## Quantum Cohomology of $G/P$

Last update: 16 December 2013

## Lecture 6: February 26, 1997

(The following is the beginning of Lecture 4 given on Feb. 19).

### Schubert Cells in $G/P$

Recall that a closed subgroup $P$ of $G$ is called a standard parabolic subgroup if $P\supset B\text{.}$

Let $P\subset G$ be a standard parabolic subgroup. Then $\exists$ subset $J\subset I$ s.t. $P=BWJB$ where $WJ=⟨rj⟩j∈J$ is the subgroup of $W$ by $\left\{{r}_{j}:j\in J\right\}\text{.}$ Set $WP = WJ WP = { u∈W:u Thus ${W}^{P}$ is the set of minimum representatives of the coset space $W/{W}^{P}\text{.}$ We have $G/P=⨆w∈WP BwP$ ????? $Bwp≃ℂℓ(w)$ ????? is called the Schubert Cell corresponding to $w\text{.}$

Each $Ewp$ is $T\text{-stable}$ and $G/P=⨆w∈WP BwP$ takes $G/P$ into a $CW\text{-complex.}$

????? $XwP= closure of BwP in G/P$ ????? is a complex projective variety called the Schubert variety ????? have $XwP= ⋃v∈WPv≤w BvP$ ????? $w\in {W}^{P},$ let $iwP: XwP ↪ G/P$ ????? $\left[{X}_{w}^{P}\right]\in {H}_{2\ell \left(w\right)}\left({X}_{w}^{P},Z\right)\text{.}$ Set $σwP= (iwP)* [XwP]∈ H2ℓ(w) (G/P).$

### Schubert Basis for ${H}_{•}\left(G/P,ℤ\right)$ and ${H}^{•}\left(G/P,ℤ\right)$

Fact: $\left\{{\sigma }_{w}^{P}:w\in W\right\}$ is a basis for ${H}_{•}\left(G/P,ℤ\right)\text{.}$

Notation: The dual basis of ${H}^{•}\left(G/P,ℤ\right)$ dual to $\left\{{\sigma }_{w}^{P}:w\in W\right\}$ is denoted by ${σPw:w∈W} .$

Remark: ${H}^{\text{odd}}\left(G/P\right)=0\text{.}$

(Here starts Lecture 6)

### Schubert Basis for ${\text{Hom}}_{S}\left({H}^{T}\left(G/P\right),S\right)$ and ${H}^{T}\left(G/P\right)\text{:}$

Definition: For $w\in {W}^{P},$ put $σ(w)P= (iwP)* ∫[XwP]∈ HomS(HT(G/P),S).$ Then $\left\{{\sigma }_{\left(w\right)}^{P}:w\in W\right\}$ is a basis for ${\text{Hom}}_{S}\left({H}^{T}\left(G/P\right),S\right)\text{.}$ There is then a unique basis ${ σP(w):w∈W }$ of ${H}^{T}\left(G/P\right)$ (over $S\text{)}$ s.t. $⟨σP(v),σ(w)P⟩ =δv,w.$ Bot $\left\{{\sigma }_{\left(w\right)}^{P}\right\}$ and $\left\{{\sigma }_{P}^{\left(w\right)}\right\}$ are called Schubert basis.

????? basis $\left\{{\sigma }_{P}^{\left(w\right)}:w\in W\right\}$ of ${H}^{T}\left(G/P\right)$ is characterized by ????? properties:

 (1) $\text{deg}\left({\sigma }_{P}^{\left(w\right)}\right)=2\ell \left(w\right)$ (2) Under evaluation at $0\text{:}$ $ℤ⊗SHT(G/P) ⟶ H*(G/P)$ we have $σP(w) ⟼ σPw.$ (3) ${\left({i}_{w}^{P}:{X}_{w}^{P}\to G/P\right)}^{*}\left({\sigma }_{P}^{\left(v\right)}\right)=0$ if $v\nleqq w\text{.}$

?????, we look at

• The action of $\underset{_}{A}$ on ${\text{Hom}}_{S}\left({H}^{T}\left(G/P\right),S\right)$ in the basis $\left\{{\sigma }_{\left(w\right)}^{P}\right\}\text{.}$
• The action of $\underset{_}{A}$ on ${H}^{T}\left(G/P\right)$ in the basis $\left\{{\sigma }_{P}^{\left(w\right)}\right\}\text{.}$
• The ring of ????? characteristic operators $\underset{_}{{\stackrel{ˆ}{A}}_{c}}$ expressed in terms of the $\underset{_}{A}\text{-action}$ on ${H}^{T}\left(K/T\right)={H}^{T}\left(G/B\right)\text{.}$
• The Hopf algebroid structure on ${H}^{T}\left(K/T\right)\text{.}$

### Another set of elements $\left\{{\psi }_{w}^{P}:w\in {W}^{P}\right\}$ in ${\text{Hom}}_{S}\left({H}^{T}\left(G/P\right),S\right)\text{:}$

For $w\in W,$ consider the $T\text{-equivariant}$ map $jwP: pt ⟶ G/P: pt ⟼ wP$ set $ψwP= (jwP)*∈ HomS(HT(G/P),S).$ Of course ${\psi }_{w}^{P}={\psi }_{{w}_{1}}^{P}$ if $w\in {w}_{1}{W}_{p}\text{.}$ We think of ${\psi }_{w}^{P}$ as localizing at the $T\text{-fixed}$ pt $wP\text{.}$

Warning: $\left\{{\psi }_{w}^{P}:w\in {W}^{P}\right\}$ is NOT an $S\text{-basis}$ for ${\text{Hom}}_{S}\left({H}^{T}\left(G/P\right)\right)$ because $σ(ri)P= 1αi ψidP-1αi ψriP.$

Remark: Expressing ${\psi }_{w}^{P}$ as a linear combination over $S$ of the ${\sigma }_{\left(v\right)}^{P}$ we get the $D\text{-matrix}$ in Kostant-Kumar. Will do this later.

Properties: Consider the $G\text{-equivariant}$ map $πP: G/B ⟶ G/P: gB ⟼ gP.$ Then $(πP)* ψwB = ψwP w∈W (πP)* ψP(w) = ψB(w) w∈WP.$

### Action of $\underset{_}{A}$ on ${\text{Hom}}_{{S}^{1}}\left({H}^{T}\left(G/P\right),S\right)$ in the basis $\left\{{\sigma }_{\left(w\right)}^{P}:w\in {W}^{P}\right\}$

Proposition 1: $Ai·σ(w)P= { σ(riw)P if w

 Proof. Let $iwP: XwP ↪ G/P.$ Recall that $σ(w)P= (iwP)* ∫[XwP]$ From the fact stated at the end of last lecture, $Ai·σ(w)P= μ*∫σi*[XwP] ∈HomS(G/P,S)$ where $μ: Ki×TXwP ⟶ G/P: (ki,x) ⟼ kix.$ It follows that (?) $Ai·σ(w)P= { σ(riw)P if w $\square$

?????ave: for $v,w\in W,$ $v·ψwP= ψvwP$

### Action of $\underset{_}{A}$ on ${H}^{T}\left(G/P\right)$ in the basis $\left\{{\sigma }_{P}^{\left(w\right)}:w\in {W}^{P}\right\}$

Proposition 2: For $v\in W,$ $w\in {W}^{P},$ $Av·σP(w)= { ϵ(v)σP(vw) if ℓ(v-1)+ ℓ(vw)=ℓ(w) ⇔vw∈?????, 0 otherwise.$

 Proof. Let's first check that $Ai·σP(w)= { -σP(riw) if 1+ℓ(riw) =ℓ(w) (ie. riw?????, 0 otherwise.$ From the previous Proposition 1, if ${r}_{i}w $\text{(}⇒$ ${r}_{i}\text{?????}$ $Ai·σ(riw)P =σ(w)P.$ But $(Aif)(z)= Ai·f(z)-ri ·f(Ai·z) z∈HT(G/P)$ by definition, so by letting $f={\sigma }_{\left({r}_{i}w\right)}^{P}$ and $z\in {\sigma }_{P}^{\left(v\right)},$ we get $δw,v=0-ri· σ(riw)P (Ai·σP(v))$ or $( Ai·σP(v), σ(riw)P ) =-δw,v ⇒Ai· σP(w)=- σP(riw)$ otherwise follows. $\square$

Remark: Recall that $ε=ψidB= σ(id)B∈ HomS(HT(G/B),S).$ We can identify $A_≃HomS (HT(G/B),S) (*)$ by $a ⟼ fa: fa(z) = ε(a·z).$ Then this is an identification of $S\text{-modules,}$ and from Proposition 2, $fAW=ϵ(w) σ(w-1)B$ ie. $Aw ⟼ ϵ(w)σ(w-1)B$ Thus by Proposition 1, we see that under the identification $\left(*\right),$ the (left) $\underset{_}{A}\text{-action}$ on ${\text{Hom}}_{S}\left({H}^{T}\left(G/B\right),S\right)$ becomes the (left) action $\underset{_}{A}$ on $\underset{_}{A}$ by $a·b=b(*a)$ where, recall from lecture 2, that $*S = S, *w = w-1, *Aw = ϵ(w) Aw-1.$ (The $*$ in Lecture 2 is defined to be $*S=S,$ $*w=ϵ\left(w\right){w}^{-1}$ and $*{A}_{w}={A}_{{w}^{-1}}\text{).}$

### The ring $\underset{_}{\overset{ˆ}{A}}$ of characteristic operators again

Set $ε = ψidB∈ HomS(HT(G/B),S) = σB(id).$ So $ε(σB(w))= δw,idw∈W.$

Proposition:

 (1) Every characteristic operator $a\in \underset{_}{\stackrel{ˆ}{A}}$ can be uniquely written as $a=∑w∈W swAwsw∈ S.$ In fact, $sw=ε (a·(ϵ(w)σB(w-1)))$ (Recall $ϵ\left(w\right)={\left(-1\right)}^{\ell \left(w\right)}\text{).}$ (2) $a$ is compactly supported iff only finitely many $w\text{'s}$ occur in the sum. (ie. at $f$ only finitely many ${s}_{w}\text{'s}$ are ?????

 Proof. (1) For any $a\in \stackrel{ˆ}{A},$ write $a′=a-∑w∈Wε (a·(ϵ(w)σB(w-1))) Aw$ Then $a\prime \in \underset{_}{\stackrel{ˆ}{A}}\text{.}$ Thus to show $a\prime =0$ it is enough to show that $ε(a′·z)=0$ for any $z\in {H}^{T}\left(G/B\right)\text{.}$ (See Lecture 5). Since both $a\prime$ and $\epsilon$ are $S\text{-linear,}$ it is enough to show that $ε(a′·σB(v))=0$ for all $v\in W\text{.}$ Now $a′-σB(v) = a·σB(v)- ∑w∈Wε (a·ϵ(w)σB(w-1)) Aw·σB(v) = a·σB(v)- ∑w∈Wℓ(w-1)+ℓ(wv)=ℓ(v)ε (a·ϵ(w)σB(w-1)) ϵ(w)σB(wv) = a·σB(v)- ∑w∈Wℓ(w-1)+ℓ(wv)=ℓ(v)ε (a·σB(w-1)) σB(v)$ $*$ $ΔσB(v) = ∑ u,w∈W uw=v ℓ(v)=ℓ(u)+ℓ(w) σB(u)⊗ σB(w) a = ((ε∘a)⊗id) ∘ΔX$ ?????e Corollary 3 in Lecture 5). $⇒a′·σB(v) = 0 ⇒a′ ≡ 0.$ Uniqueness is clear. If $a$ has compact support, since any compact subset of $K$ is contained in some ${K}_{w}$ where ${K}_{w}={K}_{{i}_{1}}{K}_{{i}_{2}}\cdots {K}_{{i}_{r}}$ if $w={r}_{{i}_{1}}{r}_{{i}_{2}}\cdots {r}_{{i}_{r}}$ [red], we see that there are only finitely many $w\text{'s}$ involved in the expression $a=∑w∈WswAw.$ $\square$

Note: The following remark was crossed out in the scanned notes.

Remark: We can think of $\underset{_}{A}$ as ${\text{Hom}}_{S}\left({H}^{T}\left(K/T\right),S\right),$ or the ????? of ${H}^{T}\left(K/T\right)$ via the pairing: $(a,z) =def ε(a·z).$ Let's check then that $\underset{_}{A}$ action on ${\text{Hom}}_{S}\left({H}^{T}\left(K/T\right),S\right)$ becomes the $\underset{_}{A}\text{-action}$ on $\underset{_}{A}$ by left multiplications: For $a\in \underset{_}{A},$ use ${f}_{a}\in {\text{Hom}}_{S}\left({H}^{T}\left(K/T\right),S\right)$ to denote the element given by $fa(z)= (a,z)=ε (a·z).$ For $i\in I\text{.}$ we want to check $Ai·fa = fAia.$

### The Hopf Algebroid Structure on ${H}^{T}\left(K/T\right)$

Recall from Lecture 5 that ${H}^{T}\left(K/T\right)$ is a Hopf algebroid over $S\text{.}$ We now express the structure maps for this Hopf algebroid in the basis $\left\{{\sigma }_{B}^{\left(w\right)}:w\in W\right\}\text{.}$

First, recall that we have ring homomorphisms $πL: S ⟶ HT(K/T) πR: S ⟶ HT(K/T).$ This gives two $S\text{-module}$ structures on ${H}^{T}\left(K/T\right)\text{.}$ The map ${\pi }_{L}$ is nothing but the characteristic homomorphism $\text{ch}$ in Lecture 5. The map ${\pi }_{R}$ is a little more mysterious. It gives the 2nd $S\text{-module}$ structure on ${H}^{T}\left(K/T\right)$ in Lecture 4.

