Quantum Cohomology of $G/P$

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

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. $$P=B{W}_{J}B$$ where $$W_J={⟨r_j⟩}_{j\in J}$$ is the subgroup of $W$ by $\{r_j:j\in J\}\text{.}$ Set $$\begin{array}{ccc}{W}_P& =& W_J\\ W^P& =& \{u\in W:u<uv\hspace{0.17em}\text{for all}\hspace{0.17em}v\in W_P,v\ne \text{id}\}\text{.}\end{array}$$ Thus $W^P$ is the set of minimum representatives of the coset space $W/W^P\text{.}$ We have $$G/P=\underset{w\in W^P}{⨆}BwP$$ ????? $$Bwp\simeq {ℂ}^{\ell \left(w\right)}$$ ????? is called the Schubert Cell corresponding to $w\text{.}$

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

????? $$X_w^P=\text{closure of}\hspace{0.17em}BwP\hspace{0.17em}\text{in}\hspace{0.17em}G/P$$ ????? is a complex projective variety called the Schubert variety ????? have $$X_w^P=\bigcup _{\underset{v\le w}{v\in W^P}}BvP$$ ????? $w\in W^P,$ let $$\begin{array}{cccc}{i}_w^P:& X_w^P& \hookrightarrow & G/P\end{array}$$ ????? $\left[X_w^P\right]\in H_{2\ell \left(w\right)}(X_w^P,Z)\text{.}$ Set $${\sigma }_w^P={\left(i_w^P\right)}_{*}\left[X_w^P\right]\in H_{2\ell \left(w\right)}(G/P)\text{.}$$

Schubert Basis for $H_{•}(G/P,\mathbb{Z})$ and $H^{•}(G/P,\mathbb{Z})$

Fact: $\{{\sigma }_w^P:w\in W\}$ is a basis for $H_{•}(G/P,\mathbb{Z})\text{.}$

Notation: The dual basis of $H^{•}(G/P,\mathbb{Z})$ dual to $\{{\sigma }_w^P:w\in W\}$ is denoted by $$\{{\sigma }_P^w:w\in W\}\text{.}$$

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

(Here starts Lecture 6)

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

Definition: For $w\in W^P,$ put $${\sigma }_{\left(w\right)}^P={\left(i_w^P\right)}_{*}{\int }_{\left[X_w^P\right]}\in {\text{Hom}}_S(H^T(G/P),S)\text{.}$$ Then $\{{\sigma }_{\left(w\right)}^P:w\in W\}$ is a basis for ${\text{Hom}}_S(H^T(G/P),S)\text{.}$ There is then a unique basis $$\{{\sigma }_P^{\left(w\right)}:w\in W\}$$ of $H^T(G/P)$ (over $S\text{)}$ s.t. $$⟨{\sigma }_P^{\left(v\right)},{\sigma }_{\left(w\right)}^P⟩={\delta }_{v,w}\text{.}$$ Bot $\left\{{\sigma }_{\left(w\right)}^P\right\}$ and $\left\{{\sigma }_P^{\left(w\right)}\right\}$ are called Schubert basis.

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

(1)	$\text{deg}\left({\sigma }_P^{\left(w\right)}\right)=2\ell \left(w\right)$
(2)	Under evaluation at $0\text{:}$ $$\begin{array}{ccc}\mathbb{Z}{\otimes }_S{H}^T(G/P)& ⟶& H^{*}(G/P)\end{array}$$ we have $$\begin{array}{ccc}{\sigma }_P^{\left(w\right)}& ⟼& {\sigma }_P^w\text{.}\end{array}$$
(3)	${(i_w^P:X_w^P\to G/P)}^{*}\left({\sigma }_P^{\left(v\right)}\right)=0$ if $v\nleqq w\text{.}$

?????, we look at

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

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

For $w\in W,$ consider the $T\text{-equivariant}$ map $$\begin{array}{ccccccc}{j}_w^P:& \text{pt}& ⟶& G/P:& \text{pt}& ⟼& wP\end{array}$$ set $${\psi }_w^P={\left(j_w^P\right)}^{*}\in {\text{Hom}}_S(H^T(G/P),S)\text{.}$$ 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: $\{{\psi }_w^P:w\in W^P\}$ is NOT an $S\text{-basis}$ for ${\text{Hom}}_S\left(H^T(G/P)\right)$ because $${\sigma }_{\left(r_i\right)}^P=\frac{1}{{\alpha }_i}{\psi }_{\text{id}}^P-\frac{1}{{\alpha }_i}{\psi }_{r_i}^P\text{.}$$

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 $$\begin{array}{ccccccc}{\pi }_P:& G/B& ⟶& G/P:& gB& ⟼& gP\text{.}\end{array}$$ Then $$\begin{array}{cccc}{\left({\pi }_P\right)}_{*}{\psi }_w^B& =& {\psi }_w^P& w\in W\\ {\left({\pi }_P\right)}^{*}{\psi }_P^{\left(w\right)}& =& {\psi }_B^{\left(w\right)}& w\in W^P\text{.}\end{array}$$

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

Proposition 1: $$A_i·{\sigma }_{\left(w\right)}^P=\{\begin{array}{cc}{\sigma }_{\left(r_{i}w\right)}^P& \text{if}\hspace{0.17em}w<r_{i}w\hspace{0.17em}\text{and}\hspace{0.17em}{r}_{i}w\in W^P\\ 0& \text{otherwise}\end{array}$$

		Proof.
	Let $$\begin{array}{cccc}{i}_w^P:& X_w^P& \hookrightarrow & G/P\text{.}\end{array}$$ Recall that $${\sigma }_{\left(w\right)}^P={\left(i_w^P\right)}_{*}{\int }_{\left[X_w^P\right]}$$ From the fact stated at the end of last lecture, $$A_i·{\sigma }_{\left(w\right)}^P=\mu *{\int }_{{\sigma }_i*\left[X_w^P\right]}\in {\text{Hom}}_S(G/P,S)$$ where $$\begin{array}{ccccccc}\mu :& K_i{\times }^T{X}_w^P& ⟶& G/P:& (k_i,x)& ⟼& k_{i}x\text{.}\end{array}$$ It follows that (?) $$A_i·{\sigma }_{\left(w\right)}^P=\{\begin{array}{cc}{\sigma }_{\left(r_{i}w\right)}^P& \text{if}\hspace{0.17em}w<r_{i}w\hspace{0.17em}\text{and}\hspace{0.17em}{r}_{i}w\in W^P,\\ 0& \text{otherwise.}\end{array}$$ $\square$

?????ave: for $v,w\in W,$ $$v·{\psi }_w^P={\psi }_{vw}^P$$

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

Proposition 2: For $v\in W,$ $w\in W^P,$ $$A_v·{\sigma }_P^{\left(w\right)}=\{\begin{array}{cc}ϵ\left(v\right){\sigma }_P^{\left(vw\right)}& \text{if}\hspace{0.17em}\ell \left(v^{-1}\right)+\ell \left(vw\right)=\ell \left(w\right)\iff vw\in \text{?????},\\ 0& \text{otherwise.}\end{array}$$

		Proof.
	Let's first check that $$A_i·{\sigma }_P^{\left(w\right)}=\{\begin{array}{cc}-{\sigma }_P^{\left(r_{i}w\right)}& \text{if}\hspace{0.17em}1+\ell \left(r_{i}w\right)=\ell \left(w\right)\hspace{0.17em}\text{(ie.}\hspace{0.17em}{r}_{i}w\text{?????},\\ 0& \text{otherwise.}\end{array}$$ From the previous Proposition 1, if $r_{i}w<w$ $\text{(}\Rightarrow$ $r_i\text{?????}$ $$A_i·{\sigma }_{\left(r_{i}w\right)}^P={\sigma }_{\left(w\right)}^P\text{.}$$ But $$\left(A_{i}f\right)\left(z\right)=A_i·f\left(z\right)-r_i·f(A_i·z)z\in H^T(G/P)$$ by definition, so by letting $f={\sigma }_{\left(r_{i}w\right)}^P$ and $z\in {\sigma }_P^{\left(v\right)},$ we get $${\delta }_{w,v}=0-r_i·{\sigma }_{\left(r_{i}w\right)}^P(A_i·{\sigma }_P^{\left(v\right)})$$ or $$\begin{array}{c}(A_i·{\sigma }_P^{\left(v\right)},{\sigma }_{\left(r_{i}w\right)}^P)=-{\delta }_{w,v}\\ \Rightarrow A_i·{\sigma }_P^{\left(w\right)}=-{\sigma }_P^{\left(r_{i}w\right)}\end{array}$$ otherwise follows. $\square$

Remark: Recall that $$\epsilon ={\psi }_{\text{id}}^B={\sigma }_{\left(\text{id}\right)}^B\in {\text{Hom}}_S(H^T(G/B),S)\text{.}$$ We can identify $$\begin{array}{cc}\underset{\_}{A}\simeq {\text{Hom}}_S(H^T(G/B),S)& (*)\end{array}$$ by $$\begin{array}{cccccc}a& ⟼& f_a:& f_a\left(z\right)& =& \epsilon (a·z)\text{.}\end{array}$$ Then this is an identification of $S\text{-modules,}$ and from Proposition 2, $$f_{A_W}=ϵ\left(w\right){\sigma }_{\left(w^{-1}\right)}^B$$ ie. $$\begin{array}{ccc}{A}_w& ⟼& ϵ\left(w\right){\sigma }_{\left(w^{-1}\right)}^B\end{array}$$ Thus by Proposition 1, we see that under the identification $(*),$ the (left) $\underset{\_}{A}\text{-action}$ on ${\text{Hom}}_S(H^T(G/B),S)$ becomes the (left) action $\underset{\_}{A}$ on $\underset{\_}{A}$ by $$a·b=b(*a)$$ where, recall from lecture 2, that $$\begin{array}{ccc}*S& =& S,\\ *w& =& w^{-1},\\ *A_w& =& ϵ\left(w\right)A_{w^{-1}}\text{.}\end{array}$$ (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 $$\begin{array}{ccc}\epsilon & =& {\psi }_{\text{id}}^B\in {\text{Hom}}_S(H^T(G/B),S)\\ & =& {\sigma }_B^{\left(\text{id}\right)}\text{.}\end{array}$$ So $$\epsilon \left({\sigma }_B^{\left(w\right)}\right)={\delta }_{w,\text{id}}w\in W\text{.}$$

