi }_{i}:SL\left(2,ℂ\right)\to G$ and for $u\in ℂ$ $\text{(}ℤ\text{?)}$ $xi(u) = ϕi (1u01), yi(u) = ϕi (10u1).$

### The finite case:

In this case, a theorem of Kostant says that the element ${U}_{w}\left(h\right)\in {U}_{-}$ is unique for any given $w\in W$ and $h\in \underset{_}{h}\text{.}$

Example: For $𝔤=sl\left(3\right),$ $h=\text{diag}\left({x}_{1},{x}_{2},{x}_{3}\right),$ 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: ${U}_{{w}_{0}}\left(t{\rho }^{\vee }\right)=\text{exp}\left(tf\right),$ $t\in ℂ,$ where $\left\{e,f,2{\rho }^{\vee }\right\}$ is a TDS.

### The affine case:

In this case, define ${U}_{{r}_{i}}\left(h\right)={y}_{i}\left(⟨{\alpha }_{i},h⟩\right)$ for $i\in {I}_{\text{af}}$ and use $Uvw(h) = Uv(w·h) Uw(h)$ to extend to any $w\text{.}$ This is well-defined because of the braid relations: assume that $2<{m}_{ij}<\infty$ and $⟨{\alpha }_{j},{\alpha }_{i}^{\vee }⟩=-1$ for $i\ne j\text{.}$ Set $a=⟨{\alpha }_{i},h⟩,$ $b=⟨{\alpha }_{j},h⟩\text{.}$ 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 $\text{(}{U}_{w}\left(\text{?????}\right)$ 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 ${m}_{ij}=\infty \text{?}$ This is what is needed in the affine case?

Remark: In the affine case, the element ${U}_{w}\left(h\right)\in {U}_{-}$ is not necessarily unique for a given $\left(w,h\right)\text{.}$ For example, when $t·h=h,$ have ${U}_{t}\left(h\right)\in Z\left(e+h\right)\cap {U}_{-}\text{.}$

### The action of $W$ on $\left(e+\underset{_}{h}\right)×Z$ and on $S\otimes 𝒪\left(Z\right)$

Suppose that ${U}_{-}$ acts on a scheme $Z\text{.}$ Then $W$ acts on $\left(e+\underset{_}{h}\right)×Z$ by $w((e+h,z)) = (e+wh,Uw(h)·z).$ Assume that $Z$ is affine. Then $W$ acts on $𝒪\left(\left(e+\underset{_}{h}\right)×Z\right)\simeq S\otimes 𝒪\left(Z\right)\text{:}$ $(w·p1) (e=h,z) = p1(w-1·(e+h,z)) ,p1=s⊗p∈ S⊗𝒪(Z).$

Lemma: For $s\otimes p\in S\otimes 𝒪\left(Z\right),$ $ri·(s⊗p) = ∑n≥0 (ri·(αins)) ⊗fi(n)·p.$

 Proof. By definition, $ri·(s⊗p) (e+h,z) = (s⊗p) (ri·(e+h,z)) = (s⊗p) (e+ri·h,Uri(h)·z) = s(ri·h) p(Uri(h)·z) = (ri·s)(h) (Uri(h)-1·p)(z) = (ri·s)(h) ( exp(-⟨αi,h⟩fi·p) (z) ) = (ri·s)(h) ( ∑n≥0 -⟨αi,h⟩nn! fin·p ) (z) = ∑n≥0 ((-αi)nri·s) (h)(fi(n)·p)(z) = ∑n≥0ri· (αins)(h) (fi(n)·p)(z) ⇒ ri· (s⊗p) = ∑n≥0 (ri·(αins)) ⊗fi(n)·p.$ $\square$

Consequently, we get an integrable $\underset{_}{A}\text{-module}$ structure on $S\otimes 𝒪\left(Z\right)$ by $Ai·(s⊗p) = 1αi (1-ri)· (s⊗p) = (Ai·s)⊗p+ ∑n≥1ri· (αin-1s) ⊗fi(n)·p.$ For each $p\in 𝒪\left(Z\right),$ this is a finite sum.