Proposition: The elements $\left\{{\sigma }_{B}^{\left(w\right)}:w\in W\right\}$ is also a basis for the second $S\text{-module}$ on ${H}^{T}\left(K/T\right)$ defined by ${\pi }_{R}\text{.}$

Remark: I (Lu) suspect that ${\pi }_{R}$ has a lot to do with the Bruhat-Poisson structure on $K/T\text{.}$

The next theorem expresses the structure maps for the Hopf algebroid structure on ${H}^{T}\left(K/T\right)$ in the basis $\left\{{\sigma }_{B}^{\left(w\right)}:w\in W\right\}\text{.}$

Theorem: (Recall notation from Lecture 5):

 1) For $\lambda \in {h}_{ℤ}^{*},$ $πR(λ)= πL(λ)+ ∑i∈I ⟨λ,αi∨⟩ σB(ri)$ 2) $\epsilon \left({\sigma }_{B}^{\left(w\right)}\right)={\delta }_{w,\text{id}}$ 3) $\Delta {\sigma }_{B}^{\left(w\right)}=\sum _{\underset{w=uv \text{[red]}}{u,v\in W}}{\sigma }_{B}^{\left(u\right)}\otimes {\sigma }_{B}^{\left(v\right)}\phantom{\rule{2em}{0ex}}$ $\text{(}w=uv$ [red] means $w=uv$ ????? 4) $c\left({\sigma }_{B}^{\left(w\right)}\right)=ϵ\left(w\right){\sigma }_{B}^{\left({w}^{-1}\right)}$ 5) For any $K\text{-space}$ $X$ and $\sigma \in {H}^{T}\left(X\right)$ $ΔX(σ)= ∑w∈Wϵ(w) σB(w-1)⊗ (Aw·σ) ∈HT(K/T)⊗S HT(X)$

Proof.

We first prove 5). 5) is due to the general fact if an algebra $A$ acts on a space $M,$ then using a basis ${a}_{1},\cdots {a}_{n},\cdots$ of $A$ and the dual basis ${\xi }_{1},\dots {\xi }_{n},\cdots$ of ${A}^{*},$ the co-module map is nothing but $ΔM: M ⟶ A*⊗M: ΔM(m) = ∑iξi⊗ai·m$ ????? in our example, we are identifying ${H}^{T}\left(K/T\right)$ with ${\underset{_}{A}}^{*}$ ????? the pairing $(a,z)=ε(a·z) a∈A_, z∈ HT(K/T).$ ????? this pairing; we have $\left\{{A}_{w}:w\in W\right\}$ as a basis for $\underset{_}{A}$ ????? dual basis in ${H}^{T}\left(K/T\right)$ is $\left\{ϵ\left(w\right){\sigma }_{B}^{\left({w}^{-1}\right)}:w\in W\right\}$ (see page 6-8). Thus for any $\sigma \in {H}^{T}\left(X\right)$ $ΔX(σ)= ∑w∈Wϵ(w) σB(w-1)⊗ (Aw·σ)$ Peterson gave the following proof in class:

Since $\left\{ϵ\left(w\right){\sigma }_{B}^{\left({w}^{-1}\right)}:w\in W\right\}$ is a basis for ${H}^{T}\left(K/T\right),$ we know $ΔX(σ)= ∑w∈Wϵ(w) σB(w-1)⊗ ϕw$ for some ${\varphi }_{w}\in {H}^{T}\left(X\right)$ for each $w\in W\text{.}$ Need to show ${\varphi }_{w}={A}_{w}·\sigma \text{.}$ To do this, let $v\in W$ and calculate ${A}_{v}·\sigma \text{.}$ We have $Av·σ = (ε⊗id)ΔX (Av·σ) = (εAv⊗id) ΔX(σ) (see Lecture 5, Corollary 1) = ∑w∈Wε (Av·ϵ(w)σB(w-1)) ∘ϕw = ε(Av·ϵ(v)σB(w-1)) ϕv = ϕv.$ This finishes the proof of 5)

Remark: What is quoted as Corollary 1 in Lecture 5 is the fact that the action of $\underset{_}{A}$ on ${H}^{T}\left(X\right)$ is obtained by the comodule map $ΔX: HT(X) ⟶ HT(K/T)⊗SHT(X)$ by $a·σ=(a,σ(1)) σ(2)if ΔX σ=σ(1)⊗σ(2)$ and $\left(a,z\right)=\epsilon \left(a·z\right)$ is the pairing between ${H}^{T}\left(K/T\right)$ and $\underset{_}{A}\text{.}$ This is just like in the Hopf algebra case.

We now prove 3). This is just a special case of 5) for $X=K/T\text{.}$ Indeed, ????? 5), we get $ΔσB(w)= ∑u1∈Wϵ (u1)σB(u1-1) ⊗Au1·σB(w).$ But $Au1·σB(2)= { ϵ(u1)σB(u1w) if ℓ(u1-1) +ℓ(u1w)= ℓ(w), 0 otherwise.$ ????? $ΔσB(w) = ∑u1∈Ww=u1-1·(u1w) [red]ϵ (u1)σB(u1-1) ⊗ϵ(u1) σB(u1w) = ∑ u=u1-1∈w v=u1w∈W w=uv [red] σB(u)⊗ σB(v).$ This finishes the proof of 3).

2) is is clear from definition since $\epsilon ={\sigma }_{\left(\text{id}\right)}^{B}\text{.}$

It remains to prove 1) and 4).

To prove 1), we need the following Lemma:

Lemma: For any $\sigma \in {H}^{T}\left(K/T\right),$ $σ=∑w∈WπR (ε(Aw·σ)) ϵ(w)σB(w-1).$

 Proof. Write $σ=∑w∈WπR (sw)ϵ(w) σB(w-1)$ for some ${s}_{w}\in S$ for each $w\in W\text{.}$ Using $ε∘πR=idS$ and $(Av,ϵ(w)σB(w-1)) (=ε(Av·ε(w)σB(w-1))) =δv,w$ we get $ε(Av·σ) | | (Av,σ) = ∑w∈Wε πR(sw) (Av·ϵ(w)σB(w-1)) =επR(sv) = sv ⇒σ = ∑w∈WπR(ε(Av·σ))ϵ(w)σB(w-1).$ This proves the Lemma. Remark: In proving the Lemma, we used the fact that $S\text{-valued}$ ????? the pairing $\left( \right)$ between $\underset{_}{A}$ and ${H}^{T}\left(K/T\right)$ defined by $(a,σ) = ε(a·σ)$ satisfies $(πR(s)a,σ)= ε(πR(s)) (a,σ)=s(a,σ)$ and $ε(πR(s))=s ∀s∈S.$ It says that $\epsilon :{H}^{T}\left(K/T\right)\to S$ is not only an $S\text{-map}$ for the first $S\text{-module}$ structure on ${H}^{T}\left(K/T\right),$ (defined by ${\pi }_{L}\text{)}$ but also for the 2nd $S\text{-module}$ structure ${H}^{T}\left(K/T\right)$ defined by ${\pi }_{R}\text{.}$ Is this really true? Recall that ${\pi }_{R}:S\to {H}^{T}\left(K/T\right)$ is the pullback of the map $(Eu×K)/(T×T) ⟶ E/T [e,k] ⟼ e·k.$ It is not clear why $\epsilon :{H}^{T}\left(K/T\right)\to S$ is ${\pi }_{R}\left(S\right)\text{-linear.}$ $\square$

Now we prove 1): By Lemma $πL(λ)=∑w∈W πR(ε(Aw·πL(λ))) ϵ(w)σB(w-1)$ But $Aw·πL(λ)= πL(Aw·λ)= { πL(λ) w=id, ⟨λ,αi∨⟩ w=ri, 0 otherwise$ $⇒πL(λ) = πR(ε(πL(λ))) +∑i∈IπR (ε(⟨λ,αi∨⟩)) (-1)σB(ri) = πR(λ)- ∑i∈I ⟨λ,αi∨⟩ σB(ri) ⇒πR(λ) = πL(λ)+ ∑i∈I ⟨λ,αi∨⟩ σB(ri).$

Remark: This is an interesting formula. Understand what this says for Kostant's Harmonic form ${s}^{{r}_{i}}$ later.

It remains to prove 4), ie. $c(σB(w))= ϵ(w) σB(w-1).$

The following is the proof given by Peterson. It is kind of strange.

W>e first prove that $c(σB(w))= ±σB(w-1)$ We'll determine the sign later. For $w\in W,$ let $Ew(2)= {(e,ek):e∈Eu,k∈Kw}.$ Then $H*(Eu(2)/T×T) ≃HT(XwB).$ (Why? This is saying that we do not distinguish ${X}_{w}^{B}$ and its Bott-Samelson resolution?)

Recall that $t: E(2) ⟶ E(2): (e1,e2) ⟼ (e2,e1).$ So $t(E2(2)) = Ew-1(2).$ But $Rw= { σ∈HT(E(2)/T×T) : deg σ=2ℓ(w)and σ|Ev(2)/T×T =0forv∈Ws.t.v≱w } .$ We know $Rw = ℤσB(w) c(Rw) = Rw-1 ⇒c(σB(w)) = ±σB(w-1).$ Now show that $c\left({\sigma }_{B}^{\left(w\right)}\right)=ϵ\left(w\right){\sigma }_{B}^{\left({w}^{-1}\right)}\text{.}$

$w=\text{id}$ OK.

$w={r}_{i}$ OK.

For $\ell \left(w\right)\ge 2,$ assume sign $=·ϵ\left(v\right)$ for $\ell \left(v\right)<\ell \left(w\right)\text{.}$ Since $(c⊗c)·T∘Δ= Δ·c$ where $T(σ⊗σ′)= σ′⊗σ$ we get, from $Δ(σB(w))= ∑w=uv [red] σB(u)⊗ σB(v)$ that $Δ(cσB(w)) = ∑vu?????[red] c(σB(v))⊗ c(σB(u)) = c(σB(w))⊗1+ 1⊗c(σB(w))+ ∑w=uv[red]u≠1v≠1 ϵ(u)ϵ(v) σB(v-1)⊗ σB(u-1).$ But $\Delta \left(ϵ\left(w\right){\sigma }_{B}^{\left(w\right)}\right)=ϵ\left(w\right){\sigma }_{B}^{\left({w}^{-1}\right)}\otimes 1+1\otimes ϵ\left(w\right){\sigma }_{B}^{\left({w}^{-1}\right)}+\text{same sum}\ne 0$ $⇒$ must have $Δ(σB(w))= ϵ(w) σB(w-1).$ This proves 4).

This completes the proof of the theorem.

$\square$

### ?????rable $\underset{_}{A}\text{-modules}$$\text{(}⇔$ actions of $𝒰=\text{Spec} {H}^{T}\left(K/T\right)\text{)}$

Definition: Let $X$ be an affine scheme over $\underset{_}{h}=\text{Spec} S$ with structure homomorphism ${\pi }_{X}:S\to 𝒪\left(X\right)\text{.}$ An $\underset{_}{A}\text{-module}$ structure on $𝒪\left(X\right)$ is said to be integrable if for all $s\in S$ and $p\in 𝒪\left(X\right)\text{.}$

 1) $s·p={\pi }_{X}\left(s\right)p$ 2) ${\pi }_{X}:s\to 𝒪\left(X\right)$ and $m:𝒪\left(X\right){\otimes }_{S}𝒪\left(X\right)\to 𝒪\left(X\right)$
are both $\underset{_}{A}\text{-module}$ maps
 3) For each $p\in 𝒪\left(X\right),$ ${A}_{w}·p=0$ for all but finitely many $w\in W\text{.}$

Example: $𝒰$ as a scheme over $\underset{_}{h}=\text{Spec} S$ with structure homomorphism ${\pi }_{L}$ (?) Is this an example? Maybe not, because ${H}^{T}\left(K/T\right){\otimes }_{S}{H}^{T}\left(K/T\right)$ we use ${\pi }_{R}$ to define the $S\text{-module}$ structure on the first copy of ${H}^{T}\left(K/T\right)\text{.}$ (OK. Because in the multiplication of ${H}^{T}\left(K/T\right){\otimes }_{S}{H}^{T}\left(K/T\right),$ even the $S\text{-structure}$ on the first copy is defined by ${\pi }_{L}\text{).}$ $Integrable A_-module structure on 𝒪(X) ↕ action ϕ: 𝒰×h_X⟶X.$ One way:

If $\varphi :𝒰{×}_{\underset{_}{h}}X\to X$ is an action, have $ϕ*: 𝒪(X) ⟶ HT(K/T)⊗𝒪(X)$ Then for $a\in \underset{_}{A},$ define $p\in P$ $a·p=m· ( πX(ε(a·p(1))) ⊗p(2) ) if ϕ*p= p(1)⊗p(2).$ The other way, given $\underset{_}{A}\text{-action}$ on $𝒪\left(X\right),$ define $ϕ*(p)=∑w∈W c(σG/B(w)) ⊗(Aw·p)$ This is the map giving the action $ϕ: 𝒰×h_X ⟶ X.$

Next, we look at the 2nd action of $\underset{_}{A}$ on ${H}^{T}\left(K/T\right)\text{.}$

Notation: The action of $\underset{_}{A}$ on ${H}^{T}\left(K/T\right)$ that we have been talking about way along will from now on be denoted by ${a}_{L}·\text{.}$ the second action that we will now introduce now will be denoted by ${a}_{R}·\text{.}$