Proposition:

(1)	Every characteristic operator $a\in \underset{\_}{\hat{A}}$ can be uniquely written as $$a=\sum _{w\in W}{s}_w{A}_w{s}_w\in S\text{.}$$ In fact, $$s_w=\epsilon (a·\left(ϵ\left(w\right){\sigma }_B^{\left(w^{-1}\right)}\right))$$ (Recall $ϵ\left(w\right)={(-1)}^{\ell \left(w\right)}\text{).}$
(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 $a\in \hat{A},$ write $$a\prime =a-\sum _{w\in W}\epsilon (a·\left(ϵ\left(w\right){\sigma }_B^{\left(w^{-1}\right)}\right))A_w$$ Then $a\prime \in \underset{\_}{\hat{A}}\text{.}$ Thus to show $a\prime =0$ it is enough to show that $$\epsilon (a\prime ·z)=0$$ for any $z\in H^T(G/B)\text{.}$ (See Lecture 5). Since both $a\prime$ and $\epsilon$ are $S\text{-linear,}$ it is enough to show that $$\epsilon (a\prime ·{\sigma }_B^{\left(v\right)})=0$$ for all $v\in W\text{.}$ Now $$\begin{array}{ccc}a\prime -{\sigma }_B^{\left(v\right)}& =& a·{\sigma }_B^{\left(v\right)}-\sum _{w\in W}\epsilon (a·ϵ\left(w\right){\sigma }_B^{\left(w^{-1}\right)})A_w·{\sigma }_B^{\left(v\right)}\\ & =& a·{\sigma }_B^{\left(v\right)}-\sum _{\underset{\ell \left(w^{-1}\right)+\ell \left(wv\right)=\ell \left(v\right)}{w\in W}}\epsilon (a·ϵ\left(w\right){\sigma }_B^{\left(w^{-1}\right)})ϵ\left(w\right){\sigma }_B^{\left(wv\right)}\\ & =& a·{\sigma }_B^{\left(v\right)}-\sum _{\underset{\ell \left(w^{-1}\right)+\ell \left(wv\right)=\ell \left(v\right)}{w\in W}}\epsilon (a·{\sigma }_B^{\left(w^{-1}\right)}){\sigma }_B^{\left(v\right)}\end{array}$$ $*$ $$\begin{array}{ccc}\Delta {\sigma }_B^{\left(v\right)}& =& \sum _{\underset{\underset{\ell \left(v\right)=\ell \left(u\right)+\ell \left(w\right)}{uw=v}}{u,w\in W}}{\sigma }_B^{\left(u\right)}\otimes {\sigma }_B^{\left(w\right)}\\ a& =& ((\epsilon \circ a)\otimes \text{id})\circ {\Delta }_X\end{array}$$ ?????e Corollary 3 in Lecture 5). $$\Rightarrow a\prime ·{\sigma }_B^{\left(v\right)}& =& 0\\ \Rightarrow a\prime & \equiv & 0\text{.}$$ 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 $$a=\sum _{w\in W}{s}_w{A}_w\text{.}$$ $\square$

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

Remark: We can think of $\underset{\_}{A}$ as ${\text{Hom}}_S(H^T(K/T),S),$ or the ????? of $H^T(K/T)$ via the pairing: $$\begin{array}{ccc}(a,z)& \stackrel{\text{def}}{=}& \epsilon (a·z)\text{.}\end{array}$$ Let's check then that $\underset{\_}{A}$ action on ${\text{Hom}}_S(H^T(K/T),S)$ becomes the $\underset{\_}{A}\text{-action}$ on $\underset{\_}{A}$ by left multiplications: For $a\in \underset{\_}{A},$ use $f_a\in {\text{Hom}}_S(H^T(K/T),S)$ to denote the element given by $$f_a\left(z\right)=(a,z)=\epsilon (a·z)\text{.}$$ For $i\in I\text{.}$ we want to check $$\begin{array}{ccc}{A}_i·f_a& =& f_{A_{i}a}\text{.}\end{array}$$

The Hopf Algebroid Structure on $H^T(K/T)$

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

First, recall that we have ring homomorphisms $$\begin{array}{cccc}{\pi }_L:& S& ⟶& H^T(K/T)\\ {\pi }_R:& S& ⟶& H^T(K/T)\text{.}\end{array}$$ This gives two $S\text{-module}$ structures on $H^T(K/T)\text{.}$ 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 $H^T(K/T)$ in Lecture 4.

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

Theorem: (Recall notation from Lecture 5):

1)	For $\lambda \in h_{\mathbb{Z}}^{*},$ $${\pi }_R\left(\lambda \right)={\pi }_L\left(\lambda \right)+\sum _{i\in I}⟨\lambda ,{\alpha }_i^{\vee }⟩{\sigma }_B^{\left(r_i\right)}$$
2)	$\epsilon \left({\sigma }_B^{\left(w\right)}\right)={\delta }_{w,\text{id}}$
3)	$\Delta {\sigma }_B^{\left(w\right)}=\sum _{\underset{w=uv\hspace{0.17em}\text{[red]}}{u,v\in W}}{\sigma }_B^{\left(u\right)}\otimes {\sigma }_B^{\left(v\right)}$ $\text{(}w=uv$ [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 $\sigma \in H^T\left(X\right)$ $${\Delta }_X\left(\sigma \right)=\sum _{w\in W}ϵ\left(w\right){\sigma }_B^{\left(w^{-1}\right)}\otimes (A_w·\sigma )\in H^T(K/T){\otimes }_S{H}^T\left(X\right)$$