### The groupoid scheme $𝒰\prime =\left(e+\underset{_}{h}\right)×{U}_{-}\text{:}$

Define $PL=p1: 𝒰′ ⟶ e+h_: (e+h,u) ⟼ e+h$ and $PR: 𝒰′ ⟶ e+h_: (e+h,u) ⟼ proj. of u-1·(e+h) to e+h_ in e+b_-=e+h_+n-.$ These are the source and target maps for the groupoid struture on $𝒰\prime \text{.}$ Other structure maps: $identities i′: e+h_ ↪ 𝒰′: e+h ⟼ (e+h,1) multiplication: μ′: 𝒰′×e+h_𝒰′ ⟶ 𝒰′: (e+h,u)·(e+h′,u′) = (e+h,u′) if PR(e+h,u)=e+h′=PL(e+h′,u′). inverse: ι: 𝒰′ ⟶ 𝒰′: (e+h,u) ⟼ (PR(e+h,u),u-1).$ The idea now is to embed $𝒰$ as a subgroupoid scheme of $𝒰\prime \text{.}$ Here the groupoidscheme structure on $𝒰$ is the one defined in Lecture 5. To this end, we use the integrable $\underset{_}{A}\text{-module}$ structure on $𝒰\prime \text{.}$

### The groupoid morphism $𝒰\to 𝒰\prime \text{:}$

Consider the ${W}_{L}$ action on $\left(e+\underset{_}{h}\right)×{U}_{-}\text{:}$ $wL· (e+h,u) = (e+wh,Uw(h)u).$ It satisfies $PR·wL = PR$ by the definition of ${U}_{w}\text{.}$ By the discussion in Lecture 17, we have an integrable ${\underset{_}{A}}_{L}\text{-module}$ structure on $𝒪\left(𝒰\prime \right)\text{.}$ In other words we have a groupoid action $ϕ: 𝒰×h_𝒰′ ⟶ 𝒰′ PR∘p2 ↓ PR∘p2 PR ↓ PR e+h_ ≃ e+h_$ Also have $𝒰×h_𝒰′×h_𝒰′ ⟶id×μ′ 𝒰×h_𝒰′ ϕ×id ↓ ϕ×id ϕ ↓ ϕ 𝒰′×h_𝒰′ ⟶μ′ 𝒰′$ where $\mu \prime :𝒰\prime {×}_{h}𝒰\prime \to 𝒰\prime$ is the multiplication morphism for $𝒰\prime \text{.}$ These imply that the following composition is a morphism of groupoid schemes over $\underset{_}{h}\text{:}$ $𝒰 = 𝒰×h_h_ ⟶id×i′ 𝒰×h_𝒰′ ⟶ϕ 𝒰′$ where $i\prime :\underset{_}{h}& ↪& 𝒰\prime$ is the identity morphism for $𝒰\prime \text{.}$

### The groupoid isomoprhism $𝒰\prime \simeq 𝒰\prime {×}_{e+{\underset{_}{b}}_{-}}\left(e+\underset{_}{h}\right)$

Define $PR′: 𝒰′ ⟶ e+b_-: (e+h,u) ⟼ u-1·(e+h) ∈ e+b_-.$ Form $𝒰′×e+b_- (e+h_) ≔ 𝒰″$ using ${P}_{R}^{\prime }$ and $e+\underset{_}{h}↪e+{\underset{_}{b}}_{-}$ (the inclusion). We think of $𝒰\prime {×}_{e+{\underset{_}{b}}_{-}\left(e+\underset{_}{h}\right)}=𝒰″$ as the subset of $𝒰\prime \text{:}$ ${ (e+h,u): u-1· (e+h)∈e+h } .$ We claim that the morphism $𝒰\to 𝒰\prime$ factors through $𝒰″\text{.}$ To prove this, we look at $𝒪(𝒰′) ⟶ 𝒪(𝒰) ≃ HT(G/B).$ For each $w\in W,$ recall that we have ${\psi }_{w}:{H}^{T}\left(G/B\right)\to S\text{.}$