### The second action of $\underset{_}{A}$ on ${H}^{T}\left(K/T\right)$

Define a second action of $\underset{_}{A}$ on ${H}^{T}\left(K/T\right)$ by $aR·= c·(aL·)∘c$

Properties:

 1) ${a}_{L}\circ {b}_{R}={b}_{R}\circ {a}_{L}$ $\forall a,b\in \underset{_}{A}$ 2) $\Delta \circ {a}_{L}=\left({a}_{L}\otimes \text{id}\right)\circ \Delta$ $\Delta \circ {b}_{R}=\left(\text{id}\otimes {b}_{R}\right)\circ \Delta$ 3) for $s\in S,$ $a\in \underset{_}{A}$ and $z\in {H}^{T}\left(K/T\right)$ $sL·z = πL(s)z sR·z = πR(s)z$ and $aL·πL(s) = πL(a·s) aR·πR(s) = πR(a·s) a·ε(z) = ε(a(1)La(2)R·z) if Δa= a(1)⊗a(2) ⟺w ·ε(z) = ε(wLwR·z)$
Thus, in the basis $\left\{{\sigma }_{B}^{\left(w\right)}:w\in W\right\}\text{.}$
• ${{A}_{v}}_{R}·{\sigma }_{B}^{\left(w\right)}=\left\{\begin{array}{cc}{\sigma }_{B}^{\left(w{v}^{-1}\right)}& \text{if} \ell \left(w{v}^{-1}\right)+\ell \left(v\right)=\ell \left(w\right),\\ 0& \text{otherwise.}\end{array}$
• Any $z\in {H}^{T}\left(K/T\right)$ can be written as $z=∑w∈W (πL(ε(AwR·z))) σB(w).$
• Any $a\in \underset{_}{\stackrel{ˆ}{A}}$ can be written as $a=∑w∈Wε (aR·σG/B(w)) Aw.$
• $\forall w\in W,$ $ε∘AwR = σ(w)B ε∘wR = ψwB$ (Recall: $\epsilon \circ {{A}_{w}}_{L}=ϵ\left(w\right){\sigma }_{\left({w}^{-1}\right)}^{B}$ see page 6-8).

## Lecture 7: March 4, 1997

Recall formulas from last time: $AvR· σB(w)= { σ(wv-1) if ℓ(wv-1) +ℓ(v)=ℓ(w), 0 otherwise.$ For any $a\in \underset{_}{\stackrel{ˆ}{A}}$ $a=∑w∈Wε (aR·σB(w)) Aw$ ????? $z\in {H}^{T}\left(K/T\right)$ $z=∑w∈WπL (ε(AwR·z)) σB(w)$ ????? $ε∘AwR = σ(w)B ε∘wR = ψwB$ Given $w\in W,$ $\exists$ ${d}_{u,w}\in S\text{?????}$ of degree $\ell \left(u\right)$ for each $u\le w$ s.t. $w=∑u≤wdu,w Au$ moreover $dw,w= ∏α∈Δ+redw-1α<0 (-α)=ϵ(w) ∏α∈Δ+redw-1α<0α$

 Proof. Induction on $\ell \left(w\right)\text{:}$ $ℓ(w)=0 w=id id=id. ℓ(w)=1 w=ri ri=1-αiAi OK.$ Assume $w={r}_{i}{w}_{1}>{w}_{1}\text{.}$ Assume $w1=∑u≤w1 du,w1Au du,w1∈ Sℓ(u) (h*)$ Then $w = riw1= (1-αiAi) ∑u≤w1 du,w1 Au = ∑u≤w1 du,w1 Au- ∑u∈w1 αi(Aidu,w1) Au$ Since $Aidu,w1= (ri·du,w1) Ai+Ai· du,w1$ $⇒w = ∑u≤w1 du,w1 Au-∑u≤w1 αi(ri·ru,w1) AiAu+αi (Ai·du,w1) = ∑u≤w1 ( du,w1-αiAi ·du,w1 ) Au-∑u≤w1 αi(ri·du,w1) AiAu = ∑u≤w1 (ri·du,w1) Au- ∑u≤w1riu>u αi(ri·du,w1) Ariu$ $du,w = ri·du,w1 if u≤w1 dri·u,w = -αi(ri·du,w1) if u≤w1, riu>u$ shows that ${d}_{u,w}\in {S}^{\ell \left(u\right)}\left({h}_{ℤ}^{*}\right)$ for any $u\le w\text{.}$ Moreover, $driw1,w= -αi(ri·dw1,w1).$ Assume $dw1,w1=ϵ (w1) ∏α∈Δ+redw1-1α<0 α$ Then $dw,w = -α1 (ri·dw1,w1) = ϵ(w) ∏α∈Δ+redw?????-1α<0α$ $\square$

Remark: Sara Billey's formula gives an express for each ${d}_{u,w}\text{.}$ Will come back to this later.

Corollary:

 1) ${\psi }_{w}^{B}=\sum _{u\le w}{d}_{u,w}{\sigma }_{\left(w\right)}^{B}$ 2) $\bigcap _{w\in W}\text{ker} {\psi }_{w}^{B}=0\text{.}$ 3) ${H}^{T}\left(K/T\right)$ is reduced, ie. the only nilpotent elemtn 4) ${H}^{T}\left(G/P\right)\simeq {\left({H}^{T}\left(K/T\right)\right)}^{{\left({W}_{P}\right)}_{R}}$ is also reduced.

 Proof. 1) follows from $ε∘AwR = σ(w)B ε∘wR = ψ(w)B$ 2) If $z\in \bigcap _{w\in W}\text{ker} {\psi }_{w}^{B}$ then ${\psi }_{w}^{B}\left(z\right)=0$ $\forall w\text{.}$ Since the matrix $D=\left({d}_{u,w}\right)$ is upper-triangular it is invertible $⇒{\sigma }_{\left(w\right)}^{B}\left(z\right)=0\text{.}$ But $\left\{{\sigma }_{\left(w\right)}^{B}\right\}$ is a basis for ${\text{Hom}}_{S}\left({H}^{T}\left(K/T\right),S\right)$ $⇒z=0\text{.}$ If $z\in {H}^{T}\left(K/T\right)$ is s.t. ${z}^{m}=0$ for some $m\ge 1$ then for each $w\in W$ $ε(wR·zm)=0$ But $wR·zm= (wR·z)m$ $⇒$ $ε((wR·z)m) = 0 (ε(wn·z))m = 0 ⇒ε(wR·z) = 0$ ie. $z\in \text{ker} {\psi }_{w}^{B}=0$ $\forall w$ $⇒z=0.$ Clear. $\square$

Proposition: The action ${a}_{R}·$ of $\underset{_}{A}$ on ${H}^{T}\left(K/T\right)$ descends to an action on ${H}^{•}\left(K/T\right)$ via the map $ℤ⊗SHT(K/T) ⟶H•(K/T)$ where the $S\text{-module}$ structure on ${H}^{T}\left(K/T\right)$ is defined by ${\pi }_{L}\text{.}$

 Proof. This is because the $S$ action defined by ${\pi }_{L}$ commutes with ${a}_{R}·$ for any $a\in \underset{_}{A}\text{.}$ $\square$

Remark: The incuded action of ${A}_{w}\in \underset{_}{A}$ on ${H}^{•}\left(K/T\right)$ is by the BGG-operators.

### ?????ne Constants" for the multiplication on ${H}^{T}\left(K/T\right)$

For $u,v,w\in W,$ define ${a}_{w}^{u,v}\in S$ by $ΔAw= ∑u,v∈W awu,v Au⊗Av$ $\text{(}\Delta$ cocommutative $⇒{a}_{w}^{uv}={a}_{w}^{vu}\text{)}$

Proposition: $σB(u) σB(v)= ∑w∈W πL (awu,v) σB(w)$

 Proof. We know that $σB(u) σB(v)= ∑w∈WπL (ε(AwR·σB(u)σB(v))) σB(w)$ Then $AwR· (σB(u)σB(v))= ∑u′,v′∈W πR (awu′,v′) (Au′R·σB(u)) (Av′R·σB(v))$ and $ε (AwR·(σB(u)σB(v))) = ∑u′,v′∈W awu′,v′ε (Au′R·σB(u)) ε(Av′R·σB(v)) = ∑u′,v′∈W awu′,v′ε (σ(u′)B,σB(u)) (σ(v′)B,σB(v)) = ∑u′,v′∈W awu′,v′ε δu′,u δv′,v = awu,v$ $⇒ σB(u) σB(v)= ∑w∈W πL (awu,v) σB(w).$ $\square$

Special properties of the ${a}_{w}^{u,v}\text{'s:}$

 (1) ${a}_{w}^{u,v}=0$ unless $u\le w,$ $v\le w$

 Proof. This is seen from the definition: $ΔAi = Ai⊗1+ri⊗Ai = Ai⊗1+ (1-αiAi)⊗Ai = 1⊗Ai+Ai⊗1- Ai⊗αiAi = 1⊗Ai+Ai⊗1- Ai⊗αiAi ΔAiAj = (1⊗Ai+Ai⊗1-Ai⊗αiAi) (1⊗Aj+Aj⊗1-Aj⊗αjAj) = 1⊗AiAj+ Aj⊗Ai+ Ai⊗Aj+ AiAj⊗1 -Ai⊗αiAiAj -AiAj⊗αiAi -AiAj⊗αiAiαjAj -Aj⊗AiαjAj -AiAj⊗αjAj +AiAj⊗αiAiαjAj$ so clear from induction on $\ell \left(w\right)\text{.}$ $\square$

Proposition: ${a}_{w}^{u,v}$ is a homogeneous polynomial of degree $ℓ(u)+ℓ(v)- ℓ(w)inS.$

 Proof. $deg σB(u) σB(v) = deg(πLawu,vσB(w)) 2ℓ(u)+2ℓ(v) = 2(deg awu,v in S) +2ℓ(w). ⇒ deg (awu,v in S) = ℓ(u)+ℓ(v)-ℓ(w).$ $\square$

Proposition: For $w,v\in W,$ $v\le w$ $dv,w=awv,w$ where, recall ${d}_{v,w}\in S$ are defined by $w=∑v≤w dv,wAv$ ?????e $w=∑v≤w awv,wAv.$

 Proof. Write $w⊗w=∑u1,u2≤w Swu1,u2 Au1⊗Au2.$ $⇒wε (wR·σB(w)) = ∑u1,u2∈w Swu1,u2 au1ε (Au2R·σB(w)) = ∑u1,u2≤w Swu1,u2 Au1δu2,w = ∑u1≤w Swu1,w Au1.$ But $ε(wR·σB(w)) =dw,w.$ $⇒dw,ww = ∑u1≤w Swu1,w Au1 ⇒Swu1w = dw,wdu1,w$ On the other hand, $w = ∑du,wAu.$ $⇒w⊗w = ∑u1,u2 (∑vdv,w{a}_{v}^{{u}_{1},{u}_{2}}) Au1⊗Au2 ⇒Swu1u2 = ∑vdv,w avu1,u2 ⇒Swu1,w = ∑vdv,w avu1,w= dw,w awu1,w.$ By ${S}^{{u}_{1}w}={d}_{w,w}{d}_{{u}_{1},w}$ and ${d}_{w,w}\ne 0$ get $du1,w= awu1,w$ $\square$

(Very strange proof).

Proposition: For $w\in W,$ $∑w∈uv [red] ϵ(u)σB(u-1) σB(v)= δw,id (1) ∑w∈uv [red] σB(u) ϵ(v) σB(v)= δw,id (2)$

 Proof. $(1) ⟺ m∘ (c⊗id)∘Δ =ε, (2) ⟺ m∘ (id⊗c)∘Δ =ε.$ $\square$

Remark: This will also be true for quantum cohomology.