		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 $$\begin{array}{cccc}{\Delta }_M:& M& ⟶& A^{*}\otimes M:\\ & {\Delta }_M\left(m\right)& =& \sum _i{\xi }_i\otimes a_i·m\end{array}$$ ????? in our example, we are identifying $H^T(K/T)$ with ${\underset{\_}{A}}^{*}$ ????? the pairing $$(a,z)=\epsilon (a·z)a\in \underset{\_}{A},\hspace{0.17em}z\in H^T(K/T)\text{.}$$ ????? this pairing; we have $\{A_w:w\in W\}$ as a basis for $\underset{\_}{A}$ ????? dual basis in $H^T(K/T)$ is $\{ϵ\left(w\right){\sigma }_B^{\left(w^{-1}\right)}:w\in W\}$ (see page 6-8). Thus for any $\sigma \in H^T\left(X\right)$ $${\Delta }_X\left(\sigma \right)=\sum _{w\in W}ϵ\left(w\right){\sigma }_B^{\left(w^{-1}\right)}\otimes (A_w·\sigma )$$ Peterson gave the following proof in class: Since $\{ϵ\left(w\right){\sigma }_B^{\left(w^{-1}\right)}:w\in W\}$ is a basis for $H^T(K/T),$ we know $${\Delta }_X\left(\sigma \right)=\sum _{w\in W}ϵ\left(w\right){\sigma }_B^{\left(w^{-1}\right)}\otimes {\varphi }_w$$ 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 $$\begin{array}{ccc}{A}_v·\sigma & =& (\epsilon \otimes \text{id}){\Delta }_X(A_v·\sigma )\\ & =& (\epsilon A_v\otimes \text{id}){\Delta }_X\left(\sigma \right)\text{(see Lecture 5, Corollary 1)}\\ & =& \sum _{w\in W}\epsilon (A_v·ϵ\left(w\right){\sigma }_B^{\left(w^{-1}\right)})\circ {\varphi }_w\\ & =& \epsilon (A_v·ϵ\left(v\right){\sigma }_B^{\left(w^{-1}\right)}){\varphi }_v\\ & =& {\varphi }_v\text{.}\end{array}$$ 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 $$\begin{array}{cccc}{\Delta }_X:& H^T\left(X\right)& ⟶& H^T(K/T){\otimes }_S{H}^T\left(X\right)\end{array}$$ by $$a·\sigma =(a,{\sigma }^{\left(1\right)}){\sigma }^{\left(2\right)}\text{if}\hspace{0.17em}{\Delta }_X\sigma ={\sigma }^{\left(1\right)}\otimes {\sigma }^{\left(2\right)}$$ and $(a,z)=\epsilon (a·z)$ is the pairing between $H^T(K/T)$ 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 $$\Delta {\sigma }_B^{\left(w\right)}=\sum _{u_1\in W}ϵ\left(u_1\right){\sigma }_B^{\left(u_1^{-1}\right)}\otimes A_{u_1}·{\sigma }_B^{\left(w\right)}\text{.}$$ But $$A_{u_1}·{\sigma }_B^{\left(2\right)}=\{\begin{array}{cc}ϵ\left(u_1\right){\sigma }_B^{\left(u_{1}w\right)}& \text{if}\hspace{0.17em}\ell \left(u_1^{-1}\right)+\ell \left(u_{1}w\right)=\ell \left(w\right),\\ 0& \text{otherwise.}\end{array}$$ ????? $$\begin{array}{ccc}\Delta {\sigma }_B^{\left(w\right)}& =& \sum _{\underset{w=u_1^{-1}·\left(u_{1}w\right)\hspace{0.17em}\text{[red]}}{u_1\in W}}ϵ\left(u_1\right){\sigma }_B^{\left(u_1^{-1}\right)}\otimes ϵ\left(u_1\right){\sigma }_B^{\left(u_{1}w\right)}\\ & =& \sum _{\underset{w=uv\hspace{0.17em}\text{[red]}}{\underset{v=u_{1}w\in W}{u=u_1^{-1}\in w}}}{\sigma }_B^{\left(u\right)}\otimes {\sigma }_B^{\left(v\right)}\text{.}\end{array}$$ 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 $\sigma \in H^T(K/T),$ $$\sigma =\sum _{w\in W}{\pi }_R\left(\epsilon (A_w·\sigma )\right)ϵ\left(w\right){\sigma }_B^{\left(w^{-1}\right)}\text{.}$$ Proof. Write $$\sigma =\sum _{w\in W}{\pi }_R\left(s_w\right)ϵ\left(w\right){\sigma }_B^{\left(w^{-1}\right)}$$ for some $s_w\in S$ for each $w\in W\text{.}$ Using $$\epsilon \circ {\pi }_R={\text{id}}_S$$ and $$(A_v,ϵ\left(w\right){\sigma }_B^{\left(w^{-1}\right)})(=\epsilon (A_v·\epsilon \left(w\right){\sigma }_B^{\left(w^{-1}\right)}))={\delta }_{v,w}$$ we get $$\begin{array}{ccc}\underset{\underset{(A_v,\sigma )}{\left|\hspace{0.17em}\right|}}{\epsilon (A_v·\sigma )}& =& \sum _{w\in W}\epsilon {\pi }_R\left(s_w\right)(A_v·ϵ\left(w\right){\sigma }_B^{\left(w^{-1}\right)})=\epsilon {\pi }_R\left(s_v\right)\\ & =& s_v\\ \Rightarrow \sigma & =& \sum _{w\in W}{\pi }_R\left(\epsilon (A_v·\sigma )\right)ϵ\left(w\right){\sigma }_B^{\left(w^{-1}\right)}\text{.}\end{array}$$ This proves the Lemma. Remark: In proving the Lemma, we used the fact that $S\text{-valued}$ ????? the pairing $\left(\hspace{0.17em}\right)$ between $\underset{\_}{A}$ and $H^T(K/T)$ defined by $$\begin{array}{ccc}(a,\sigma )& =& \epsilon (a·\sigma )\end{array}$$ satisfies $$({\pi }_R\left(s\right)a,\sigma )=\epsilon \left({\pi }_R\left(s\right)\right)(a,\sigma )=s(a,\sigma )$$ and $$\epsilon \left({\pi }_R\left(s\right)\right)=s\forall s\in S\text{.}$$ It says that $\epsilon :H^T(K/T)\to S$ is not only an $S\text{-map}$ for the first $S\text{-module}$ structure on $H^T(K/T),$ (defined by ${\pi }_L\text{)}$ but also for the 2nd $S\text{-module}$ structure $H^T(K/T)$ defined by ${\pi }_R\text{.}$ Is this really true? Recall that ${\pi }_R:S\to H^T(K/T)$ is the pullback of the map $$\begin{array}{ccc}(E_u\times K)/(T\times T)& ⟶& E/T\\ [e,k]& ⟼& e·k\text{.}\end{array}$$ It is not clear why $\epsilon :H^T(K/T)\to S$ is ${\pi }_R\left(S\right)\text{-linear.}$ $\square$ Now we prove 1): By Lemma $${\pi }_L\left(\lambda \right)=\sum _{w\in W}{\pi }_R\left(\epsilon (A_w·{\pi }_L\left(\lambda \right))\right)ϵ\left(w\right){\sigma }_B^{\left(w^{-1}\right)}$$ But $$A_w·{\pi }_L\left(\lambda \right)={\pi }_L(A_w·\lambda )=\{\begin{array}{cc}{\pi }_L\left(\lambda \right)& w=\text{id},\\ ⟨\lambda ,{\alpha }_i^{\vee }⟩& w=r_i,\\ 0& \text{otherwise}\end{array}$$ $$\begin{array}{ccc}\Rightarrow {\pi }_L\left(\lambda \right)& =& {\pi }_R\left(\epsilon \left({\pi }_L\left(\lambda \right)\right)\right)+\sum _{i\in I}{\pi }_R\left(\epsilon \left(⟨\lambda ,{\alpha }_i^{\vee }⟩\right)\right)(-1){\sigma }_B^{\left(r_i\right)}\\ & =& {\pi }_R\left(\lambda \right)-\sum _{i\in I}⟨\lambda ,{\alpha }_i^{\vee }⟩{\sigma }_B^{\left(r_i\right)}\\ \Rightarrow {\pi }_R\left(\lambda \right)& =& {\pi }_L\left(\lambda \right)+\sum _{i\in I}⟨\lambda ,{\alpha }_i^{\vee }⟩{\sigma }_B^{\left(r_i\right)}\text{.}\end{array}$$ 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. $$c\left({\sigma }_B^{\left(w\right)}\right)=ϵ\left(w\right){\sigma }_B^{\left(w^{-1}\right)}\text{.}$$ The following is the proof given by Peterson. It is kind of strange. W>e first prove that $$c\left({\sigma }_B^{\left(w\right)}\right)=\pm {\sigma }_B^{\left(w^{-1}\right)}$$ We'll determine the sign later. For $w\in W,$ let $$E_w^{\left(2\right)}=\{(e,ek):e\in E_u,k\in K_w\}\text{.}$$ Then $$H^{*}(E_u^{\left(2\right)}/T\times T)\simeq H^T\left(X_w^B\right)\text{.}$$ (Why? This is saying that we do not distinguish $X_w^B$ and its Bott-Samelson resolution?) Recall that $$\begin{array}{ccccccc}t:& E^{\left(2\right)}& ⟶& E^{\left(2\right)}:& (e_1,e_2)& ⟼& (e_2,e_1)\text{.}\end{array}$$ So $$\begin{array}{ccc}t\left(E_2^{\left(2\right)}\right)& =& E_{w^{-1}}^{\left(2\right)}\text{.}\end{array}$$ But $$R_w=\{\sigma \in H^T(E^{\left(2\right)}/T\times T):\hspace{0.17em}\text{deg}\hspace{0.17em}\sigma =2\ell \left(w\right)\text{and}\sigma {|}_{E_v^{\left(2\right)}/T\times T}=0\text{for}v\in W\text{s.t.}v\ngeqq w\}\text{.}$$ We know $$\begin{array}{ccc}{R}_w& =& \mathbb{Z}{\sigma }_B^{\left(w\right)}\\ c\left(R_w\right)& =& R_{w^{-1}}\\ \Rightarrow c\left({\sigma }_B^{\left(w\right)}\right)& =& \pm {\sigma }_B^{\left(w^{-1}\right)}\text{.}\end{array}$$ 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 $$(c\otimes c)·T\circ \Delta =\Delta ·c$$ where $$T(\sigma \otimes \sigma \prime )=\sigma \prime \otimes \sigma$$ we get, from $$\Delta \left({\sigma }_B^{\left(w\right)}\right)=\sum _{w=uv\hspace{0.17em}\text{[red]}}{\sigma }_B^{\left(u\right)}\otimes {\sigma }_B^{\left(v\right)}$$ that $$\begin{array}{ccc}\Delta \left(c{\sigma }_B^{\left(w\right)}\right)& =& \sum _{vu\text{?????}\text{[red]}}c\left({\sigma }_B^{\left(v\right)}\right)\otimes c\left({\sigma }_B^{\left(u\right)}\right)\\ & =& c\left({\sigma }_B^{\left(w\right)}\right)\otimes 1+1\otimes c\left({\sigma }_B^{\left(w\right)}\right)+\sum _{\underset{\underset{v\ne 1}{u\ne 1}}{w=uv\text{[red]}}}ϵ\left(u\right)ϵ\left(v\right){\sigma }_B^{\left(v^{-1}\right)}\otimes {\sigma }_B^{\left(u^{-1}\right)}\text{.}\end{array}$$ But $\Delta \left(ϵ\left(w\right){\sigma }_B^{\left(w\right)}\right)=ϵ\left(w\right){\sigma }_B^{\left(w^{-1}\right)}\otimes 1+1\otimes ϵ\left(w\right){\sigma }_B^{\left(w^{-1}\right)}+\text{same sum}\ne 0$ $\Rightarrow$ must have $$\Delta \left({\sigma }_B^{\left(w\right)}\right)=ϵ\left(w\right){\sigma }_B^{\left(w^{-1}\right)}\text{.}$$ This proves 4). This completes the proof of the theorem. $\square$

?????rable $\underset{\_}{A}\text{-modules}$ $\text{(}\iff$ actions of $𝒰=\text{Spec}\hspace{0.17em}{H}^T(K/T)\text{)}$

Definition: Let $X$ be an affine scheme over $\underset{\_}{h}=\text{Spec}\hspace{0.17em}S$ 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 $m:𝒪\left(X\right){\otimes }_S𝒪\left(X\right)\to 𝒪\left(X\right)$

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 $\underset{\_}{h}=\text{Spec}\hspace{0.17em}S$ with structure homomorphism ${\pi }_L$ (?) Is this an example? Maybe not, because $H^T(K/T){\otimes }_S{H}^T(K/T)$ we use ${\pi }_R$ to define the $S\text{-module}$ structure on the first copy of $H^T(K/T)\text{.}$ (OK. Because in the multiplication of $H^T(K/T){\otimes }_S{H}^T(K/T),$ even the $S\text{-structure}$ on the first copy is defined by ${\pi }_L\text{).}$ $$\begin{array}{c}\text{Integrable}\hspace{0.17em}\underset{\_}{A}\text{-module structure on}\hspace{0.17em}𝒪\left(X\right)\\ \updownarrow \\ \text{action}\hspace{0.17em}\varphi :𝒰{\times }_{\underset{\_}{h}}X⟶X\text{.}\end{array}$$ One way:

If $\varphi :𝒰{\times }_{\underset{\_}{h}}X\to X$ is an action, have $$\begin{array}{cccc}{\varphi }^{*}:& 𝒪\left(X\right)& ⟶& H^T(K/T)\otimes 𝒪\left(X\right)\end{array}$$ Then for $a\in \underset{\_}{A},$ define $p\in P$ $$a·p=m·({\pi }_X\left(\epsilon (a·p_{\left(1\right)})\right)\otimes p_{\left(2\right)})\text{if}\hspace{0.17em}{\varphi }^{*}p=p_{\left(1\right)}\otimes p_{\left(2\right)}\text{.}$$ The other way, given $\underset{\_}{A}\text{-action}$ on $𝒪\left(X\right),$ define $${\varphi }^{*}\left(p\right)=\sum _{w\in W}c\left({\sigma }_{G/B}^{\left(w\right)}\right)\otimes (A_w·p)$$ This is the map giving the action $$\begin{array}{cccc}\varphi :& 𝒰{\times }_{\underset{\_}{h}}X& ⟶& X\text{.}\end{array}$$