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 $\left\{{\psi }_{w}:w\in W\right\}$ is a basis for ${\text{Hom}}_{S}\left({H}^{T}\left(K/T\right),S\right),$ we conclude that the morphism $𝒰\to 𝒰\prime$ factors through $𝒰″$ to give $𝒰 ⟶ 𝒰″.$

Theorem: $𝒰 ⟶∼ 𝒰″$ as groupoid schemes over $\underset{_}{h}\text{.}$

## Lecture 18: April 29, 1997

Last time we had morphisms of groupoid schemes over $\underset{_}{h}$ $Spec HT(K/T)=𝒰 ⟶ (e+h_)×𝒰-≕𝒰′ ⤏ ↗ [(e+h_)×𝒰-] ×e+b_- (e+h_)=𝒰″$

Consider the corresponding ring homomorphism $𝒪(𝒰′) = S⊗𝒪(𝒰-) ⟶ 𝒪(U) = HT(K/T). (*)$

Definition: $w\in W$ is called ${G}^{\vee }\text{-abelian}$ if the following equivalent conditions hold.

 (1) ${r}_{i}{r}_{j}{r}_{i},$ where ${a}_{ij}=-1,$ does not occur as a consecutive subexpression for any reduced expression of $w\text{.}$ (2) ${𝒰}_{-}^{\vee }\cap w{B}^{\vee }{w}^{-1}$ is commutative.

### Lifting of ${\sigma }_{G/B}^{\left(w\right)}$ for ${G}^{\vee }\text{-abelian}$$w$ to $𝒪\left({𝒰}_{-}\right)$

Consider the quotient of $𝒰{\left({\underset{_}{n}}_{-}\right)}_{ℤ}$ by the 2-sided ideal generated by $\left\{{f}_{i}^{\left(m\right)} | i\in I, m\ge 2\right\}\text{.}$ The resulting ring with identity $𝒰{\left({\underset{_}{n}}_{-}\right)}_{ℤ}/⟨{f}_{i}^{\left(2\right)};i\in I⟩$ is given by generators $\left\{{f}_{i} | i\in I\right\}$ and relations: $fifi=0, fifjfi=0 if aij=-1, andfifj=fjfi if aij=0.$ For ${G}^{\vee }\text{-abelian}$ $w$ with reduced expression ${r}_{{i}_{1}}\cdots {r}_{{i}_{N}},$ put $fw = fi1⋯ fiN.$ These ${f}_{w}$ define a basis of $𝒰{\left({\underset{_}{n}}_{-}\right)}_{ℤ}/⟨{f}_{i}^{\left(2\right)};i\in I⟩\text{.}$ The dual basis gives us elements in $𝒪\left({𝒰}_{-}\right)\text{:}$ $fw*∈Hom (𝒰(n_-)ℤ/⟨fi(2);i∈I⟩,ℤ) ⊂Hom(𝒰(n_-)ℤ,ℤ) =𝒪(𝒰-).$

Claim: Under the homomorphism $\left(*\right)$ $S⊗𝒪(𝒰-) ⟶ HT(G/B), 1⊗fw* ⟼ σG/B(w).$