Remark: Fix ${e}_{0}\in {E}_{u}\text{.}$ Define $i: K/T ⟶ Eu/T: kT ⟼ e0kT.$ Then $i×i: K/T×K/T ⟶ Eu(2)/T×T.$ Consequently, $(i×i)*: HT(K/T) ⟶ HT(K/T)⊗ℤHT(K/T).$ We have $(i×i)* σB(w) = ∑w∈uv [red] ϵ(u) σβu-1⊗ σBv.$

### The Finite Case

Proposition: In the finite case, we have $A_L = EndA_R (HT(K/T)) A_R = EndA_L (HT(K/T)).$ ${H}^{T}\left(K/T\right)$ is a free ${\underset{_}{A}}_{L}$ (as well as ${\underset{_}{A}}_{R}\text{)}$ module with one generator ${\sigma }_{B}^{\left({w}_{0}\right)},$ where ${w}_{0}$ is the longest element in $W\text{.}$ If $\varphi \in {\text{End}}_{{\underset{_}{A}}_{L}}\left({H}^{T}\left(K/T\right)\right)$ then $\exists$ $a\in \underset{_}{A}$ s.t. $ϕ(σB(w0))= aR·σB(w0).$

Claim: $\forall z\in {H}^{T}\left(K/T\right),$ $ϕ(z)=aR·z.$

 Proof. For any $z\in {H}^{T}\left(K/T\right),$ $\exists$ $b\in \underset{_}{A}$ s.t. $z={b}_{L}·{\sigma }_{B}^{\left({w}_{0}\right)}$ $⇒ϕ(z) = ϕ(bL·σB(w0)) = bL·ϕ(σB(w0)) (ϕ∈EndA_L) = bL·aR· σB(w0) = aR·bL· σB(w0) = aR·z.$ $\square$

### The space ${H}^{T}\left(K\right)$ with $K$ acting on $K$ by conjugations

Consider now $K$ as a $K\text{-space}$ by conjugations. The map $p: K ⟶ K/T$ is $T\text{-equivariant}$ (but not $K\text{-equivariant).}$ Thus $p*: H(K/T) ⟶ HT(K)$ is an $S\text{-module}$ map: $p*(πL(s)z) =π(s)p*(z)$ where $π = [k→pt]*: S ⟶ HT(K).$ Now $\underset{_}{A}$ acts on both ${H}^{T}\left(K\right)$ and ${H}^{T}\left(K/T\right)$ by characteristic operations. But since $p$ is not a $K\text{-map,}$ ${p}^{*}$ does not intertwine the $\underset{_}{A}\text{-actions}$ on ${H}^{T}\left(K\right)$ and on ${H}^{T}\left(K/T\right)\text{.}$ We have, nevertheless, the following:

Proposition: For $a\in \underset{_}{A}$ with $\Delta a={a}_{\left(1\right)}\otimes {a}_{\left(2\right)},$ and for all $z\in {H}^{T}\left(K/T\right)$ $a·p*(z)= p*(a(1)La(2)R·z)$ In particular, for $s\in S$ and $w\in W$ $π(s)p*(z) = p*(πL(s)z) =p*(πR(s)z) w·p*(z) = p*(wLwR·z) Aw·p*(z) = p* ( ∑u≤wv≤w πL(awuv) AuLAvR·z )$

Proposition: For any $K\text{-space}$ $X$ with action map $μX: K×X ⟶ X$ the pullback $μX*: HT(X) ⟶ HT(K×X)$ is the composition $HT(X) ⟶ΔX HT(K/T)⊗SHT(X) ⟶p*⊗id HT(K)⊗SHT(X) ≃ HT(K×X).$

### The Pontrayagin action of the ring ${H}_{*}\left(K\right):$

$μK: K×K ⟶ K: (k1,k2) ⟼ k1k2$ gives a map $μK*: H*(K)⊗H*(K) ⟶ H*(K).$ This defines a ring structure on ${H}_{*}\left(K\right)\text{.}$ Now for any $K\text{-space}$ $X$ with $μX: K×X ⟶ X$ get $μX*: H*(K)⊗H*(X) ⟶ H*(X)$ which defines an action of ${H}_{*}\left(K\right)$ on ${H}_{*}\left(X\right)\text{.}$

????? at the special case $X=K/T$ with $μX = μK/T: K×K/T ⟶ K/T.$ ${\underset{_}{A}}_{R}$ acts on ${H}_{*}\left(K/T\right),$ and this action commutes with the Pontryagin action of ${H}_{*}\left(K\right)$ on ${H}_{*}\left(K/T\right)\text{.}$

Define a ring structure on ${H}_{*}\left(K/T\right)$ by $σvσw= { σvw if ℓ(v)+ ℓ(w)= ℓ(vw), 0 otherwise.$ Then $μK/T* ( σ ⫙ H*(K) × σ′ ⫙ H*(K/T) ) = p*(σ)σ′.$ Consequently, $p*: H*(K) ⟶ H*(K/T)$ is a ring homomorphism.

Theorem (Peterson-Kac): Over any field $𝔽$

 1) ${p}^{*}\left({H}^{*}\left(K/T\right),𝔽\right)$ is a Hopf subalgebra of ${H}^{*}\left(K𝔽\right)\text{.}$ 2) ${p}_{*}\left({H}_{*}\left(K/T\right),𝔽\right)={H}_{*}{\left(K/T,𝔽\right)}^{S}=\left\{\sigma :\lambda \cap \sigma =0 \forall \lambda \in {h}_{ℤ}^{*}\right\}\text{.}$ 3) If ${m}_{ij}=\infty$ for all $i\ne j,$ then ${p}^{*}\left({H}^{*}\left(K/T\right),ℚ\right)\simeq$ the dual of a tensor algebra as a Hopf algebra.

### Poincare Duality in the finite case

Define $\underset{_}{A}\text{-module}$ homomorphism $PD: HT(G/P) ⟶ HomS(HT(G/P),S), PD(z)(y)= ∫[G/P]yz∈S$ Consider the case $P=B\text{:}$ $∫[G/B]=ε∘ Aw0R.$ In general, $∫[G/P] σP(w)= δw,w0wP$ where ${w}_{P}$ is the longest element in ${W}_{P},$ so ${w}_{0}{w}_{P}$ is the longest element in ${W}^{P}\text{.}$

Recall that (from Lecture 2) $ΔAw0= ∑w∈WAw⊗ w0Aw0w$ $PD(σP(w))= w0· σ(w0wwP)P$ Also $w0Lw0R· σB(w)=ϵ (w)σB(w0ww0).$ It follows that $PD$ is an $S\text{-module}$ isomorphism.

### The Euler Class:

For $z\in {H}^{T}\left(G/P\right),$ consider the operator ${M}_{z}$ on ${H}^{T}\left(G/P\right)$ by $y⟼zy\text{.}$ The Euler Class ${\chi }_{G/P}\in {H}^{T}\left(G/P\right)$ is defined by the property: $trace Mz= ∫[G/P] χG/P·z.$

Proposition: $χG/P= ∑w∈WP σP(w) [w0·σP(w0wwP)].$

 Proof. By the definition of trace and using the "dual basis" $\left\{{\sigma }_{\left(w\right)}^{P}\right\}$ of $\left\{{\sigma }_{P}^{\left(w\right)}\right\},$ we have $Mz = ∑w∈WP ( σ(w)P,z σP(w) ) σ(w)P = PD(w0·σP(w0wwP))$ ????? $?????Mz = ∑w∈WP ( PD(w0·σP(w0wwP)) ,z∈σP(w) ) = ∑w∈WP ∫[G/P]z σP(w) (w0·σP(w0wWP))$ ????? $χG/P=∑w∈WP σP(w) (w0·σP(w0wwP)).$ $\square$

We will use $PD$ to denote its inverse as well.

Lemma: For $v,w\in {W}^{P}$ $σP(v)PD (σ(w)P)=0$ unless $v\le w\text{.}$

So ${\chi }_{G/P}$ is the trace of a rank $1$ upper triangular matrix. Also $σP(w)PD (σ(w)P)= w·PD(σ(id)P).$

Facts:

 1) ${\chi }_{G/P}$ has image $\prod _{\underset{{w}_{0}{w}_{P}>0}{\alpha >0}}{\alpha }_{R}·1$ in ${H}^{T}\left(K/T\right)$ 2) ${\chi }_{G/P}$ is $W\text{-invariant}$ under the left action 3) Image of ${\chi }_{G/P}$ in ${H}^{*}\left(G/P\right)$ is $|{W}^{P}|{\sigma }_{P}^{{w}_{0}{w}_{P}}$

### Some facts on the classifcying spaces

$H*(BT) ↪πR HT(G/B) ↪ ⤣ H*(BT)wP ↪ HT(G/B)(wP)R ≅ HT(G/P)$ ????? $ℚ,$ we have $H*(BT)wP ≡H*(BK∩P) ≃HT(G/P).$ In fact

 1) $H*(BK∩P,ℚ) ≃ (ℚ⊗ℤHT(G/P))W (?)$ 2) $S⊗H*(BK) H*(BK∩P) ≃ HT(G/P) Z⊗SHT(G/P) ≃ H*(G/P)$

### Open Problems

(1) In what sense does the diagonal map $K ⟶ K×K: k ⟼ (k,k)$ correspond to the co-product $Δ: A_ ⟶ A_⊗SA_$ Given homomorphism ${K}_{1}\to {K}_{2}$ with ${T}_{1}\to {T}_{2},$ ${N}_{1}\to {N}_{2}$ can easily calculate $HT2(K2/T2) ⟶ HT1(K1/T1)$
(2) Conjecture: For each $u,v,w\in W,$ the $ϵ\left(uvw\right){a}_{w}^{u,v}$ is a polynomial in the ${\alpha }_{j}\text{'s}$ $i\in I$ with ${ℤ}_{+}\text{-coefficient.}$

True for:
 (1) $\ell \left(u\right)+\ell \left(v\right)=\ell \left(w\right)$ - Kumar (2) $v=w$ or $u=w$ - Sara Billey.
(3) Similar models for $K\text{-theory}$ (done?). Cobordism: $HT(G/P) ⟶ KT(G/P). BGG-operators ⟶ Demazure operators$
(4) Find combinatorial interpretation of the coefficients of $ϵ\left(uvw\right){a}_{w}^{u,v}$
(5) Find combinatorial interpretation of the structure constants of ${H}^{{S}^{1}}\left(\text{Grass}\left(k,n\right)\right)$ with ${S}^{1}$ acting by $\text{exp}\left(t{\rho }^{\vee }\right)\text{.}$
(6) Prove Little-Richardson Rule for $\sigma$ where $\sigma$ is a diagram automorphism of $f$ dim $G$ and $\sigma$ is admissible, re. $⟨{\alpha }_{{\sigma }^{k}\left(i\right)},{\alpha }_{i}^{\vee }⟩\ne 0⇒{\sigma }^{k}\left(i\right)=1\text{.}$ (In this case ${G}^{\sigma }$ has the structure of a Kac-Moody group. $\lambda \in {h}_{ℤ}^{*}$ $\sigma \left(\lambda \right)=\lambda$ $\lambda$ minuscule $\alpha \in {\Delta }_{+}$ $⇒0\le ⟨\lambda ,{\alpha }^{\vee }⟩\le 1$ $⇒{H}^{*}\left(G/{P}_{\lambda }\right)\to {H}^{*}\left({G}^{\sigma }/\left({G}^{\sigma }\cap P\right)\right)$ ?)
(7) Study more of the Bruhat Graph $(G/B)T ⟷ W$ Vertices: $W,$ edges $w\to w{r}_{\alpha }$ $\alpha >0$ $T\text{-stable}$ curves $\left(\equiv P\prime \right)$ in $G/P$

Full subgraphs correspond to ${X}_{w}^{P}$ with vertices $v\le w\text{.}$ $v\to v{r}_{\alpha }$ iff $v,v{r}_{\alpha }\le w\text{.}$
(8) Theorem (Carrell-Peterson): The Kazdan-Lusztig Polynomial ${P}_{v,w}=1$ $⇔$ for this graph, have the same # of edges emanate from each point.
(9) Study directed Bruhat graphs: $w⟶α∨wrα if w

## Lecture 8: March 11, 1997

Recall picture for the next two lectures

Let $K\text{:}$ compact simple Lie group
$\Omega K\text{:}$ base preserving algebraic loops in $K$
Then $T\subset K$ acts on $\Omega K$ by conjugation: $(t·k)(z)=tk (z)t-1.$ Roughly, the diagonal embedding $ΩK ⟶ ΩK×ΩK$ gives a co-product $HT(ΩK) ⟶ HT(ΩK)⊗SHT(ΩK)$ and the multiplication map for the group structure on $\Omega K\text{:}$ $ΩK×ΩK ⟶ ΩK$ gives a product $HT(ΩK)⊗SHT(ΩK) ⟶ HT(ΩK)$ In fact, ${H}_{T}\left(\Omega K\right)$ is a commutative and cocommutative Hopf algebra over $S\text{.}$ We will identify this Hopf algebra structure using ${\underset{_}{A}}_{\text{af}}\text{.}$ In fact, we have a map $ΩK ⟶ Gaf/Baf$ which gives $HT(ΩK) ⟶ HT(Gaf/Baf) = A_af.$ Under this, we will identify $HT(ΩK)≃ Zaf(S) (centralizer of S in A_af)$ and describe ${Z}_{\text{af}}\left(S\right)$ using the affine Weyl group ${W}_{\text{af}}\text{.}$

Notation: For a variety $X$ over $ℂ,$ use $X∼=Mor(ℂ×,X).$ Let $G$ be a finite dimensional connected simple algebraic group over $ℂ\text{.}$ We then have the finite root datum $I, αi,∈hℤ∨, αi∨∈hℤ. Δ+, Π, W, g_, h_, b_, …$ Let $\theta$ be the highest root. From these we form the following Kac-Moody root datum:

• ${\left({h}_{ℤ}\right)}_{\text{af}}={h}_{ℤ},$ ${\left({h}_{ℤ}\right)}_{\text{af}}^{\vee }={h}_{ℤ}^{\vee }$
• ${I}_{\text{af}}=I\cup \left\{0\right\}$
• ${Q}_{\text{af}}=\underset{i\in {\xi }_{\text{af}}}{⨁}ℤ{\alpha }_{i}=ℤ{\alpha }_{0}+Q=ℤ\delta +Q,\phantom{\rule{2em}{0ex}}\delta ={\alpha }_{0}+\theta$ $Qaf ⟶ (hℤ)af: α0 ⟼ -θ αi ⟼ αi, i≠0,i∈I.$ $Πaf = Π∪{-θ} Πaf∨ = Π∨∪ {-θ∨} δ⫙Qaff = α0+θ⟼0 ϵ(hℤ)af∨ =hℤ∨ie. ⟨δ,h⟩=0 ∀h∈hℤ.$
Corresponding to this root datum, we have the following Kac-Moody Lie algebra ${\underset{_}{𝔤}}_{\text{af}}\text{:}$ $𝔤_af = 𝔤_⊗ℂℂ [t,t-1]= 𝔤_∼ ei = ei⊗1 fi = fi⊗1 e0 = e-θ⊗t f0 = fθ⊗t-1 ⇒[e0,f0] = [e-θ⊗1,eθ⊗1]=-?????$ Roots are in ${Q}_{\text{af}}\text{.}$ They are all those in ${Q}_{\text{af}}$ of the form $α+nδ n∈ℤ,α∈Δ, or α=0.$ The root spaces are $(𝔤_af)α+nδ= { {𝔤}_{\alpha }⊗tn if α∈Δ,n∈ℤ, h_⊗tn α=0,n∈ℤ$ so $Δre= { α∈nδ:α∈Δ,n∈ℤ }$ and all $n\delta \text{'s}$ $n\in ℤ,$ are "imaginary roots". They have multiplicity $={\text{dim}}_{ℂ} \underset{_}{h}\text{.}$