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

Notation: The action of $\underset{\_}{A}$ on $H^T(K/T)$ 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 $H^T(K/T)$

Define a second action of $\underset{\_}{A}$ on $H^T(K/T)$ by $$a_R·=c·(a_L·)\circ c$$

Properties:

1)	$a_L\circ b_R=b_R\circ a_L$ $\forall a,b\in \underset{\_}{A}$
2)	$\Delta \circ a_L=(a_L\otimes \text{id})\circ \Delta$ $\Delta \circ b_R=(\text{id}\otimes b_R)\circ \Delta$
3)	for $s\in S,$ $a\in \underset{\_}{A}$ and $z\in H^T(K/T)$ $$\begin{array}{ccc}{s}_L·z& =& {\pi }_L\left(s\right)z\\ s_R·z& =& {\pi }_R\left(s\right)z\end{array}$$ and $$\begin{array}{ccc}{a}_L·{\pi }_L\left(s\right)& =& {\pi }_L(a·s)\\ a_R·{\pi }_R\left(s\right)& =& {\pi }_R(a·s)\\ a·\epsilon \left(z\right)& =& \epsilon ({a_{\left(1\right)}}_L{a_{\left(2\right)}}_R·z)\text{if}\hspace{0.17em}\Delta a=a_{\left(1\right)}\otimes a_{\left(2\right)}\\ ⟺w·\epsilon \left(z\right)& =& \epsilon (w_L{w}_R·z)\end{array}$$

Thus, in the basis $\{{\sigma }_B^{\left(w\right)}:w\in W\}\text{.}$

• ${A_v}_R·{\sigma }_B^{\left(w\right)}=\{\begin{array}{cc}{\sigma }_B^{\left(w{v}^{-1}\right)}& \text{if}\hspace{0.17em}\ell \left(w{v}^{-1}\right)+\ell \left(v\right)=\ell \left(w\right),\\ 0& \text{otherwise.}\end{array}$
• Any $z\in H^T(K/T)$ can be written as $$z=\sum _{w\in W}\left({\pi }_L\left(\epsilon ({A_w}_R·z)\right)\right){\sigma }_B^{\left(w\right)}\text{.}$$
• Any $a\in \underset{\_}{\hat{A}}$ can be written as $$a=\sum _{w\in W}\epsilon (a_R·{\sigma }_{G/B}^{\left(w\right)})A_w\text{.}$$
• $\forall w\in W,$ $$\begin{array}{ccc}\epsilon \circ {A_w}_R& =& {\sigma }_{\left(w\right)}^B\\ \epsilon \circ w_R& =& {\psi }_w^B\end{array}$$ (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: $${A_v}_R·{\sigma }_B^{\left(w\right)}=\{\begin{array}{cc}{\sigma }^{\left(w{v}^{-1}\right)}& \text{if}\hspace{0.17em}\ell \left(w{v}^{-1}\right)+\ell \left(v\right)=\ell \left(w\right),\\ 0& \text{otherwise.}\end{array}$$ For any $a\in \underset{\_}{\hat{A}}$ $$a=\sum _{w\in W}\epsilon (a_R·{\sigma }_B^{\left(w\right)})A_w$$ ????? $z\in H^T(K/T)$ $$z=\sum _{w\in W}{\pi }_L\left(\epsilon ({A_w}_R·z)\right){\sigma }_B^{\left(w\right)}$$ ????? $$\begin{array}{ccc}\epsilon \circ {A_w}_R& =& {\sigma }_{\left(w\right)}^B\\ \epsilon \circ w_R& =& {\psi }_w^B\end{array}$$ Given $w\in W,$ $\exists$ $d_{u,w}\in S\text{?????}$ of degree $\ell \left(u\right)$ for each $u\le w$ s.t. $$w=\sum _{u\le w}{d}_{u,w}{A}_u$$ moreover $$d_{w,w}=\prod _{\underset{w^{-1}\alpha <0}{\alpha \in {\Delta }_{+}^{\text{red}}}}(-\alpha )=ϵ\left(w\right)\prod _{\underset{w^{-1}\alpha <0}{\alpha \in {\Delta }_{+}^{\text{red}}}}\alpha$$

		Proof.
	Induction on $\ell \left(w\right)\text{:}$ $$\begin{array}{ccc}\ell \left(w\right)=0& w=\text{id}& \text{id}=\text{id.}\\ \ell \left(w\right)=1& w=r_i& r_i=1-{\alpha }_i{A}_i& \text{OK.}\end{array}$$ Assume $w=r_i{w}_1>w_1\text{.}$ Assume $$w_1=\sum _{u\le w_1}{d}_{u,w_1}{A}_u{d}_{u,w_1}\in S^{\ell \left(u\right)}\left(h^{*}\right)$$ Then $$\begin{array}{ccc}w& =& r_i{w}_1=(1-{\alpha }_i{A}_i)\sum _{u\le w_1}{d}_{u,w_1}{A}_u\\ & =& \sum _{u\le w_1}{d}_{u,w_1}{A}_u-\sum _{u\in w_1}{\alpha }_i\left(A_i{d}_{u,w_1}\right)A_u\end{array}$$ Since $$A_i{d}_{u,w_1}=(r_i·d_{u,w_1})A_i+A_i·d_{u,w_1}$$ $$\begin{array}{ccc}\Rightarrow w& =& \sum _{u\le w_1}{d}_{u,w_1}{A}_u-\sum _{u\le w_1}{\alpha }_i(r_i·r_{u,w_1})A_i{A}_u+{\alpha }_i(A_i·d_{u,w_1})\\ & =& \sum _{u\le w_1}(d_{u,w_1}-{\alpha }_i{A}_i·d_{u,w_1})A_u-\sum _{u\le w_1}{\alpha }_i(r_i·d_{u,w_1})A_i{A}_u\\ & =& \sum _{u\le w_1}(r_i·d_{u,w_1})A_u-\sum _{\underset{r_{i}u>u}{u\le w_1}}{\alpha }_i(r_i·d_{u,w_1})A_{r_{i}u}\end{array}$$ $$\begin{array}{ccc}{d}_{u,w}& =& r_i·d_{u,w_1}\text{if}\hspace{0.17em}u\le w_1\\ d_{r_i·u,w}& =& -{\alpha }_i(r_i·d_{u,w_1})\text{if}\hspace{0.17em}u\le w_1,\hspace{0.17em}{r}_{i}u>u\end{array}$$ shows that $d_{u,w}\in S^{\ell \left(u\right)}\left(h_{\mathbb{Z}}^{*}\right)$ for any $u\le w\text{.}$ Moreover, $$d_{r_i{w}_1,w}=-{\alpha }_i(r_i·d_{w_1,w_1})\text{.}$$ Assume $$d_{w_1,w_1}=ϵ\left(w_1\right)\prod _{\underset{w_1^{-1}\alpha <0}{\alpha \in {\Delta }_{+}^{\text{red}}}}\alpha$$ Then $$\begin{array}{ccc}{d}_{w,w}& =& -{\alpha }_1(r_i·d_{w_1,w_1})\\ & =& ϵ\left(w\right)\prod _{\underset{w_{\text{?????}}^{-1}\alpha <0}{\alpha \in {\Delta }_{+}^{\text{red}}}}\alpha \end{array}$$ $\square$

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)	$\bigcap _{w\in W}\text{ker}\hspace{0.17em}{\psi }_w^B=0\text{.}$
3)	$H^T(K/T)$ is reduced, ie. the only nilpotent elemtn
4)	$H^T(G/P)\simeq {\left(H^T(K/T)\right)}^{{\left(W_P\right)}_R}$ is also reduced.

		Proof.
	1) follows from $$\begin{array}{ccc}\epsilon \circ {A_w}_R& =& {\sigma }_{\left(w\right)}^B\\ \epsilon \circ w_R& =& {\psi }_{\left(w\right)}^B\end{array}$$ 2) If $z\in \bigcap _{w\in W}\text{ker}\hspace{0.17em}{\psi }_w^B$ 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 $\Rightarrow {\sigma }_{\left(w\right)}^B\left(z\right)=0\text{.}$ But $\left\{{\sigma }_{\left(w\right)}^B\right\}$ is a basis for ${\text{Hom}}_S(H^T(K/T),S)$ $\Rightarrow z=0\text{.}$ If $z\in H^T(K/T)$ is s.t. $z^m=0$ for some $m\ge 1$ then for each $w\in W$ $$\epsilon (w_R·z^m)=0$$ But $$w_R·z^m={(w_R·z)}^m$$ $\Rightarrow$ $$\begin{array}{ccc}\epsilon \left({(w_R·z)}^m\right)& =& 0\\ {\left(\epsilon (w_n·z)\right)}^m& =& 0\\ \Rightarrow \epsilon (w_R·z)& =& 0\end{array}$$ ie. $z\in \text{ker}\hspace{0.17em}{\psi }_w^B=0$ $\forall w$ $$\Rightarrow z=0\text{.}$$ Clear. $\square$

Proposition: The action $a_R·$ of $\underset{\_}{A}$ on $H^T(K/T)$ descends to an action on $H^{•}(K/T)$ via the map $$\mathbb{Z}{\otimes }_S{H}^T(K/T)⟶H^{•}(K/T)$$ where the $S\text{-module}$ structure on $H^T(K/T)$ 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 $H^{•}(K/T)$ is by the BGG-operators.