 Proof. Write ${f}_{w}^{*}$ for $1\otimes {f}_{w}^{*}\text{.}$ The statement is clear for the identity elements: ${f}_{1}^{*}↦{\sigma }_{G/B}^{\left(1\right)}\text{.}$ Suppose ${r}_{i}w\le w\text{.}$ Then ${r}_{i}w$ is again ${G}^{\vee }\text{-abelian,}$ 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, ${A}_{j}·{f}_{w}^{*}=0$ if $w\le {r}_{j}w\text{.}$ Define $x\in {H}^{T}\left(G/B\right)$ by $fw* ⟼ σG/B(w)+x.$ Then $Ai·fw* ⟼ AiσG/B(w)+Aix.$ We can assume by induction that ${f}_{{r}_{i}w}↦{\sigma }_{G/B}^{\left({r}_{i}w\right)}$ whenever ${r}_{i}w\le w\text{.}$ Therefore ${A}_{i}·x=0$ in this case. Also ${A}_{j}·x=0$ for ${r}_{j}w\ge w,$ by the above. So $x=0\text{.}$ $\square$

### Miniscule representations

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

 (1) All weights lie in the same $W\text{-orbit.}$ (2) The representation has highest weight $\lambda$ such that $0\le ⟨\lambda ,{\alpha }^{\vee }⟩\le 1$ for all $\alpha \in {\varphi }^{+}\text{.}$

Let $V=V\left(\lambda \right)$ be a miniscule representation of $G$ with highest weight ${v}^{+}\in V\left(\lambda \right)\text{.}$ The stabilizer of the $\lambda$ weight space is the parabolic subgroup $P={P}_{\lambda }=B{W}_{\lambda }B$ (where ${W}_{\lambda }$ is the stabilizer of $\lambda$ in $W\text{).}$ The weights of $V\left(\lambda \right)$ are precisely $\left\{w·\lambda | w\in {W}^{P}\right\}\text{.}$

Lemma: All $w\in {W}^{P},$ for $P$ as above, are ${G}^{\vee }\text{-abelian,}$ and $\left\{{v}_{w}={f}_{w}·{v}^{+} | w\in {W}^{P}\right\}$ gives a basis of $V\left(\lambda \right)\text{.}$

 Proof. ${W}^{P}$ is characterised as $WP = { w∈W | α∈ ϕ+, w,α∨ <0 ⇒ ⟨λ,α∨⟩ =1 } .$ Therefore ${𝒰}_{-}^{\vee }\cap w{B}^{\vee }{w}^{-1}$ (for $w\in {W}^{P}\text{)}$ is generated by $1\text{-parameter}$ subgroups ${𝒰}_{-{\alpha }^{\vee }}^{\vee }=\text{exp} {𝔤}_{-{\alpha }^{\vee }}^{\vee }$ for which $⟨\lambda ,{\alpha }^{\vee }⟩=1\text{.}$ Any two such subgroups ${𝒰}_{-\alpha }^{\vee },{𝒰}_{-\beta }^{\vee }$ commute, since $⟨\lambda ,{\alpha }^{\vee }+{\beta }^{\vee }⟩=2$ and thus ${\alpha }^{\vee }+{\beta }^{\vee }$ is not a root of ${𝔤}^{\vee }$ (by condition (2) for miniscule $\lambda \text{).}$ So $w$ is ${G}^{\vee }\text{-abelian.}$ That ${f}_{w}·{v}^{+}\in {V}_{w·\lambda }$ is proved inductively. Let $w={r}_{i}w\prime$ with $\ell \left(w\right)=\ell \left(w\prime \right)+1\text{.}$ Then $w\prime \in {W}^{P}$ and $fw·v+= fifw′· v+∈Vw′·λ-αi.$ On the other hand ${r}_{i}w\prime ·\lambda =w\prime ·\lambda -⟨{\alpha }_{i}^{\vee },w\prime ·\lambda ⟩{\alpha }_{i}\text{.}$ We have $⟨{\alpha }_{i}^{\vee },w\prime ·\lambda ⟩=⟨{\left(w\prime \right)}^{-1}{\alpha }_{i}^{\vee },\lambda ⟩=1,$ since ${r}_{i}w\prime =w$ lies in ${W}^{P}$ and takes the positive weight ${\left(w\prime \right)}^{-1}{\alpha }_{i}^{\vee }$ to $-{\alpha }_{i}^{\vee }\text{.}$ Thus $w·\lambda =w\prime ·\lambda -{\alpha }_{i}$ and ${f}_{w}·{v}^{+}\in {V}_{w·\lambda }$ (and ${f}_{w}·{v}^{+}$ is nonzero). $\square$