The positive roots are $(Δaf)+ = { α+nδ:n>0 or n≥0 α∈Δ+ } , (Δaf)+re = { α+nδ:n≥0 α∈Δ+ } .$

### The affine Weyl group ${W}_{\text{af}}\text{:}$

By definition, $Waf=W⋉Γ$ the semi-direct product, where $\Gamma \simeq {Q}^{\vee }$ with $Q∨ ⟶ Γ: h ⟼ th.$ $wthw-1 = tw·h thth′ = th+h′$ The reason why this is the same as the group generated by the reflections ${r}_{0},{r}_{i},$ $i\in I$ is because $tθ∨= r0rθ$ For $w\in W,$ $w·(α+nδ) = w·α+nδ (⇒ wδ=δ) th· (α+nδ) = α+nδ- ⟨α,h⟩δ$ $(soth·α=α- ⟨α,h⟩δ, th·(nδ)= nδ-⟨α,h⟩ δ).$

The Kac-Moody group: $Gaf=G∼=Mor (ℂ×,G) (Laurent series in t)$ set $P0 = Mor(ℂ,G) (power series in t) Baf = {g∈Mor(ℂ,G):g(0)∈B} ⊂P0 Uaf+ = {g∈Mor(ℂ,G):g(0)∈U+} Kaf = {g∈Gaf:g(S1)⊂K} ΩK = {k∈Kaf:k(1)=id} Taf = T G ≃ const. loops⊂Gaf$ ${K}_{\text{af}}$ acts on $\Omega K$ by $k·k′=kk′k (1)-1$ Then $iΩ: ΩK ⟶ Gaf/P0: k ⟼ k·* *=P0$ is a ${K}_{\text{af}}\text{-equivariant}$ map. This map is also a home????? because $Gaf = KafBaf Kaf∩Baf=T = (ΩK)K Baf=(ΩK) P0$

### The compact involution on ${G}_{\text{af}}\text{:}$

$(wKaf)(g) (t)=wK (g(t‾-1)) g∈Mor(ℂ×,G) =Gaf$ where ${w}_{k}:G\to G$ is the compact involution on $G$ corresponding to $K\text{.}$

The normalizer ${N}_{\text{af}}$ of ${H}_{\text{af}}=H$ in ${G}_{\text{af}}$ is $Naf=N∼=Mor (ℂ×,N)$ where, recall, $N$ is the normalizer of $H$ in $G,$ so also have $Naf = semi-direct product of N and ΩT ΩT = {g∈ΩK:g(S1)⊂T}$ so $g\in \text{ΩT}$ must be a homomorphism from ${S}^{1}$ to $T\text{.}$ Thus $ΩT≃Γ≃Q∨$ where $Q∨ ⟶∼ ΩT: h ⟼ hˆ: hˆ(z)=zh,z∈ℂ×.$ This way we also see $W⋉Γ ⟶∼ Waf: (w,th) ⟼ (W,hˆ-1H) ∈ Naf/H.$

### The nil-Hecke rings $\underset{_}{A}$ and ${\underset{_}{A}}_{\text{af}}\text{:}$

Since ${\left({h}_{ℤ}\right)}_{\text{af}}={h}_{ℤ},$ we have ${S}_{\text{af}}=S\text{.}$ Let $\underset{_}{A}$ be the nil-Hecke ring defined by $W\text{.}$ Let ${\underset{_}{A}}_{\text{af}}$ be the nil-Hecke ring defined by ${W}_{\text{af}}\text{.}$ Then we have the embedding $A_ ↪ A_af: s ⟼ s Ai ⟼ Ai, i∈I,(i≠0).$ Recall that if $\beta =w{a}_{i}^{ϵ{\Delta }^{\text{re}}}$ with $i\in I,$ then we define $Aβ∨ = wAαiw-1= wAiw-1 rβ = 1-βAβ∨$ (It is not obvious how to write ${A}_{{\beta }^{\vee }}$ in terms of the ${A}_{i}\text{'s).}$

Define a ring homomorphism $ev: A_af ⟶ A_: ev|S = id ev(Aβ∨) = Aβ‾∨ ev(wth) = w$ where if $\beta =\alpha +n,$ $\stackrel{‾}{\beta }=\alpha \text{.}$ This is well-defined.

The embedding $\underset{_}{A}↪{\underset{_}{A}}_{\text{af}}$ is a section of ev.

Now identify $ΩK ⟶iΩ∼ Gaf/P0$ we see that $\Omega K$ is a Kac-Moody $G/P,$ so we have all we discussed before, namely:

• set ${W}_{\text{af}}^{-}={W}_{\text{af}}^{{P}_{0}}$
• For each $x\in {W}_{\text{af}}^{-},$ have Schubert variety ${\underset{_}{\overset{‾}{X}}}_{x}^{\Omega }$ and inclusion $ixΩ: X_‾xΩ ⟶ ΩK$ so have Schubert basis $σxΩ ∈ H2ℓ(x) (ΩK) σΩx ∈ H2ℓ(x) (ΩK) σ(x)Ω ∈ HomS (HT(ΩK),S) σΩ(x) ∈ HT(ΩK)$
• Also for $x\in {W}_{\text{af}},$ have ${\psi }_{x}^{\Omega }\in {\text{Hom}}_{S}\left({H}^{T}\left(\Omega K\right),S\right)\text{.}$ (It is possible that ${\psi }_{x}^{\Omega }={\psi }_{y}^{\Omega }$ for $x\ne y\text{).}$
• Have ${\underset{_}{A}}_{\text{af}}\text{-module}$ structures on ${H}^{T}\left(\Omega K\right)$ and ${\text{Hom}}_{S}\left({H}^{T}\left(\Omega K\right),S\right)$.

In the Schubert basis $Ax·σ(y)Ω= { σ(xy)Ω if xy∈Waf-, ℓ(xy)=ℓ(x)+ ℓ(y), 0 otherwise.$ Define $HT(ΩK)=S-span of {σ(x)Ω:x∈?????} ⊂HomS(HT(ΩK),S)$ In our special case at hand, not only do we have ${G}_{\text{af}}/{B}_{\text{af}}\to {G}_{\text{af}}/\text{?????}$ but also: $\Omega K↪{G}_{\text{af}}/{B}_{\text{af}}\text{.}$ Thus have $HT(ΩK) ⟶ A_af.$ Next time, write the images of ${\sigma }_{\left(x\right)}^{\Omega },$ for $x\in {W}_{\text{af}}^{-},$ in ${\underset{_}{A}}_{\text{af}}$ under the above embedding and identify ${H}_{T}\left(\Omega K\right)$ as a subalgebra of ${\underset{_}{A}}_{\text{af}}\text{.}$

### About ${W}_{\text{af}}^{-}$ and ${W}_{\text{af}}/W$

Recall that ${W}_{\text{af}}^{-}={W}_{\text{af}}^{{P}_{0}}$ is the set of minimal representatives of the coset space ${W}_{\text{af}}/{W}_{{P}_{0}}={W}_{\text{af}}/W\text{.}$ $\text{(}{W}_{{P}_{0}}=W\text{).}$ $x∈Waf- ⇔ x0∀i∈I.$ Write $x=w{t}_{-h}\text{.}$ Then $x·αi = w·t-h·αi = w· (αi+⟨h,αi⟩δ) = wαi+ ⟨h,αi⟩δ x∈Waf- ⟺ wαi+⟨h,αi⟩ δ>0∀i∈I ⟺ ⟨h,αi⟩≥0 and when ⟨h,αi⟩ =0 must have wαi>0 ⟺ h is dominant and when ⟨h,αi⟩=0 must have w Now for $h$ dominant, set $Wh = the subgroup of W generated by ⟨ri:⟨h,αi⟩=0⟩ = {w∈W:wh=h}.$ Set ${P}_{h}=B{W}_{h}B\supset B$ parabolic. Then ${W}_{h}={W}_{{P}_{h}}\text{.}$ Let ${W}^{h}={W}^{{P}_{h}}$ be the set of minimal representatives of the coset space $W/{W}_{h},$ ie. $w∈Wh ⇔ w so $w∈Wh ⇔ For each i with ⟨h,αi⟩=0 have w Thus we have proved $Waf- = { wt-h:h dominant (ie. ⟨h,αi⟩≥0 ∀i∈I and w∈Wh } = { wt-h:h dominant and if ⟨h,αi⟩=0 for i∈I must have wαi>0 } .$ The map $Waf- ⟼ Waf/W wt-h ⟼ wt-h/W$ is of course a bijection.

Now another model for ${W}_{\text{af}}/W$ is $\Gamma \simeq {Q}^{\vee }\text{:}$ $Γ ⟶∼ Waf/W th ⟼ th/W.$ In other words, each coset ${W}_{\text{af}}/W$ has a unique translation element ${t}_{-h}$ in it, namely $wt-h/W=w t-hw-1/W =t-w·h/W.$

Thus:

 (1) each coset in ${W}_{\text{af}}/W$ has a unique minimal representative. (2) each coset in ${W}_{\text{af}}/W$ has a unique translation element as a representative. (3) Let $x\in {W}_{\text{af}}^{-}\text{.}$ Then $x$ is the minimal representative for the coset $xW\text{.}$ We know that $x$ must be of the form $x=w{t}_{-h}$ where $h$ is dominant and $w\in {W}^{h}\text{.}$ The translation element in this coset is ${t}_{-w-h},$ so $w{t}_{-h}\le {t}_{-w·h}\text{.}$ (4) When $h$ is dominant and regular, we have $wt-h∈Waf-$ for all $w\in W\text{.}$ So for different ${w}_{1},{w}_{2}\in W,$ the two elements ${w}_{1}{t}_{-h}$ and ${w}_{2}{t}_{-h}$ lie in two different cosets in ${W}_{\text{af}}/W\text{.}$ (5) A special case is when $x∈Waf-∩Γ.$ This is the case iff the minimal representative for $xW,$ namely $x$ itself, coincides with the translational representative of $xW\text{.}$ Write $x=w{t}_{-h}$ where $h$ is dominant and $w\in {W}^{h}\text{.}$ Then $x=t-w·h⟺ wt-h=t-w·h⟺ w=1$ so $Waf-∩P = {t-h:h is dominant}.$ (6) Let's now calculate the length $\ell \left({t}_{-h}\right)$ when $h$ is dominant. Recall that $\alpha +n\delta >0$ $⇔$ either $n>0$ or $n=0,\alpha >0\text{.}$ Now we need to see for $\alpha +n\delta >0,$ when do we have $t-h·(α+nδ)<0.$ Now $t-h·(α+nδ) =α+(n+⟨h,α⟩) δ$ If $n>0,\alpha <0,$ then ${t}_{-h}·\left(\alpha +n\delta \right)<0$ for $n=0,1,\dots ,⟨h,\alpha ⟩-1\text{.}$ If $n>0,\alpha =0,$ then ${t}_{-h}·\left(\alpha +n\delta \right)n\delta <0\text{.}$ If $n>0,\alpha >0,$ then ${t}_{-h}·\left(\alpha +n\delta \right)<0\text{.}$ If $n=0,\alpha >0,$ then ${t}_{-h}·\left(\alpha +n\delta \right)<0\text{.}$ Then the only case when $\alpha +n\delta >-$ and ${t}_{-h}·\left(\alpha +n\delta \right)<0$ is when $\alpha =-\beta <0$ (so $\beta >0\text{)}$ $n=0,1,\dots ,⟨h,\beta ⟩-1$ The number of such element is $\sum _{\beta >0}⟨h,\beta ⟩=⟨h,2\rho ⟩\text{.}$ Hence $ℓ(t-h)= ⟨h,2ρ⟩= ∑β>0 ⟨h,β⟩$ for $h$ dominant. Let's notice that $the sum of all {α+nδ>0;t-h(α+nδ)<0} = ∑β>0 ( -β-β+δ+ (-β+2δ) +…+ (-β+(⟨h,β⟩-1)δ) ) = ∑β>0 ( -⟨h,β⟩β +12⟨h,β⟩ (⟨h,β⟩-1) δ ) .$ (7) For any $x=w{t}_{-h}\in {W}_{\text{af}}^{-},$ $t={t}_{-{h}_{1}}\in {\Gamma }^{-}={W}_{\text{af}}^{-}\cap \Gamma$ we have $xt=w{t}_{-\left(h+{h}_{1}\right)}\in {W}_{\text{af}}^{-}$ and $\ell \left(xt\right)=\ell \left(x\right)+\ell \left(t\right)\text{.}$ (8) Can prove that for $x=w{t}_{-h}\in {W}_{\text{af}}^{-},$ $\alpha +n\delta >0$ be st. $x·\left(\alpha +n\delta \right)=w\alpha +\left(n+⟨h,\alpha ⟩\right)\delta <0$ $⟺$ either $\alpha <0,w\alpha >0$ and $n=1,2,\dots ,-⟨\alpha ,h⟩-1$ or $\alpha <0,w\alpha <0$ and $n=1,2,\dots ,-⟨\alpha ,h⟩\text{.}$ In other words ${ α+nδ>0:wt-h ·(α+nδ)<0 } = { -β+nδ:β>0,wβ<0, n=1,…,⟨β,h⟩-1 } ∪ { -β+nδ:β>0, wβ>0,n=1,2,… ⟨β,h⟩ } .$ Consequently, $ℓ(wt-h)= ⟨2ρ,h⟩- ℓ(w).$