?????ne Constants" for the multiplication on $H^T(K/T)$

For $u,v,w\in W,$ define $a_w^{u,v}\in S$ by $$\Delta A_w=\sum _{u,v\in W}{a}_w^{u,v}{A}_u\otimes A_v$$ $\text{(}\Delta$ cocommutative $\Rightarrow a_w^{uv}=a_w^{vu}\text{)}$

Proposition: $${\sigma }_B^{\left(u\right)}{\sigma }_B^{\left(v\right)}=\sum _{w\in W}{\pi }_L\left(a_w^{u,v}\right){\sigma }_B^{\left(w\right)}$$

		Proof.
	We know that $${\sigma }_B^{\left(u\right)}{\sigma }_B^{\left(v\right)}=\sum _{w\in W}{\pi }_L\left(\epsilon ({A_w}_R·{\sigma }_B^{\left(u\right)}{\sigma }_B^{\left(v\right)})\right){\sigma }_B^{\left(w\right)}$$ Then $${A_w}_R·\left({\sigma }_B^{\left(u\right)}{\sigma }_B^{\left(v\right)}\right)=\sum _{u\prime ,v\prime \in W}{\pi }_R\left(a_w^{u\prime ,v\prime }\right)({A_{u\prime }}_R·{\sigma }_B^{\left(u\right)})({A_{v\prime }}_R·{\sigma }_B^{\left(v\right)})$$ and $$\begin{array}{ccc}\epsilon ({A_w}_R·\left({\sigma }_B^{\left(u\right)}{\sigma }_B^{\left(v\right)}\right))& =& \sum _{u\prime ,v\prime \in W}{a}_w^{u\prime ,v\prime }\epsilon ({A_{u\prime }}_R·{\sigma }_B^{\left(u\right)})\epsilon ({A_{v\prime }}_R·{\sigma }_B^{\left(v\right)})\\ & =& \sum _{u\prime ,v\prime \in W}{a}_w^{u\prime ,v\prime }\epsilon ({\sigma }_{\left(u\prime \right)}^B,{\sigma }_B^{\left(u\right)})({\sigma }_{\left(v\prime \right)}^B,{\sigma }_B^{\left(v\right)})\\ & =& \sum _{u\prime ,v\prime \in W}{a}_w^{u\prime ,v\prime }\epsilon {\delta }_{u\prime ,u}{\delta }_{v\prime ,v}\\ & =& a_w^{u,v}\end{array}$$ $$\Rightarrow {\sigma }_B^{\left(u\right)}{\sigma }_B^{\left(v\right)}=\sum _{w\in W}{\pi }_L\left(a_w^{u,v}\right){\sigma }_B^{\left(w\right)}\text{.}$$ $\square$

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: $$\begin{array}{ccc}\Delta A_i& =& A_i\otimes 1+r_i\otimes A_i\\ & =& A_i\otimes 1+(1-{\alpha }_i{A}_i)\otimes A_i\\ & =& 1\otimes A_i+A_i\otimes 1-A_i\otimes {\alpha }_i{A}_i\\ & =& 1\otimes A_i+A_i\otimes 1-A_i\otimes {\alpha }_i{A}_i\\ \Delta A_i{A}_j& =& (1\otimes A_i+A_i\otimes 1-A_i\otimes {\alpha }_i{A}_i)(1\otimes A_j+A_j\otimes 1-A_j\otimes {\alpha }_j{A}_j)\\ & =& 1\otimes A_i{A}_j+A_j\otimes A_i+A_i\otimes A_j+A_i{A}_j\otimes 1\\ & & -A_i\otimes {\alpha }_i{A}_i{A}_j-A_i{A}_j\otimes {\alpha }_i{A}_i-A_i{A}_j\otimes {\alpha }_i{A}_i{\alpha }_j{A}_j\\ & & -A_j\otimes A_i{\alpha }_j{A}_j-A_i{A}_j\otimes {\alpha }_j{A}_j+A_i{A}_j\otimes {\alpha }_i{A}_i{\alpha }_j{A}_j\end{array}$$ so clear from induction on $\ell \left(w\right)\text{.}$ $\square$

Proposition: $a_w^{u,v}$ is a homogeneous polynomial of degree $$\ell \left(u\right)+\ell \left(v\right)-\ell \left(w\right)\text{in}S\text{.}$$

		Proof.
	$$\begin{array}{ccc}\text{deg}\hspace{0.17em}{\sigma }_B^{\left(u\right)}{\sigma }_B^{\left(v\right)}& =& \text{deg}\left({\pi }_L{a}_w^{u,v}{\sigma }_B^{\left(w\right)}\right)\\ 2\ell \left(u\right)+2\ell \left(v\right)& =& 2\left(\text{deg}\hspace{0.17em}{a}_w^{u,v}\hspace{0.17em}\text{in}\hspace{0.17em}S\right)+2\ell \left(w\right)\text{.}\\ \Rightarrow \text{deg}\hspace{0.17em}\left(a_w^{u,v}\hspace{0.17em}\text{in}\hspace{0.17em}S\right)& =& \ell \left(u\right)+\ell \left(v\right)-\ell \left(w\right)\text{.}\end{array}$$ $\square$

Proposition: For $w,v\in W,$ $v\le w$ $$d_{v,w}=a_w^{v,w}$$ where, recall $d_{v,w}\in S$ are defined by $$w=\sum _{v\le w}{d}_{v,w}{A}_v$$ ?????e $$w=\sum _{v\le w}{a}_w^{v,w}{A}_v\text{.}$$

		Proof.
	Write $$w\otimes w=\sum _{u_1,u_2\le w}{S}_w^{u_1,u_2}{A}_{u_1}\otimes A_{u_2}\text{.}$$ $$\begin{array}{ccc}\Rightarrow w\epsilon (w_R·{\sigma }_B^{\left(w\right)})& =& \sum _{u_1,u_2\in w}{S}_w^{u_1,u_2}{a}_{u_1}\epsilon ({A_{u_2}}_R·{\sigma }_B^{\left(w\right)})\\ & =& \sum _{u_1,u_2\le w}{S}_w^{u_1,u_2}{A}_{u_1}{\delta }_{u_2,w}\\ & =& \sum _{u_1\le w}{S}_w^{u_1,w}{A}_{u_1}\text{.}\end{array}$$ But $$\epsilon (w_R·{\sigma }_B^{\left(w\right)})=d_{w,w}\text{.}$$ $$\begin{array}{ccc}\Rightarrow d_{w,w}w& =& \sum _{u_1\le w}{S}_w^{u_1,w}{A}_{u_1}\\ \Rightarrow S_w^{u_{1}w}& =& d_{w,w}{d}_{u_1,w}\end{array}$$ On the other hand, $$\begin{array}{ccc}w& =& \sum d_{u,w}{A}_u\text{.}\end{array}$$ $$\begin{array}{ccc}\Rightarrow w\otimes w& =& \sum _{u_1,u_2}\left(\sum _v{d}_{v,w}$ a_v^{u_1,u_2}$\right)A_{u_1}\otimes A_{u_2}\\ \Rightarrow S_w^{u_1{u}_2}& =& \sum _v{d}_{v,w}{a}_v^{u_1,u_2}\\ \Rightarrow S_w^{u_1,w}& =& \sum _v{d}_{v,w}{a}_v^{u_1,w}=d_{w,w}{a}_w^{u_1,w}\text{.}\end{array}$$ By $S^{u_{1}w}=d_{w,w}{d}_{u_1,w}$ and $d_{w,w}\ne 0$ get $$d_{u_1,w}=a_w^{u_1,w}$$ $\square$

(Very strange proof).

Proposition: For $w\in W,$ $$\begin{array}{cc}\sum _{w\in uv\hspace{0.17em}\text{[red]}}ϵ\left(u\right){\sigma }_B^{\left(u^{-1}\right)}{\sigma }_B^{\left(v\right)}={\delta }_{w,\text{id}}& \text{(1)}\\ \sum _{w\in uv\hspace{0.17em}\text{[red]}}{\sigma }_B^{\left(u\right)}ϵ\left(v\right){\sigma }_B^{\left(v\right)}={\delta }_{w,\text{id}}& \text{(2)}\end{array}$$

		Proof.
	$$\begin{array}{ccc}\text{(1)}& ⟺& m\circ (c\otimes \text{id})\circ \Delta =\epsilon ,\\ \text{(2)}& ⟺& m\circ (\text{id}\otimes c)\circ \Delta =\epsilon \text{.}\end{array}$$ $\square$

Remark: This will also be true for quantum cohomology.

Remark: Fix $e_0\in E_u\text{.}$ Define $$\begin{array}{ccccccc}i:& K/T& ⟶& E_u/T:& kT& ⟼& e_{0}kT\text{.}\end{array}$$ Then $$\begin{array}{cccc}i\times i:& K/T\times K/T& ⟶& E_u^{\left(2\right)}/T\times T\text{.}\end{array}$$ Consequently, $$\begin{array}{cccc}{(i\times i)}^{*}:& H^T(K/T)& ⟶& H^T(K/T){\otimes }_{\mathbb{Z}}{H}^T(K/T)\text{.}\end{array}$$ We have $$\begin{array}{ccc}{(i\times i)}^{*}{\sigma }_B^{\left(w\right)}& =& \sum _{w\in uv\hspace{0.17em}\text{[red]}}ϵ\left(u\right){\sigma }_{\beta }^{u^{-1}}\otimes {\sigma }_B^v\text{.}\end{array}$$

The Finite Case

Proposition: In the finite case, we have $$\begin{array}{ccc}{\underset{\_}{A}}_L& =& {\text{End}}_{{\underset{\_}{A}}_R}\left(H^T(K/T)\right)\\ {\underset{\_}{A}}_R& =& {\text{End}}_{{\underset{\_}{A}}_L}\left(H^T(K/T)\right)\text{.}\end{array}$$ $H^T(K/T)$ 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 $\varphi \in {\text{End}}_{{\underset{\_}{A}}_L}\left(H^T(K/T)\right)$ then $\exists$ $a\in \underset{\_}{A}$ s.t. $$\varphi \left({\sigma }_B^{\left(w_0\right)}\right)=a_R·{\sigma }_B^{\left(w_0\right)}\text{.}$$

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

		Proof.
	For any $z\in H^T(K/T),$ $\exists$ $b\in \underset{\_}{A}$ s.t. $z=b_L·{\sigma }_B^{\left(w_0\right)}$ $$\begin{array}{ccc}\Rightarrow \varphi \left(z\right)& =& \varphi (b_L·{\sigma }_B^{\left(w_0\right)})\\ & =& b_L·\varphi \left({\sigma }_B^{\left(w_0\right)}\right)(\varphi \in {\text{End}}_{{\underset{\_}{A}}_L})\\ & =& b_L·a_R·{\sigma }_B^{\left(w_0\right)}\\ & =& a_R·b_L·{\sigma }_B^{\left(w_0\right)}\\ & =& a_R·z\text{.}\end{array}$$ $\square$