Corollary: All matrix coefficients in $𝒪\left({𝒰}_{-}\right)$ of the miniscule representation $V\left(\lambda \right)$ go to Schubert basis elements in ${H}^{T}\left(G/B\right)$ under the homomorphism $\left(*\right)$ (matrix coefficients with respect to $\left\{{v}_{w}\right\},$ that is).

 Proof. This follows since ${f}_{i}^{\left(2\right)}$ acts on $V\left(\lambda \right)$ by $0\text{.}$ $\square$

Example: Consider the standard representation $V\left({\rho }_{1}\right)$ of ${SL}_{3}\text{.}$ It is clearly miniscule. The homomorphism $𝒪\left({𝒰}_{-}\right)\to {H}^{T}\left(G/B\right)$ 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 ${\pi }_{L}$ and ${\pi }_{R}:𝒪\left(\underset{_}{h}\right)\to {H}^{T}\left(G/B\right)$ correspond to $hL= ( πL(ρ1) πL(ρ2-ρ1) πL(-ρ2) ) , hR= ( πR(ρ1) πR(ρ2-ρ1) πR(-ρ2) )$ in $\underset{_}{h}\left({H}^{T}\left(G/B\right)\right)\text{.}$

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 $S\otimes 𝒪\left({𝒰}_{-}\right)\to {H}^{T}\left(G/B\right)$ gives rise to (after applying ${\otimes }_{S}ℤ$ and dualizing) a map ${H}_{*}\left(G/B\right)\to 𝒰\left({\underset{_}{n}}_{-}\right)\text{.}$ So to any representation $V$ with highest weight ${v}^{+}$ one can define a subspace of $V$ by applying the image of ${H}_{*}\left(G/B\right)$ in $𝒰\left({\underset{_}{n}}_{-}\right)$ to ${v}^{+}\text{.}$ If ${v}^{+}$ is of weight $\lambda$ then the map ${H}_{*}\left(G/B\right)\stackrel{{v}^{+}}{\to }V$ factors through ${H}_{*}\left(G/B\right)\to {H}_{*}\left(G/{P}_{\lambda }\right)\text{.}$ It seems natural to ask whether the resulting map ${H}_{*}\left(G/{P}_{\lambda }\right)\to V$ is injective. If $\lambda$ is miniscule then this map is in fact bijective.

There is also a similar construction for ${H}^{*}\left(\Omega K\right)\text{.}$ It will be shown later that ${H}^{*}\left(\Omega K\right)\cong 𝒰\left({\underset{_}{n}}_{+}^{e}\right)\text{.}$ 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 ${\underset{_}{s}}_{+}$ be the centralizer in ${\underset{_}{n}}_{+}^{\text{af}}$ of $e+{e}_{0}=e+t{e}_{-\theta }\text{.}$ Any representation of ${G}_{\text{af}}$ with miniscule highest weight ${\rho }_{i}$ is isomorphic to $V\left({\rho }_{0}^{\text{af}}\right),$ the representation with highest weight ${\rho }_{0}$ (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 $𝒰\left({\underset{_}{s}}_{+}\right)\to {V}^{*}\left({\rho }_{0}^{\text{af}}\right)$ on the right hand side is bijective. Hence the composition is surjective and so is $𝒰\left({\underset{_}{n}}_{+}^{e}\right)\to {V}^{*}\left({\rho }_{i}\right)\text{.}$

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

Lemma: We have the following inclusion of $𝔽\text{-valued}$ points (not schematically) $ZG(e)⊆B.$