## Lecture 9: March 12, 1997

Recall the ${\underset{_}{A}}_{\text{af}}\text{-action}$ on ${\text{Hom}}_{S}\left({H}^{T}\left(\Omega K\right),S\right)\text{:}$ $Ax·σ(y)Ω = { σ(xy)Ω if xy∈Waf- ℓ(x)+ℓ(y)= ℓ(xy), 0 otherwise, w·ψt = ψwt t′·ψt = ψt′t$ $t,t\prime \in \Gamma ,w\in W\text{.}$

Define $HT(ΩK)= ∑x∈Waf-s σ(x)Ω$ as the ${\underset{_}{A}}_{\text{af}}\text{-submodule}$ of ${\text{Hom}}_{S}\left({H}^{T}\left(\Omega K\right),S\right)$ spanned over $S$ by $\left\{{\sigma }_{\left(x\right)}^{\Omega }:x\in {W}_{\text{af}}^{-}\right\}\text{.}$ For $x\in {W}_{\text{af}}^{-},$ set $Fx= ∑y∈Waf-y≤x sσ(y)Ω.$ Then $ixΩ: X_‾xΩ ⟶ ΩK$ gives $HomS(HT(X_‾xΩ),S) ⟶∼ Fx HomS(Fx,S) ⟶∼ HT(X_‾xΩ).$

### Structure on ${F}_{x}$

 (1) $\left\{1\otimes {\psi }_{t}:t\in \Gamma ,t\le x{w}_{0}\right\}$ is a free $S\text{-basis}$ for $\text{Frac}\left(S\right){\otimes }_{S}{F}_{x}$ where $Frac(S) = the fractional field of S x∈Γ- ⇔ the minimal rep. of xW coincides with the translational representative of xW.$ (2) Set $Γ-=Γ∩Waf-= {t-h:h∈h_ℤ dominant} see end of Lecture 8 on Waf-≃ Waf/W≃Γ.$ Then: ${\underset{_}{\overset{‾}{X}}}_{t}^{\Omega }$ is $K\text{-stable,}$ so ${F}_{t}$ is an $\underset{_}{A}\text{-submodule}$ of ${H}_{T}\left(\Omega K\right),$ ${\sigma }_{\left(t\right)}^{\Omega }\in {\left[{H}_{T}\left(\Omega K\right)\right]}^{\underset{_}{A}}$ ie. ${\sigma }_{\left(t\right)}^{\Omega }$ is $\underset{_}{A}\text{-invariant.}$

 Proof. To show that ${\underset{_}{\overset{‾}{X}}}_{t}^{\Omega }$ is $K\text{-stable,}$ it is enough to show $P0t·P0⊂ X_‾tΩ⟺ t-1B-t∈P0$ But for any $\alpha \in {\Delta }_{+}$ $t-h·α=α+ ⟨h,α⟩δ∈ Δ(P0/baf) (ie. a root for P0)$ $⇒t-1B-t∈P0 ⇒X_‾tΩ is J-stable ⇒Ft is A_-submod. of HT(ΩK).$ Next, we need to show that $\forall i\in I,$ $\forall$ $\ell \left({r}_{i}t\right)<\ell \left(t\right)+1,$ ${A}_{i}·{\sigma }_{\left(t\right)}^{\Omega }=0$ $\forall$ ?????. But ${A}_{i}·{\sigma }_{\left(t\right)}^{\Omega }=0$ unless ${r}_{i}t\in {W}_{\text{af}}^{-}\text{.}$ So just need to show that ${r}_{i}t\notin {W}_{\text{af}}^{-}\text{.}$ for any $i\in I\text{.}$ This is not possible. Suppose ${r}_{i}t\in {W}_{\text{af}}^{-}$ for some $i\text{.}$ Then ${r}_{i}$ must satisfy $\text{"}⟨h,{\alpha }_{j}⟩=0$ for some $j\in I$ $⇒$ ${r}_{i}{\alpha }_{j}>0\text{"}\text{.}$ Since ${r}_{i}{\alpha }_{i}<0,$ must have $⟨h,{\alpha }_{i}⟩>0\text{.}$ If $\ell \left({r}_{i}t\right)=\ell \left(t\right)+1,$ then $t<{r}_{i}t$ or ${t}^{-1}<{t}^{-1}{r}_{i}$ $⇒$ ${t}^{-1}{\alpha }_{i}>0\text{.}$ But ${t}^{-1}·{\alpha }_{i}={t}_{h}·{\alpha }_{i}={\alpha }_{i}-⟨h,{\alpha }_{i}⟩\delta ,$ Since $⟨h,{\alpha }_{i}⟩>0$ $⇒$ ${t}_{h}·{\alpha }_{i}<0\text{.}$ Contractiction. Hence ${A}_{i}·{\sigma }_{\left(t\right)}^{\Omega }=0$ $\forall i\in I\text{.}$ $\square$

### Hopf algebra structure on ${H}_{T}\left(\Omega K\right)$

Proposition: ${H}_{T}\left(\Omega K\right)$ is a Hopf algebra over $S,$ commutative and cocommutative.

 Proof (outline) and structure maps. The $T\text{-equivariant}$ multiplication map $m: ΩK×ΩK ⟶ ΩK$ induces the product map: $μ: HT(ΩK)⊗HT(ΩK) ⟶ HT(ΩK).$ Since $m(X_‾xΩ×X_‾tΩ) ⊂X_‾xtΩ$ we actually have $μ: Fx⊗Ft ⟶ Fxt.$ The diagonal imbedding $ΩK ⟶ ΩK×ΩK$ induces the co-product: $Δ: HT(ΩK) ⟶ HT(ΩK)⊗HT(ΩK).$ Clearly $ΔFx⊂Fx⊗Fx$ Co-commutativity is clear. As for commutativity of $\mu ,$ one can give a couple of reasons. One reason is that over $\text{Frac}\left(S\right),$ ${F}_{x}$ has a basis $\left\{1\otimes {\psi }_{t}:t\in \Gamma ,t\le x{w}_{0}\right\}$ and ${\psi }_{t}{\psi }_{t\prime }={\psi }_{t\prime }{\psi }_{t}={\psi }_{t\prime t}\text{.}$ Another reason is because $\Omega K$ is a double loop space so its (at least ordinary) homology is commutative. unit: ${\psi }_{\text{id}}$ antipode: $c\left({F}_{t}\right)={F}_{\omega \left(t\right)}$ where $\omega$ is the diagram automorphism defined by $ω·αi=-αω(i), i≠0,ω(0)=0$ $\text{(}⇒$ $\omega \left(w\right)={w}_{0}w{w}_{0}$ for $w\in W$ and $\omega \left({t}_{h}\right)={t}_{-{w}_{0}·h}\text{).}$ In terms of the ${\psi }_{t}\text{'s,}$ the Hopf algebra structure is easier to express. $ε(ψt) = 1, c(ψt) = ψt-1, Δψt = ψt⊗ψt, ψtψt′ = ψtt′, ψid = 1.$ $\square$

In the following, we describe a model for ${H}_{T}\left(\Omega K\right)\text{.}$

### The map $j:{H}_{T}\left(\Omega K\right)⟶{\underset{_}{A}}_{\text{af}}$

First, we have the general fact that if $X$ is a $T\text{-space}$ and $ϕ: ΩK×X ⟶ X$ is a $T\text{-equivariant}$ map (with $T$ acting on $\Omega K$ by conjugations and on $\Omega K×X$ by the diagonal action), then each $\sigma \in {H}_{T}\left(\Omega K\right)\subset {\text{Hom}}_{S}\left({H}^{T}\left(\Omega K\right),S\right)$ defines the following composition map $HT(X) ⟶ϕ* HT(ΩK)⊗SHT(X) ⟶c(σ)⊗id S⊗SHT(X) ≃ HT(X).$ If $\varphi$ defines an action of $\Omega K$ on $X,$ then these composition maps define an ${H}^{T}\left(\Omega K\right)\text{-module}$ structure on ${H}^{T}\left(X\right)\text{.}$

Now assume that $X$ is a ${K}_{\text{af}}\text{-space.}$ By restriction to $T$ and $\Omega K,$ it is both a $T\text{-space}$ and an $\Omega K\text{-space}$ and the action map $ϕ: ΩK×X ⟶ X$ is $T\text{-equivariant.}$ Thus each $\sigma \in {H}_{T}\left(\Omega K\right)$ defines an operator on ${H}^{T}\left(X\right)\text{.}$ This is functorial in $X,$ so we get a characteristic operator. In other words, we have a map $j: HT(ΩK) ⟶ Aˆ_af.$ A calculation shows that $j\left({\psi }_{t}\right)=t\text{.}$ Thus $j\left(\sigma \right)$ is compactly supported, (?) so ie $j\left(\sigma \right)\in {\underset{_}{A}}_{\text{af}}\text{.}$ It is obvious that $j$ is a ring homomorphism. Since ${H}_{T}\left(\Omega K\right)$ is commutative and since $j$ is an $s\text{-map,}$ (?) we have $j(HT(ΩK))⊂ ZA_af(s), centralizer of s in A_af (t∈Waf ⊂A_af commutes with s).$ Set $A_Ω= ZA_af(s).$ It is a commutative $S\text{-algebra.}$ Thus we have an $S\text{-algebra}$ homomorphism $j: HT(ΩK) ⟶ A_Ω = ZA_af(S)$ Will show that it is in fact an isomorphism.

### Connection between $j:{H}_{T}\left(\Omega K\right)\to {\underset{_}{A}}_{\text{af}}$ and ${j}_{\Omega }:\Omega K\to {G}_{\text{af}}/{B}_{\text{af}}:k↦k{B}_{\text{af}}\text{.}$

Have commutative diagram $HT(ΩK) ⟶j A_af ↪ ↓ HomS(HT(ΩK),S) ⟶(jΩ)X HomS(HT(Gaf/Baf),S) a ↧ ε·aR = ε·c(a)L$ Before we find $j\left({\sigma }_{\left(x\right)}^{\Omega }\right),$ we collect some facts about the action of ${H}_{T}\left(\Omega K\right)$ on ${H}^{T}\left(X\right)$ for a ${K}_{\text{af}}\text{-space}$ $X\text{.}$

Note: The following Lemma 1 had a large cross through it in the scanned copy.

Lemma 1: For any ${K}_{\text{af}}\text{-space}$ $X,$ the action of ${\underset{_}{A}}_{\text{af}}$ on ${H}^{T}\left(X\right)$ factors through $\underset{_}{A}$ via the map (Is this right?) $ev: A_af ⟶ A_$ where, recall, $ev|S = id, ev|Aβ∨ = Aβ‾∨, ev|wth = w.$

Lemma 2: For $\sigma \in {H}^{T}\left(\Omega K\right),$ $(id⊗ev) Δ·j(σ)= j(σ)⊗1. ?$

 Proof. This is roughly due to the fact that $ΩK ↪ Kaf: k ⟼ (k,1).$ $\square$

Now for any ${\underset{_}{A}}_{\text{af}}\text{-module}$ $M$ and $\underset{_}{A}\text{-module}$ $N,$ set $M{*}_{S}N=M{\otimes }_{S}{\text{ev}}^{*}N,$ an ${\underset{_}{A}}_{\text{af}}\text{-module.}$ Then by Lemma 2, $j(σ)·(m⊗n)= j(σ)·m⊗n.$ Apply this to the action map $F: HT(ΩK)⊗SHT(X) ⟶ HT(X).$

Proposition: The above action map is an ${\underset{_}{A}}_{\text{af}}\text{-module}$ map.

 Proof. For $\sigma \in {H}_{T}\left(\Omega K\right)$ and $z\in {H}^{T}\left(X\right),$ we know $F(σ⊗z)= j(σ)·z ?$ so for $w\in W$ $w·F(σ⊗z)= w·j(σ)·z.$ In particular $w·F(ψt⊗z)= w·t·z=wtw-1 ·w·z=(w·t)· (w·z).$ On the other hand $F(w·(ψt⊗z))= F(w·ψt⊗w·z)= F(w·(ψt⊗z)).$ Also $t′·F(ψt⊗t)= t′t·z=F (ψt′t⊗z)=F (t′·(ψt⊗z)).$ $\square$

Proposition: The multiplication map $HT(ΩK)⊗SHT(ΩK) ⟶ HT(ΩK)$ is an ${\underset{_}{A}}_{\text{af}}\text{-map.}$

 Proof. This is because $σσ′=j(σ)· σ′.$ $\square$

More generally, for any ${\underset{_}{A}}_{\text{af}}\text{-module}$ $M,$ the map $ϕ: HT(ΩK)⊗SM ⟶ M σ⊗m ⟼ j(σ)·m$ is always an ${\underset{_}{A}}_{\text{af}}\text{-module}$ map.