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 $$\begin{array}{cccc}p:& K& ⟶& K/T\end{array}$$ is $T\text{-equivariant}$ (but not $K\text{-equivariant).}$ Thus $$\begin{array}{cccc}{p}^{*}:& H(K/T)& ⟶& H^T\left(K\right)\end{array}$$ is an $S\text{-module}$ map: $$p^{*}\left({\pi }_L\left(s\right)z\right)=\pi \left(s\right)p^{*}\left(z\right)$$ where $$\begin{array}{cccccc}\pi & =& {[k\to \text{pt}]}^{*}:& S& ⟶& H^T\left(K\right)\text{.}\end{array}$$ Now $\underset{\_}{A}$ acts on both $H^T\left(K\right)$ and $H^T(K/T)$ 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 $H^T(K/T)\text{.}$ 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 $z\in H^T(K/T)$ $$a·p^{*}\left(z\right)=p^{*}({a_{\left(1\right)}}_L{a_{\left(2\right)}}_R·z)$$ In particular, for $s\in S$ and $w\in W$ $$\begin{array}{ccc}\pi \left(s\right)p^{*}\left(z\right)& =& p^{*}\left({\pi }_L\left(s\right)z\right)=p^{*}\left({\pi }_R\left(s\right)z\right)\\ w·p^{*}\left(z\right)& =& p^{*}(w_L{w}_R·z)\\ A_w·p^{*}\left(z\right)& =& p^{*}(\sum _{\underset{v\le w}{u\le w}}{\pi }_L\left(a_w^{uv}\right){A_u}_L{A_v}_R·z)\end{array}$$

Proposition: For any $K\text{-space}$ $X$ with action map $$\begin{array}{cccc}{\mu }_X:& K\times X& ⟶& X\end{array}$$ the pullback $$\begin{array}{cccc}{\mu }_X^{*}:& H^T\left(X\right)& ⟶& H^T(K\times X)\end{array}$$ is the composition $$\begin{array}{ccccc}{H}^T\left(X\right)& \stackrel{{\Delta }_X}{⟶}& H^T(K/T){\otimes }_S{H}^T\left(X\right)& \stackrel{p^{*}\otimes \text{id}}{⟶}& H^T\left(K\right){\otimes }_S{H}^T\left(X\right)\\ & & & \simeq & H^T(K\times X)\text{.}\end{array}$$

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

$$\begin{array}{ccccccc}{\mu }_K:& K\times K& ⟶& K:& (k_1,k_2)& ⟼& k_1{k}_2\end{array}$$ gives a map $$\begin{array}{cccc}{{\mu }_K}_{*}:& H_{*}\left(K\right)\otimes H_{*}\left(K\right)& ⟶& H_{*}\left(K\right)\text{.}\end{array}$$ This defines a ring structure on $H_{*}\left(K\right)\text{.}$ Now for any $K\text{-space}$ $X$ with $$\begin{array}{cccc}{\mu }_X:& K\times X& ⟶& X\end{array}$$ get $$\begin{array}{cccc}{{\mu }_X}_{*}:& H_{*}\left(K\right)\otimes H_{*}\left(X\right)& ⟶& H_{*}\left(X\right)\end{array}$$ which defines an action of $H_{*}\left(K\right)$ on $H_{*}\left(X\right)\text{.}$

????? at the special case $X=K/T$ with $$\begin{array}{cccccc}{\mu }_X& =& {\mu }_{K/T}:& K\times K/T& ⟶& K/T\text{.}\end{array}$$ ${\underset{\_}{A}}_R$ acts on $H_{*}(K/T),$ and this action commutes with the Pontryagin action of $H_{*}\left(K\right)$ on $H_{*}(K/T)\text{.}$

Define a ring structure on $H_{*}(K/T)$ by $${\sigma }_v{\sigma }_w=\{\begin{array}{cc}{\sigma }_{vw}& \text{if}\hspace{0.17em}\ell \left(v\right)+\ell \left(w\right)=\ell \left(vw\right),\\ 0& \text{otherwise.}\end{array}$$ Then $$\begin{array}{ccc}{{\mu }_{K/T}}_{*}(\underset{\underset{H_{*}\left(K\right)}{⫙}}{\sigma }\times \underset{\underset{H_{*}(K/T)}{⫙}}{\sigma \prime })& =& p_{*}\left(\sigma \right)\sigma \prime \text{.}\end{array}$$ Consequently, $$\begin{array}{cccc}{p}_{*}:& H_{*}\left(K\right)& ⟶& H_{*}(K/T)\end{array}$$ is a ring homomorphism.

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

1)	$p^{*}(H^{*}(K/T),𝔽)$ is a Hopf subalgebra of $H^{*}\left(K𝔽\right)\text{.}$
2)	$p_{*}(H_{*}(K/T),𝔽)=H_{*}{(K/T,𝔽)}^S=\{\sigma :\lambda \cap \sigma =0\hspace{0.17em}\forall \hspace{0.17em}\lambda \in h_{\mathbb{Z}}^{*}\}\text{.}$
3)	If $m_{ij}=\infty$ for all $i\ne j,$ then $p^{*}(H^{*}(K/T),\mathbb{Q})\simeq$ the dual of a tensor algebra as a Hopf algebra.

Poincare Duality in the finite case

Define $\underset{\_}{A}\text{-module}$ homomorphism $$\begin{array}{c}\begin{array}{cccc}PD:& H^T(G/P)& ⟶& {\text{Hom}}_S(H^T(G/P),S),\end{array}\\ PD\left(z\right)\left(y\right)={\int }_{[G/P]}yz\in S\end{array}$$ Consider the case $P=B\text{:}$ $${\int }_{[G/B]}=\epsilon \circ {A_{w_0}}_R\text{.}$$ In general, $${\int }_{[G/P]}{\sigma }_P^{\left(w\right)}={\delta }_{w,w_0{w}_P}$$ 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) $$\Delta A_{w_0}=\sum _{w\in W}{A}_w\otimes w_0{A}_{w_{0}w}$$ $$PD\left({\sigma }_P^{\left(w\right)}\right)=w_0·{\sigma }_{\left(w_{0}w{w}_P\right)}^P$$ Also $${w_0}_L{w_0}_R·{\sigma }_B^{\left(w\right)}=ϵ\left(w\right){\sigma }_B^{\left(w_{0}w{w}_0\right)}\text{.}$$ It follows that $PD$ is an $S\text{-module}$ isomorphism.

The Euler Class:

For $z\in H^T(G/P),$ consider the operator $M_z$ on $H^T(G/P)$ by $y⟼zy\text{.}$ The Euler Class ${\chi }_{G/P}\in H^T(G/P)$ is defined by the property: $$\text{trace}\hspace{0.17em}{M}_z={\int }_{[G/P]}{\chi }_{G/P}·z\text{.}$$

Proposition: $${\chi }_{G/P}=\sum _{w\in W^P}{\sigma }_P^{\left(w\right)}[w_0·{\sigma }_P^{\left(w_{0}w{w}_P\right)}]\text{.}$$

		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 $$\begin{array}{ccc}{M}_z& =& \sum _{w\in W^P}({\sigma }_{\left(w\right)}^P,z{\sigma }_P^{\left(w\right)})\\ {\sigma }_{\left(w\right)}^P& =& PD(w_0·{\sigma }_P^{\left(w_{0}w{w}_P\right)})\end{array}$$ ????? $$\begin{array}{ccc}\text{?????}{M}_z& =& \sum _{w\in W^P}(PD(w_0·{\sigma }_P^{\left(w_{0}w{w}_P\right)}),z\in {\sigma }_P^{\left(w\right)})\\ & =& \sum _{w\in W^P}{\int }_{[G/P]}z{\sigma }_P^{\left(w\right)}(w_0·{\sigma }_P^{\left(w_{0}w{W}_P\right)})\end{array}$$ ????? $${\chi }_{G/P}=\sum _{w\in W^P}{\sigma }_P^{\left(w\right)}(w_0·{\sigma }_P^{\left(w_{0}w{w}_P\right)})\text{.}$$ $\square$

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

Lemma: For $v,w\in W^P$ $${\sigma }_P^{\left(v\right)}PD\left({\sigma }_{\left(w\right)}^P\right)=0$$ unless $v\le w\text{.}$

So ${\chi }_{G/P}$ is the trace of a rank $1$ upper triangular matrix. Also $${\sigma }_P^{\left(w\right)}PD\left({\sigma }_{\left(w\right)}^P\right)=w·PD\left({\sigma }_{\left(\text{id}\right)}^P\right)\text{.}$$

Facts:

1)	${\chi }_{G/P}$ has image $\prod _{\underset{w_0{w}_P>0}{\alpha >0}}{\alpha }_R·1$ in $H^T(K/T)$
2)	${\chi }_{G/P}$ is $W\text{-invariant}$ under the left action
3)	Image of ${\chi }_{G/P}$ in $H^{*}(G/P)$ is $\left|W^P\right|{\sigma }_P^{w_0{w}_P}$

Some facts on the classifcying spaces

$$\begin{array}{ccc}{H}^{*}\left(B_T\right)& \stackrel{{\pi }_R}{\hookrightarrow }& H^T(G/B)\\ \begin{array}{c} ↪ \end{array}& & & ⤣\\ H^{*}{\left(B_T\right)}^{w_P}& \hookrightarrow & H^T{(G/B)}^{{\left(w_P\right)}_R}& \cong & H^T(G/P)\end{array}$$ ????? $\mathbb{Q},$ we have $$H^{*}{\left(B_T\right)}^{w_P}\equiv H^{*}\left(B_{K\cap P}\right)\simeq H^T(G/P)\text{.}$$ In fact

1)	$$\begin{array}{ccc}{H}^{*}(B_{K\cap P},\mathbb{Q})& \simeq & {\left(\mathbb{Q}{\otimes }_{\mathbb{Z}}{H}^T(G/P)\right)}^W\text{(?)}\end{array}$$
2)	$$\begin{array}{ccc}S{\otimes }_{H^{*}\left(B_K\right)}{H}^{*}\left(B_{K\cap P}\right)& \simeq & H^T(G/P)\\ Z{\otimes }_S{H}^T(G/P)& \simeq & H^{*}(G/P)\end{array}$$