 Proof. Suppose $g\in {Z}_{G}\left(e\right)\text{.}$ Then, by the Bruhat decomposition, $g={b}_{1}\cap {b}_{2}$ for ${b}_{1},{b}_{2}\in B$ and $n\in {N}_{G}\left(T\right)\text{.}$ We have ${b}_{1}\cap {b}_{2}·e=e,$ hence $nb2·e = b1-1·e.$ Let $w\in W$ be the Weyl group element represented by $n\text{.}$ Then the left hand side of the above equation lies in the sum of weight spaces $\underset{\alpha \in w·{\Delta }_{+}}{⨁}{g}_{\alpha },$ while the right hand side has nonzero components in all the ${𝔤}_{{\alpha }_{i}},$ for ${\alpha }_{i}\in \Pi \text{.}$ Thus $\Pi \subseteq w·{\Delta }_{+},$ which implies that $w=\text{id.}$ $\square$

Consider the morphism $ϕ: (e+h_)×𝒰- ⟶ e+b_- (e+h,u) ⟼ u-1·(e+h).$ Let $X≔\left(e+\underset{_}{h}\right)×{𝒰}_{-}$ and $Y≔e+{\underset{_}{b}}_{-}\text{.}$ Then $𝒪\left(X\right)$ and $𝒪\left(Y\right)$ are graded polynomial rings over $ℤ$ in $N=\text{#}{\Delta }_{+}$ generators, where the grading is given as follows. For $𝒪\left(X\right)=S\otimes 𝒪\left({𝒰}_{-}\right)$ let $S$ be graded as usual by $\text{deg} {\underset{_}{h}}^{*}=2,$ and $𝒪\left({𝒰}_{-}\right)$ by $\text{deg} {𝔤}_{-\alpha }^{*}=2\text{ht}\left(\alpha \right)\text{.}$ The grading on $𝒪\left(Y\right)=𝒪\left({\underset{_}{b}}_{-}\right)$ is given by $\text{deg} {𝔤}_{-\alpha }^{*}=2\left(\text{ht}\left(\alpha \right)+1\right)\text{.}$ Then we get that $ϕ*: 𝒪(Y) ⟶ 𝒪(X)$ is a homomorphism of graded polynomial rings. Choose homogeneous generators of $𝒪\left(Y\right)$ and $𝒪\left(X\right)\text{.}$ So $𝒪\left(Y\right)=ℤ\left[{y}_{1},\dots ,{y}_{N}\right]$ and $𝒪\left(X\right)=ℤ\left[{x}_{1},\dots ,{x}_{N}\right]\text{.}$

Lemma: ${\varphi }^{*}\left({y}_{1}\right),\dots ,{\varphi }^{*}\left({y}_{N}\right)$ form a regular sequence in $𝒪\left(X\right)\otimes 𝔽\text{.}$

Proof.

Let $𝒮=⟨{\varphi }^{*}\left({y}_{1}\right),\dots ,{\varphi }^{*}\left({y}_{N}\right)⟩\text{.}$ Since the ${\varphi }^{*}\left({y}_{i}\right)$ are homogeneous elements in a graded ring it suffices to show that the depth of $𝒮$ (or equivalently $\sqrt{𝒮}\text{)}$ equals $N\text{.}$ The following claim will imply that $\sqrt{𝒮}=⟨{x}_{1},\dots ,{x}_{N}⟩$ and hence this lemma.

Claim: Let $h\in \underset{_}{h}\left(𝔽\right)$ and $u\in {𝒰}_{-}\left(𝔽\right),$ then $u-1·(e+h)=e ⇒ u=1.$

 Proof. Consider the semisimple part of $u·e=e+h\text{.}$ Since the semisimple part of $e$ is zero it must be zero. On the other hand it must be conjugate to $h\text{.}$ Hence $h={0}_{*}$ and $u·e=e\text{.}$ So $u\in {Z}_{G}\left(e\right)\left(𝔽\right)$ which by a previous lemma is contained in $B\left(𝔽\right)\text{.}$ Therefore $u=1\text{.}$ $\square$

$\square$

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.