We now look at $j\left({\sigma }_{\left(x\right)}^{\Omega }\right),$

Introduce the ideal $I\subset {\underset{_}{A}}_{\text{af}}\text{:}$ (left ideal) $I=∑w∈Ww≠id {\underset{_}{A}}_{\text{af}}Aw.$ This is the ideal of annihilators of $1\in {H}_{T}\left(\Omega K\right)$ for the action of ${\underset{_}{A}}_{\text{af}}$ on ${H}_{T}\left(\Omega K\right)\text{.}$

Proposition: For $x\in {W}_{\text{af}}^{-}$ $j(σ(x)Ω) =Ax mod I.$

 Proof. $j(σ(x)Ω)·1= σ(x)Ω1= σ(x)Ω= Ax·σ(id)Ω= Ax·1. ⇒j(σ(x)Ω) -Ax∈I.$ $\square$

Corollary 1: ${A}_{x{w}_{0}}=j\left({\sigma }_{\left(x\right)}^{\Omega }\right){A}_{{w}_{0}}$ where ${w}_{0}=$ longest in $W\text{.}$

 Proof. $j(σ(x)Ω) Aw0= (Ax+a)Aw0= AxAw0= Axw0(a∈I).$ $\square$

Corollary 2: For any $x\in {W}_{\text{af}}^{-},$ $t\in {\Gamma }_{-}$ $σ(x)Ω σ(t)Ω = σ(xt)Ω, FxFt = Fxt.$ $\text{(}\ell \left(x\right)+\ell \left({w}_{0}\right)=\ell \left(x{w}_{0}\right)$ holds for all $x\in {W}_{\text{af}}^{-}\text{).}$ This is due to the following general fact: For any parabolic $P,$ $\forall x\in {W}^{P},y\in {W}_{P},$ $\ell \left(xy\right)=\ell \left(x\right)+\ell \left(y\right)\text{.}$

 Proof. Since ${\sigma }_{\left(t\right)}^{\Omega }\in {\left[{H}_{T}\left(\Omega K\right)\right]}^{\underset{_}{A}},$ have $σ(x)Ω σ(t)Ω = j(σ(x)Ω) ·σ(t)Ω = (Ax+a)· σ(t)Ω a∈I = Ax· σ(t)Ω (a·(σ(t)Ω=0)) = σ(xt)Ω.$ (We are saying $\ell \left(x\right)+\ell \left(t\right)=\ell \left(xt\right)\text{? automatically?)}$ $\square$

Proposition: $HT(ΩK)⊗SA_ ⟶ A_af: σ⊗a ⟼ j(σ)a$ is an ${\underset{_}{A}}_{\text{af}}\text{-module}$ isomorphism, where ${\underset{_}{A}}_{\text{af}}$ acts on $\underset{_}{A}$ via $\text{ev}:{\underset{_}{A}}_{\text{af}}\to \underset{_}{A}\text{.}$

 Proof. $\square$

?????or: $j:{H}_{T}\left(\Omega K\right)\stackrel{\sim }{\to }{\underset{_}{A}}_{\Omega }$ is an isomorphism.

Thus we have a direct sum decomposition $A_af≃A_Ω +I$ as an ${\underset{_}{A}}_{\Omega }\text{-module.}$

### Structures on ${\underset{_}{A}}_{\Omega }$

• First, by identifying $A_Ω≅ A_af/I$ we get an ${\underset{_}{A}}_{\text{af}}\text{-module}$ structure on ${\underset{_}{A}}_{\Omega },$ ie., for $a\in {\underset{_}{A}}_{\text{af}}$ and $a\prime \in {\underset{_}{A}}_{\Omega },$ $a·a\prime \in {\underset{_}{A}}_{\Omega }$ is the unique element of ${\underset{_}{A}}_{\Omega }$ st $a·a′-aa′∈I.$
• By definition, $z\left({\underset{_}{A}}_{\text{af}}\right)\subset {\underset{_}{A}}_{\Omega }\simeq {Z}_{{\underset{_}{A}}_{\text{af}}}\left(S\right),$ and the action of ${\underset{_}{A}}_{\text{af}}$ on ${\underset{_}{A}}_{\Omega }$ is $Z\left({\underset{_}{A}}_{\text{af}}\right)\text{-linear.}$
• For each $x\in {W}_{\text{af}}^{-},$ $j\left({\sigma }_{\left(xe\right)}^{\Omega }\right)$ is the unique element in ${\underset{_}{A}}_{\Omega }$ such that $j(σ(x)Ω) ∈Ax+I.$ In other words, $j(σ(x)Ω) =Ax·1$ for the action of ${\underset{_}{A}}_{\text{af}}$ on ${\underset{_}{A}}_{\Omega }\text{.}$
• We can calculate the action of ${\underset{_}{A}}_{\text{af}}$ on ${\underset{_}{A}}_{\Omega }$ as follows.

Proposition: For $s\in S,$ $a\in {\underset{_}{A}}_{\Omega },$ $w\in W,$ $t\in \Gamma$ and $\beta \in {\Delta }^{\text{re}}$ $s·a = sa=as wt·a = wtaw-1 Aβ∨·a = Aβ∨a- rβaAβ‾∨ (β=α+nδ,β‾=α) = Aβ∨arβ‾ +aAβ‾∨ (Aα0∨· a‾0∨ = -θ∨).$ The proof of this proposition is not trivial. Need calculat?????

Introduce Hopf algebra (over $S\text{)}$ structure on ${\underset{_}{A}}_{\Omega }\text{:}$ $π(s) = s, ε(t) = 1, c(t) = t-1, Δ(t) = t⊗t.$

Theorem: The map $j: HT(ΩK) ⟶ A_Ω$ is an isomorphism of both ${\underset{_}{A}}_{\text{af}}\text{-modules}$ and Hopf algebra modules.

## Lecture 10: March 19, 1997

### $\Omega \text{-integrable}$${\underset{_}{A}}_{\text{af}}\text{-modules}$

We first recall the definition of the integrable $\underset{_}{A}\text{-modules}$ where $\underset{_}{A}$ is ${\underset{_}{A}}_{\text{af}}$ or ${\underset{_}{A}}_{\text{finite}},$ that was given at the end of Lecture 6:

An integrable $\underset{_}{A}\text{-module}$ is an $\underset{_}{A}\text{-module}$ structure on $𝒪\left(X\right),$ where $X$ is an affine scheme over $\underset{_}{h}=\text{Spec} S$ with structure homomorphism ${\pi }_{X}:S\to 𝒪\left(X\right)$ such that

 (1) $s·p={\pi }_{X}\left(s\right)p$ $\forall s\in S,p\in 𝒪\left(X\right)\text{.}$ (2) ${\pi }_{X}:S\to 𝒪\left(X\right)$ is an $\underset{_}{A}\text{-module}$ map. (3) $m:𝒪\left(X\right){\otimes }_{S}𝒪\left(X\right)\to 𝒪\left(X\right)$ is an $\underset{_}{A}\text{-module}$ map. (4) For each $p\in 𝒪\left(X\right),$ ${A}_{w}·p=0$ for all but finitely many $w\in W\text{.}$

Now back to our notation where $\underset{_}{A}$ denotes the nil-Hecke ring for the finite Wely group $W\text{.}$ Then condition (4) is not needed.

Definition: An $\Omega \text{-integrable}$ ${\underset{_}{A}}_{\text{af}}\text{-module}$ is by definition an affine scheme $X$ over $\underset{_}{h}=\text{Spec} S,$ with structure homomorphism ${\pi }_{X}:S\to 𝒪\left(X\right),$ and an ${\underset{_}{A}}_{\text{af}}\text{-module}$ structure on $𝒪\left(X\right)$ such that

 (1) $X$ is an integrable $\underset{_}{A}\text{-module}$ by restricting the action of ${\underset{_}{A}}_{\text{af}}$ to $\underset{_}{A}\text{;}$ (2) $m:𝒪\left(X\right){*}_{S}𝒪\left(X\right)\to 𝒪\left(X\right)$ is an ${\underset{_}{A}}_{\text{af}}\text{-map.}$
(Part of the requirement for (1) is in (2) as well).

Question: Is (2) weaker than asking $m:𝒪\left(X\right){\otimes }_{S}𝒪\left(X\right)\to 𝒪\left(X\right)$ being an ${\underset{_}{A}}_{\text{af}}\text{-map?}$ This seems to be just a different requirement. So the notion of $\Omega \text{-integrable}$ ${\underset{_}{A}}_{\text{af}}\text{-module}$ seems different from that of an integrable ${\underset{_}{A}}_{\text{af}}\text{-module.}$

Set $𝒜=\text{Spec} {H}_{T}\left(\Omega K\right)\text{.}$ Then $𝒜$ is an integrable $\underset{_}{A}\text{-module.}$ We know from Lecture 9 (page 9-9) that $m:{H}_{T}\left(\Omega K\right){*}_{S}{H}_{T}\left(\Omega K\right)\to {H}_{T}\left(\Omega K\right)$ is an ${\underset{_}{A}}_{\text{af}}\text{-module}$ map, so $𝒜$ is an $\Omega \text{-integrable}$ ${\underset{_}{A}}_{\text{af}}\text{-module.}$

Proposition: An $\Omega \text{-integrable}$ ${\underset{_}{A}}_{\text{af}}\text{-module}$ structure on $𝒪\left(X\right)$ is equivalent to

 (1) an integrable $\underset{_}{A}\text{-module}$ structure $𝒪\left(X\right)\text{;}$ and (2) an $\underset{_}{A}\text{-module}$ map $g:{H}_{T}\left(\Omega K\right)\to 𝒪\left(X\right)\text{.}$
More explicitly, given an ${\underset{_}{A}}_{\text{af}}\text{-module}$ structure on $𝒪\left(X\right),$ by restriction to $\underset{_}{A}$ we get an integrable $\underset{_}{A}\text{-module}$ structure on $𝒪\left(X\right),$ and the map $g: HT(ΩK) ⟶ 𝒪(X): g(σ) = j(σ)·1.$ Conversely, given (1) and (2), the ${\underset{_}{A}}_{\text{af}}\text{-module}$ structure on $𝒪\left(X\right)$ is defined by $(j(σ)a)·p= g(σ)(a·p).$

 Proof. Assume that the ${\underset{_}{A}}_{\text{af}}\text{-module}$ structure on $𝒪\left(X\right)$ is given. We need to show that the map $g$ is an $\underset{_}{A}\text{-map,}$ i.e., for $a\in \underset{_}{A}$ and $\sigma \in {H}_{T}\left(\Omega K\right),$ need to show $g(a·σ) = a·g(σ).$ Now $g(a·σ) = j(a·σ)·1, a·g(σ) = a·j(σ)·1= (aj(σ))·1.$ Thus we need to show $( j(a·σ)- aj(σ) ) ·1=0∈𝒪(X).$ But we know that the action of $\underset{_}{A}$ on ${H}_{T}\left(\Omega K\right)$ is characterised by the fact that $j(a·σ)- aj(σ)∈I= ∑w∈Ww≠id A_afAw.$ Since for any $i\in I,$ $Ai·1=Ai· πX(1)=πX (Ai·1)=0∈ 𝒪(X)$ we see that $b·1=0$ for any $b\in I\text{.}$ Thus $( j(a·σ)- aj(σ) ) ·1=0$ or $g:{H}_{T}\left(\Omega K\right)\to 𝒪\left(X\right)$ is an $\underset{_}{A}\text{-map.}$ Conversely, assume that we are given an integrable $\underset{_}{A}\text{-module}$ structure on $𝒪\left(X\right)$ and an $\underset{_}{A}\text{-map}$ $g:{H}_{T}\left(\Omega K\right)\to 𝒪\left(X\right)\text{.}$ Define, for $\sigma \in {H}_{T}\left(\Omega K\right)$ and $a\in \underset{_}{A},$ $p\in 𝒪\left(X\right)$ $(j(σ)a)·p= g(σ)(a·p).$ Need to show that this gives an $\Omega \text{-integrable}$ $\underset{_}{A}\text{af}\text{-module}$ structure on $𝒪\left(X\right)\text{.}$ First need to show that this is indeed and action of ${\underset{_}{A}}_{\text{af}}\text{.}$ This must follow from the fact that $HT(ΩK)*SA_ ⟶ A_af: σ⊗a ⟼ j(σ)a$ is an ${\underset{_}{A}}_{\text{af}}\text{-module}$ map. (?) In order to show $m: 𝒪(X)⊗S𝒪(X) ⟶ 𝒪(X)$ is an ${\underset{_}{A}}_{\text{af}}\text{-module}$ map, only need to show $m(j(σ)·(p1⊗p2)) =j(σ)·(p1p2).$ But $j(σ)·(p1p2)= g(σ)p1p2$ and (Remark after Lemma 2 in Lecture 9 on page 9-7) $m(j(σ)·(p1⊗p2)) = m(j(σ)·p1⊗p2) =m(g(σ)p1⊗p2) = g(σ)p1p2$ so $m(j(σ)·(p1⊗p2)) =j(σ)·(p1p2).$ $\square$

Need to fill in the proof of why $\left(j\left(\sigma \right)a\right)·p\stackrel{\text{def}}{=}g\left(\sigma \right)\left(a·p\right)$ defines an ${\underset{_}{A}}_{\text{af}}\text{-action.}$

In more geometrical terms, let $𝒰=Spec HT(K/T).$ We said in Lecture 6 that an integrable $\underset{_}{A}\text{-module}$ should be thought of as an action $\varphi :𝒰{×}_{\underset{_}{A}}X\to X\text{.}$ In this language, an $\Omega \text{-integrable}$ ${\underset{_}{A}}_{\text{af}}\text{-module}$ structure on $𝒪\left(X\right)$ $⟺$ pairs $\left(\varphi ,f\right)$ where $\varphi$ is an action of $𝒰$ on $X$ and $f:X\to 𝒜$ is a $𝒰\text{-equivariant}$ map.