Open Problems

(1)	In what sense does the diagonal map $$\begin{array}{cccccc}K& ⟶& K\times K:& k& ⟼& (k,k)\end{array}$$ correspond to the co-product $$\begin{array}{cccc}\Delta :& \underset{\_}{A}& ⟶& \underset{\_}{A}{\otimes }_S\underset{\_}{A}\end{array}$$ Given homomorphism $K_1\to K_2$ with $T_1\to T_2,$ $N_1\to N_2$ can easily calculate $$\begin{array}{ccc}{H}^{T_2}(K_2/T_2)& ⟶& H^{T_1}(K_1/T_1)\end{array}$$
(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 ${\mathbb{Z}}_{+}\text{-coefficient.}$ 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: $$\begin{array}{ccc}{H}^T(G/P)& ⟶& K_T(G/P)\text{.}\\ \text{BGG-operators}& ⟶& \text{Demazure operators}\end{array}$$
(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 $H^{S^1}\left(\text{Grass}(k,n)\right)$ 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. $⟨{\alpha }_{{\sigma }^k\left(i\right)},{\alpha }_i^{\vee }⟩\ne 0\Rightarrow {\sigma }^k\left(i\right)=1\text{.}$ (In this case $G^{\sigma }$ has the structure of a Kac-Moody group. $\lambda \in h_{\mathbb{Z}}^{*}$ $\sigma \left(\lambda \right)=\lambda$ $\lambda$ minuscule $\alpha \in {\Delta }_{+}$ $\Rightarrow 0\le ⟨\lambda ,{\alpha }^{\vee }⟩\le 1$ $\Rightarrow H^{*}(G/P_{\lambda })\to H^{*}(G^{\sigma }/(G^{\sigma }\cap P))$ ?)
(7)	Study more of the Bruhat Graph $$\begin{array}{ccc}{(G/B)}^T& ⟷& W\end{array}$$ Vertices: $W,$ edges $w\to w{r}_{\alpha }$ $\alpha >0$ $T\text{-stable}$ curves $(\equiv P\prime )$ 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 $P_{v,w}=1$ $\iff$ for this graph, have the same # of edges emanate from each point.
(9)	Study directed Bruhat graphs: $$w\stackrel{{\alpha }^{\vee }}{⟶}w{r}_{\alpha }\text{if}\hspace{0.17em}w<w{r}_{\alpha }\text{.}$$

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: $$(t·k)\left(z\right)=tk\left(z\right)t^{-1}\text{.}$$ Roughly, the diagonal embedding $$\begin{array}{ccc}\Omega K& ⟶& \Omega K\times \Omega K\end{array}$$ gives a co-product $$\begin{array}{ccc}{H}_T\left(\Omega K\right)& ⟶& H_T\left(\Omega K\right){\otimes }_S{H}_T\left(\Omega K\right)\end{array}$$ and the multiplication map for the group structure on $\Omega K\text{:}$ $$\begin{array}{ccc}\Omega K\times \Omega K& ⟶& \Omega K\end{array}$$ gives a product $$\begin{array}{ccc}{H}_T\left(\Omega K\right){\otimes }_S{H}_T\left(\Omega K\right)& ⟶& H_T\left(\Omega K\right)\end{array}$$ 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 $$\begin{array}{ccc}\Omega K& ⟶& G_{\text{af}}/B_{\text{af}}\end{array}$$ which gives $$\begin{array}{ccccc}{H}_T\left(\Omega K\right)& ⟶& H_T(G_{\text{af}}/B_{\text{af}})& =& {\underset{\_}{A}}_{\text{af}}\text{.}\end{array}$$ Under this, we will identify $$H_T\left(\Omega K\right)\simeq Z_{\text{af}}\left(S\right)\text{(centralizer of}\hspace{0.17em}S\hspace{0.17em}\text{in}\hspace{0.17em}{\underset{\_}{A}}_{\text{af}}\text{)}$$ 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 $$\stackrel{\sim }{X}=\text{Mor}({ℂ}^{\times },X)\text{.}$$ Let $G$ be a finite dimensional connected simple algebraic group over $ℂ\text{.}$ We then have the finite root datum $$I,{\alpha }_i,\in h_{\mathbb{Z}}^{\vee },{\alpha }_i^{\vee }\in h_{\mathbb{Z}}\text{.}{\Delta }_{+},\Pi ,W,\underset{\_}{g},\underset{\_}{h},\underset{\_}{b},\dots$$ Let $\theta$ be the highest root. From these we form the following Kac-Moody root datum:

• ${\left(h_{\mathbb{Z}}\right)}_{\text{af}}=h_{\mathbb{Z}},$ ${\left(h_{\mathbb{Z}}\right)}_{\text{af}}^{\vee }=h_{\mathbb{Z}}^{\vee }$
• $I_{\text{af}}=I\cup \left\{0\right\}$
• $Q_{\text{af}}=\underset{i\in {\xi }_{\text{af}}}{⨁}\mathbb{Z}{\alpha }_i=\mathbb{Z}{\alpha }_0+Q=\mathbb{Z}\delta +Q,\delta ={\alpha }_0+\theta$ $$\begin{array}{cccccc}{Q}_{\text{af}}& ⟶& {\left(h_{\mathbb{Z}}\right)}_{\text{af}}:& {\alpha }_0& ⟼& -\theta \\ & & & {\alpha }_i& ⟼& {\alpha }_i,& i\ne 0,i\in I\text{.}\end{array}$$ $$\begin{array}{ccc}{\Pi }_{\text{af}}& =& \Pi \cup \{-\theta \}\\ {\Pi }_{\text{af}}^{\vee }& =& {\Pi }^{\vee }\cup \{-{\theta }^{\vee }\}\\ \underset{\underset{Q_{aff}}{⫙}}{\delta }& =& {\alpha }_0+\theta ⟼0ϵ{\left(h_{\mathbb{Z}}\right)}_{\text{af}}^{\vee }=h_{\mathbb{Z}}^{\vee }\text{ie.}\hspace{0.17em}⟨\delta ,h⟩=0\hspace{0.17em}\forall h\in h_{\mathbb{Z}}\text{.}\end{array}$$

Corresponding to this root datum, we have the following Kac-Moody Lie algebra ${\underset{\_}{𝔤}}_{\text{af}}\text{:}$ $$\begin{array}{ccc}{\underset{\_}{𝔤}}_{\text{af}}& =& \underset{\_}{𝔤}{\otimes }_{ℂ}ℂ[t,t^{-1}]=\underset{\_}{\overset{\sim }{𝔤}}\\ e_i& =& e_i\otimes 1\\ f_i& =& f_i\otimes 1\\ e_0& =& e_{-\theta }\otimes t\\ f_0& =& f_{\theta }\otimes t^{-1}\\ \Rightarrow [e_0,f_0]& =& [e_{-\theta }\otimes 1,e_{\theta }\otimes 1]=-\text{?????}\end{array}$$ Roots are in $Q_{\text{af}}\text{.}$ They are all those in $Q_{\text{af}}$ of the form $$\alpha +n\delta \hspace{0.17em}n\in \mathbb{Z},\alpha \in \Delta ,\hspace{0.17em}\text{or}\hspace{0.17em}\alpha =0\text{.}$$ The root spaces are $${\left({\underset{\_}{𝔤}}_{\text{af}}\right)}_{\alpha +n\delta }=\{\begin{array}{cc}$ {𝔤}_{\alpha }$\otimes t^n& \text{if}\hspace{0.17em}\alpha \in \Delta ,n\in \mathbb{Z},\\ \underset{\_}{h}\otimes t^n& \alpha =0,n\in \mathbb{Z}\end{array}$$ so $${\Delta }^{\text{re}}=\{\alpha \in n\delta :\alpha \in \Delta ,n\in \mathbb{Z}\}$$ and all $n\delta \text{'s}$ $n\in \mathbb{Z},$ are "imaginary roots". They have multiplicity $={\text{dim}}_{ℂ}\hspace{0.17em}\underset{\_}{h}\text{.}$

The positive roots are $$\begin{array}{ccc}{\left({\Delta }_{\text{af}}\right)}_{+}& =& \{\alpha +n\delta :n>0\hspace{0.17em}\text{or}\hspace{0.17em}n\ge 0\hspace{0.17em}\alpha \in {\Delta }_{+}\},\\ {\left({\Delta }_{\text{af}}\right)}_{+}^{\text{re}}& =& \{\alpha +n\delta :n\ge 0\hspace{0.17em}\alpha \in {\Delta }_{+}\}\text{.}\end{array}$$

The affine Weyl group $W_{\text{af}}\text{:}$

By definition, $$W_{\text{af}}=W⋉\Gamma$$ the semi-direct product, where $\Gamma \simeq Q^{\vee }$ with $$\begin{array}{cccccc}{Q}^{\vee }& ⟶& \Gamma :& h& ⟼& t_h\text{.}\end{array}$$ $$\begin{array}{ccc}w{t}_h{w}^{-1}& =& t_{w·h}\\ t_h{t}_{h\prime }& =& t_{h+h\prime }\end{array}$$ The reason why this is the same as the group generated by the reflections $r_0,r_i,$ $i\in I$ is because $$t_{\stackrel{\vee }{\theta }}=r_0{r}_{\theta }$$ For $w\in W,$ $$\begin{array}{ccc}w·(\alpha +n\delta )& =& w·\alpha +n\delta (\Rightarrow \hspace{0.17em}w\delta =\delta )\\ t_h·(\alpha +n\delta )& =& \alpha +n\delta -⟨\alpha ,h⟩\delta \end{array}$$ $$\begin{array}{c}\text{(so}{t}_h·\alpha =\alpha -⟨\alpha ,h⟩\delta ,t_h·\left(n\delta \right)=n\delta -⟨\alpha ,h⟩\delta \text{).}\end{array}$$