### The polynomials ${j}_{x}^{y},$$x\in {W}_{\text{af}}^{-},$$y\in {W}_{\text{af}}$

For $x\in {W}_{\text{af}}^{-},$ introduce ${j}_{x}^{y}\in S,$ $y\in {W}_{\text{af}},$ by $j(σ(x)Ω)= ∑y∈Wafjxy Ay.$ In terms of the map $jΩ: Ωx ⟶ Gaf/Baf$ we have $jΩ* σGaf/Baf(y) =∑x∈Waf- jxyσΩ(x).$

### Immediate properties of the polynomial ${j}_{x}^{y}\text{'s,}$$x\in {W}_{\text{af}}^{-},$$y\in {W}_{\text{af}}\text{:}$

Property 1: $deg jxy=2 (ℓ(y)-ℓ(x)).$

This is because $deg (σ(x)Ω) = -2ℓ(x), deg Ay = -2ℓ(y).$

Property 2: $jxy=δxy if y∈Waf-.$

Property 3: $jxy=0unless y≤t≤xw0for some t∈Γ.$ $( ⇒deg jxy = 2(ℓ(y)-ℓ(x)) ≤2(ℓ(xw0)-ℓ(x)) = 2(ℓ(x)+ℓ(w0)-ℓ(x)) =2ℓ(w0) ) .$

 Proof. Since $jΩ(X_‾xΩ) ⊂πp-1 (X_‾xGaf/Baf) =X_‾xw0Gaf/Baf$ and since ${{j}_{\Omega }}_{*}\left({\psi }_{t}\right)={\psi }_{t}$ by definition, we have ${j}_{\Omega }^{*}\left(z\right)=0$ in ${H}^{T}{\underset{_}{\overset{‾}{X}}}_{x}^{\Omega }$ if ${\psi }_{t}\left(z\right)=0$ for all $t\in \Gamma$ with $t$ ?????. Property 3 now follows from this. $\square$

Proposition: For $x,z\in {W}_{\text{af}}^{-}$ $σ(x)Ω σ(z)Ω= ∑y∈Wafyz∈Waf-ℓ(y)+ℓ(z)=ℓ(yz) jxyσ(xz)Ω.$

 Proof. $σ(x)Ω σ(z)Ω = j(σ(x)Ω) ·σ(z)Ω = ∑y∈Waf jxyAy· σ(z)Ω = ∑y∈Wafyz∈Waf-ℓ(y)+ℓ(z)=ℓ(yz) jxyσ(xz)Ω.$ $\square$

Conjecture: The ${j}_{x}^{y}\text{'s}$ are polynomials in the ${\alpha }_{i}\text{'s}$ with coefficients in ${ℤ}_{+}=\left\{0,1,2,\dots \right\}\text{.}$

Remark 1: Can show ${j}_{x}^{y}\in {ℤ}_{+}$ when $\ell \left(y\right)=\ell \left(x\right)$ by making connection with quantum cohomology: these are the Gromov-Witten invariants.

Remark 2: We proved last time that $\forall x\in {W}_{\text{af}}^{-}$ and $t\in {\Gamma }^{-}={W}_{\text{af}}^{-}\cap \Gamma ,$ $σ(x)Ω σ(t)Ω= σ(xt)Ω.$ On the other hand, since ${H}_{T}\left(\Omega K\right)$ is commutative, we have $σ(x)Ω σ(t)Ω= σ(t)Ω σ(x)Ω= j(σ(t)Ω) ·σ(x)Ω.$ It follows that, for $h$ dominant $j(σ(t-h)Ω) =∑w∈WAt-w·h$ since ${\sigma }_{\left({t}_{-h}\right)}^{\Omega }$ is $\underset{_}{A}\text{-invariant,}$ we know that $j\left({\sigma }_{\left({t}_{-h}\right)}^{\Omega }\right)$ is in the center of ${\underset{_}{A}}_{\text{af}}\text{.}$

### An integral formula

Define $ev1: Kaf/T ⟶ K/T ev1(kT) = k(1)T.$

Proposition: For $x,y\in {W}_{\text{af}}^{-}$ and $w\in W,$ $jxyω(w-1) = ⟨ σGaf/Baf(yw0) ev1* (w0L·σG/B(w)) ,σ(xw0)Gaf/Baf ⟩ = ∫[X_‾Ωyw0∩X_‾xw0Ω] ev1*(w0L·σG/B(w))$ where $X_‾Ωyw0= Baf-yw0·Baf‾, X_‾xw0Ω= Bafxw0·Baf‾$ and $ω(w)=w0w w0-1$ is the diagram automorphism.

Remarks:

 1 ${{w}_{0}}_{L}·{\sigma }_{G/B}^{\left(w\right)}$ restricts to ${\sigma }_{G/B}^{w}$ under the restriction map $HT(K/T) ⟶ H(K/T).$ 2 The formula for $\ell \left(w\right)=1$ will be used later to show that $H*(ΩK) ≃ qH*(G/B).$

 Proof. The proof uses various formulas we have proved so far. $⟨ σGaf/Baf(yw0) ev1* (w0L·σG/B(w)), σ(xw0)Gaf/Baf ⟩ = ε ( (Axw0)R· ( σGaf/Baf(yw0) ev1* (w0L·σG/B(w)) ) ) (definition of ⟨ ⟩) = ε ( j(σ(x)Ω)R ·Aw0R· ( σGaf/Baf(yw0) ev1* (w0L·σG/B(w)) ) ) (Axw0=j (σ(x)Ω) Aw0 from Lecture ?????) = ε ( j(σ(x)Ω)R· ( ∑v∈W ( (Aw,v)R· σGaf/Baf(yw0) ) ( (w0Av)R· ev1*????? ) ) ) (Δλw0= ∑v∈WAw0v⊗ w0Av from Lecture ?????) = ε ( j(σ(x)Ω)R · ( ∑v∈W σGaf/Baf(yw0v-1w0) ev1* ( w0RAvR w0L· σG/B(w) ) ) ) ℓ(yw0v-1w0) +ℓ(w0v)=ℓ(yw0) ⇕ ℓ(w-v-1w0) +ℓ(w0v)= ℓ(w0) automatically satisfied. (Formula for Aw0vR· from Lecture 6 and beginning of Lecture 7), ev1* comm????? = ε ( j(σ(x)Ω)R ·∑v∈Wℓ(wv-1)+ℓ(v)=ℓ(w) σGaf/Baf(yω(v)-1) ev1* ( w0Rw0L· σG/B(wv-1) ) ) = ε ( ∑v∈Wℓ(wv-1)+ℓ(v)=ℓ(w) ( j(σ(x)Ω)R ·σGaf/Baf(yω(v)-1) ) ev1* ( w0Rw0L· σG/B(wv-1) ) ) ( (id⊗ev)Δ· j(σ)=j(σ)⊗1 in Lecture 9) = ∑v∈Wℓ(wv-1)+ℓ(v)=ℓ(w) ε ( jσ(x)ΩR· σGaf/Baf(yω(v)-1) ) ε ( ev1* ( w0Rw0L· σG/B(wv-1) ) ) (ε is a homomorphism). = ∑v∈Wℓ(wv-1)+ℓ(v)=ℓ(w) ε ( jσ(x)ΩR· σGaf/Baf(yω(v)-1) ) δv,w ( ε ( ev1* ( w0Rw0K· σG/B(wv-1) ) ) =ε(σG/B(wv-1)) =δv,w Why?) = ε ( j(σ(x)Ω)R· σGaf/Baf(yω(w)-1) ) = ⟨ j(σ(x)Ω), σGaf/Baf(yω(w)-1) ⟩ = jxyω(w)-1 .$ The fact that this is then equal to the integral is almost by definition of the Schubert basis and of the pairing $⟨ ⟩\text{.}$ $\square$

Remark: ${\underset{_}{\overset{‾}{X}}}_{x}^{\Omega }$ is rational and irreducible (?).

### The basis $\left\{{\sigma }_{\left[x\right]}:x\in {W}_{\text{af}}^{-}\right\}$ for ${H}_{T}\left(\Omega K\right)$

For $x\in {W}_{\text{af}}^{-},$ set $σ[x]=ϵ(x) c(σ(x)Ω) ∈HT(ΩK).$ This is an $S\text{-basis}$ for ${H}_{T}\left(\Omega K\right)\text{.}$

The automorphism $\nu$ of ${\underset{_}{A}}_{\text{af}}$ is used to obtain properties for this basis: $ν|Δ=id |Δ, ν|A_Ω =c.$ Can check that $ν(a)=(-1)12deg a w0ω(a)w0, a∈Waf$ where, recall, $\omega \left(w\right)={w}_{0}w{w}_{0},$ $\omega \left({t}_{h}\right)\stackrel{?}{=}{t}_{\omega \left(h\right)}=\text{?????}$ Also have $ν(a)·c(σ)= c(a·σ).$

Fact 1: $\forall x\in {W}_{\text{af}}^{-}$ $σ[x]=w0· σ(ω(x))Ω.$

 Proof. $σ[x] = ϵ(x)c(σ(x)Ω) = ϵ(x)c(Ax·1) = ϵ(x)ν(Ax)·1 = ϵ(x) (-1)ℓ(x) w0ω(Ax)w0 ·1 = ϵ(x) (-1)ℓ(x) w0Aω(x) ·1 = w0· (σ(ω(x))Ω).$ $\square$

Fact 2: For $x\in {W}_{\text{af}},$ $y\in {W}_{\text{af}}^{-}$ $ν(Ax)· σ[y]= { ϵ(x) σ[xy] if xy∈Waf-, ℓ(x)+ℓ(y)= ℓ(xy), 0 otherwise.$

 Proof. Follows from ${\sigma }_{\left[x\right]}=ϵ\left(x\right)\nu \left({A}_{x}\right)·1\text{.}$ $\square$

Fact 3: For $t\in {\Gamma }^{-},$ $x,z\in {W}_{\text{af}}^{-}$ $σ[t]= σ(ω(t))Ω.$

Fact 4: For $x,z\in {W}_{\text{af}}^{-}$ $σ[x]σ[z] ∑y∈Wafyz∈Waf-ℓ(y)+ℓ(z)=ℓ(yz) ϵ(xy)jxy σ[yz].$

### Ideals in ${H}_{T}\left(\Omega K\right)$ and ${\underset{_}{A}}_{\text{af}}$

Proposition: If $M$ is an ${\underset{_}{A}}_{\text{af}}\text{-submodule}$ of ${H}_{T}\left(\Omega K\right),$ then

 1) $M$ is an ideal of ${H}_{T}\left(\Omega K\right)$ which is stable under $\underset{_}{A}\text{;}$ 2) $j\left(M\right)\underset{_}{A}=\underset{_}{A}j\left(M\right)$ is a $2\text{-sided}$ ideal of ${\underset{_}{A}}_{\text{af}}\text{.}$

 Proof. Assume that $M$ is an ${\underset{_}{A}}_{\text{af}}\text{-submodule}$ of ${H}_{T}\left(\Omega K\right)\text{.}$ Then it is automatically $\underset{_}{A}\text{-stable.}$ If $\sigma \in {H}_{T}\left(\Omega K\right)$ and $m\in M,$ we have $σm=j(σ)·m.$ Since $M$ is ${\underset{_}{A}}_{\text{af}}\text{-stable,}$ $⇒$ $j\left(\sigma \right)·m\in M$ $⇒$ $\sigma m\in M\text{.}$ Hence $M\subset {H}_{T}\left(\Omega K\right)$ is an ideal. Now for $i\in I$ and $m\in M,$ $Aij(m) = j(m) Ai+j(Ai·m) ri ⇒A_j(m) ⊂ j(M)A_.$ Also have $j(m)Ai = Aij(m)-ri j(Ai·m) ⇒j(M)A_ ⊆ A_j(M) ⇒j(M)A_ = A_j(M).$ Thus $j\left(M\right)$ is stable under both left and right multiplications by elements in both $j\left({H}_{T}\left(\Omega K\right)\right)$ and $\underset{_}{A}\text{.}$ Hence $j\left(M\right)$ is a $2\text{-sided}$ ideal of ${\underset{_}{A}}_{\text{af}}\text{.}$ $\square$

### Examples of ideals of ${H}_{T}\left(\Omega K\right)\text{:}$

For $\beta \in {\Delta }_{+}^{\text{re}},$ let $K(β)= ∑x∈Waf-x·β<0 sσ[x].$ Since $\ell \left(zx\right)=\ell \left(z\right)+\ell \left(x\right)$ and $x·\beta <0$ $⇒$ $\left(zx\right)·\beta <0,$ the formula in Fact 2 implies that $K\left(\beta \right)$ is an ${\underset{_}{A}}_{\text{af}}\text{-stable}$ submodule of ${H}_{T}\left(\Omega K\right)\text{.}$ Hence it is an $\underset{_}{A}\text{-stable}$ ideal of ${H}_{T}\left(\Omega K\right)\text{.}$ The sum of these things will be the kernel of the map from ${H}_{T}\left(\Omega K\right)$ to $qH\left(G/B\right)\text{.}$

### Future Lectures:

• Compare ${H}_{*}\left(\Omega K\right)$ and $q{H}^{*}\left(G/B\right)\text{.}$
• Compare moduli spaces and intersection of Schubert varieties; the stable Bruhat order.
• Compare $q{H}^{*}\left(G/B\right)$ and $q{H}^{*}\left(G/P\right)\text{.}$
• Compare: ${\sigma }_{G/B}^{{r}_{i}}*$ in $q{H}^{*}\left(G/B\right),$ ${\sigma }_{\left[{r}_{i}{t}_{-h}\right]}·$ in ${H}_{T}\left(\Omega K\right),$ ${\sigma }_{G/B}^{\left[{r}_{i}\right]}·$ in ${H}^{T}\left(G/B\right)\text{.}$

## 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.