The Kac-Moody group: $$G_{\text{af}}=\stackrel{\sim }{G}=\text{Mor}({ℂ}^{\times },G)\text{(Laurent series in}\hspace{0.17em}t\text{)}$$ set $$\begin{array}{ccc}{P}_0& =& \text{Mor}(ℂ,G)\text{(power series in}\hspace{0.17em}t\text{)}\\ B_{\text{af}}& =& \{g\in \text{Mor}(ℂ,G):g\left(0\right)\in B\}\subset P_0\\ U_{\text{af}}^{+}& =& \{g\in \text{Mor}(ℂ,G):g\left(0\right)\in U^{+}\}\\ K_{\text{af}}& =& \{g\in G_{\text{af}}:g\left(S^1\right)\subset K\}\\ \Omega K& =& \{k\in K_{\text{af}}:k\left(1\right)=\text{id}\}\\ T_{\text{af}}& =& T\\ G& \simeq & \text{const. loops}\subset G_{\text{af}}\end{array}$$ $K_{\text{af}}$ acts on $\Omega K$ by $$k·k\prime =kk\prime k{\left(1\right)}^{-1}$$ Then $$\begin{array}{ccccccc}{i}_{\Omega }:& \Omega K& ⟶& G_{\text{af}}/P_0:& k& ⟼& k·*\end{array}*=P_0$$ is a $K_{\text{af}}\text{-equivariant}$ map. This map is also a home????? because $$\begin{array}{ccc}{G}_{\text{af}}& =& K_{\text{af}}{B}_{\text{af}}{K}_{\text{af}}\cap B_{\text{af}}=T\\ & =& \left(\Omega K\right)K{B}_{\text{af}}=\left(\Omega K\right)P_0\end{array}$$

The compact involution on $G_{\text{af}}\text{:}$

$$\left(w_{K_{\text{af}}}\right)\left(g\right)\left(t\right)=w_K\left(g\left({\bar{t}}^{-1}\right)\right)g\in \text{Mor}({ℂ}^{\times },G)=G_{\text{af}}$$ 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 $$N_{\text{af}}=\stackrel{\sim }{N}=\text{Mor}({ℂ}^{\times },N)$$ where, recall, $N$ is the normalizer of $H$ in $G,$ so also have $$\begin{array}{ccc}{N}_{\text{af}}& =& \text{semi-direct product of}\hspace{0.17em}N\hspace{0.17em}\text{and}\hspace{0.17em}\Omega T\\ \Omega T& =& \{g\in \Omega K:g\left(S^1\right)\subset T\}\end{array}$$ so $g\in \text{<mi style="font-style: normal;">Ω</mi><mi>T</mi>}$ must be a homomorphism from $S^1$ to $T\text{.}$ Thus $$\Omega T\simeq \Gamma \simeq Q^{\vee }$$ where $$\begin{array}{ccccccc}{Q}^{\vee }& \stackrel{\sim }{⟶}& \Omega T:& h& ⟼& \hat{h}:& \hat{h}\left(z\right)=z^h,z\in {ℂ}^{\times }\text{.}\end{array}$$ This way we also see $$\begin{array}{cccccccc}W⋉\Gamma & \stackrel{\sim }{⟶}& W_{\text{af}}:& (w,t_h)& ⟼& (W,{\hat{h}}^{-1}H)& \in & N_{\text{af}}/H\text{.}\end{array}$$

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

Since ${\left(h_{\mathbb{Z}}\right)}_{\text{af}}=h_{\mathbb{Z}},$ 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 $$\begin{array}{cccccc}\underset{\_}{A}& \hookrightarrow & {\underset{\_}{A}}_{\text{af}}:& s& ⟼& s\\ & & & A_i& ⟼& A_i,& i\in I,(i\ne 0)\text{.}\end{array}$$ Recall that if $\beta =w{a}_i^{ϵ{\Delta }^{\text{re}}}$ with $i\in I,$ then we define $$\begin{array}{ccc}{A}_{{\beta }^{\vee }}& =& w{A}_{{\alpha }_i}{w}^{-1}=w{A}_i{w}^{-1}\\ r_{\beta }& =& 1-\beta A_{{\beta }^{\vee }}\end{array}$$ (It is not obvious how to write $A_{{\beta }^{\vee }}$ in terms of the $A_i\text{'s).}$

Define a ring homomorphism $$\begin{array}{ccccccc}\text{ev}:& {\underset{\_}{A}}_{\text{af}}& ⟶& \underset{\_}{A}:& \text{ev}{|}_S& =& \text{id}\\ & & & & \text{ev}\left(A_{{\beta }^{\vee }}\right)& =& A_{{\bar{\beta }}^{\vee }}\\ & & & & \text{ev}\left(w{t}_h\right)& =& w\end{array}$$ where if $\beta =\alpha +n,$ $\bar{\beta }=\alpha \text{.}$ This is well-defined.

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

Now identify $$\begin{array}{ccc}\Omega K& \underset{i_{\Omega }}{\overset{\sim }{⟶}}& G_{\text{af}}/P_0\end{array}$$ 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 $$\begin{array}{cccc}{i}_x^{\Omega }:& {\underset{\_}{\overset{‾}{X}}}_x^{\Omega }& ⟶& \Omega K\end{array}$$ so have Schubert basis $$\begin{array}{ccc}{\sigma }_x^{\Omega }& \in & H_{2\ell \left(x\right)}\left(\Omega K\right)\\ {\sigma }_{\Omega }^x& \in & H^{2\ell \left(x\right)}\left(\Omega K\right)\\ {\sigma }_{\left(x\right)}^{\Omega }& \in & {\text{Hom}}_S(H^T\left(\Omega K\right),S)\\ {\sigma }_{\Omega }^{\left(x\right)}& \in & H^T\left(\Omega K\right)\end{array}$$
• Also for $x\in W_{\text{af}},$ have ${\psi }_x^{\Omega }\in {\text{Hom}}_S(H^T\left(\Omega K\right),S)\text{.}$ (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 ${\text{Hom}}_S(H^T\left(\Omega K\right),S)$.

In the Schubert basis $$A_x·{\sigma }_{\left(y\right)}^{\Omega }=\{\begin{array}{cc}{\sigma }_{\left(xy\right)}^{\Omega }& \text{if}\hspace{0.17em}xy\in W_{\text{af}}^{-},\hspace{0.17em}\ell \left(xy\right)=\ell {\left(x\right)}^{+}\ell \left(y\right),\\ 0& \text{otherwise.}\end{array}$$ Define $$H_T\left(\Omega K\right)=S\text{-span of}\hspace{0.17em}\{{\sigma }_{\left(x\right)}^{\Omega }:x\in \text{?????}\}\subset {\text{Hom}}_S(H^T\left(\Omega K\right),S)$$ In our special case at hand, not only do we have $G_{\text{af}}/B_{\text{af}}\to G_{\text{af}}/\text{?????}$ but also: $\Omega K\hookrightarrow G_{\text{af}}/B_{\text{af}}\text{.}$ Thus have $$\begin{array}{ccc}{H}_T\left(\Omega K\right)& ⟶& {\underset{\_}{A}}_{\text{af}}\text{.}\end{array}$$ 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 $W_{\text{af}}/W_{P_0}=W_{\text{af}}/W\text{.}$ $\text{(}{W}_{P_0}=W\text{).}$ $$\begin{array}{ccc}x\in W_{\text{af}}^{-}& \iff & x<x{r}_i\forall i\in I,(i\ne 0),\\ & \iff & x·{\alpha }_i>0\forall i\in I\text{.}\end{array}$$ Write $x=w{t}_{-h}\text{.}$ Then $$\begin{array}{ccc}x·{\alpha }_i& =& w·t_{-h}·{\alpha }_i\\ & =& w·({\alpha }_i+⟨h,{\alpha }_i⟩\delta )\\ & =& w{\alpha }_i+⟨h,{\alpha }_i⟩\delta \\ x\in W_{\text{af}}^{-}& ⟺& w{\alpha }_i+⟨h,{\alpha }_i⟩\delta >0\forall i\in I\\ & ⟺& ⟨h,{\alpha }_i⟩\ge 0\hspace{0.17em}\text{and when}\hspace{0.17em}⟨h,{\alpha }_i⟩=0\hspace{0.17em}\text{must have}\hspace{0.17em}w{\alpha }_i>0\\ & ⟺& $ h$\hspace{0.17em}\text{is dominant and when}\hspace{0.17em}⟨h,{\alpha }_i⟩=0\hspace{0.17em}\text{must have}\hspace{0.17em}w<w{r}_i\text{.}\end{array}$$ Now for $h$ dominant, set $$\begin{array}{ccc}{W}_h& =& \text{the subgroup of}\hspace{0.17em}W\hspace{0.17em}\text{generated by}\hspace{0.17em}⟨r_i:⟨h,{\alpha }_i⟩=0⟩\\ & =& \{w\in W:wh=h\}\text{.}\end{array}$$ 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. $$\begin{array}{ccc}w\in W^h& \iff & w<w{r}_i\forall r_i\in W_h\end{array}$$ so $$\begin{array}{ccc}w\in W^h& \iff & \text{For each}\hspace{0.17em}i\hspace{0.17em}\text{with}\hspace{0.17em}⟨h,{\alpha }_i⟩=0\hspace{0.17em}\text{have}\hspace{0.17em}w<w{r}_i\text{.}\end{array}$$ Thus we have proved $$\begin{array}{ccc}{W}_{\text{af}}^{-}& =& \{w{t}_{-h}:h\hspace{0.17em}\text{dominant (ie.}\hspace{0.17em}⟨h,{\alpha }_i⟩\ge 0\hspace{0.17em}\forall i\in I\hspace{0.17em}\text{and}\hspace{0.17em}w\in W^h\}\\ & =& \{w{t}_{-h}:h\hspace{0.17em}\text{dominant and if}\hspace{0.17em}⟨h,{\alpha }_i⟩=0\hspace{0.17em}\text{for}\hspace{0.17em}i\in I\hspace{0.17em}\text{must have}\hspace{0.17em}w{\alpha }_i>0\}\text{.}\end{array}$$ The map $$\begin{array}{ccc}{W}_{\text{af}}^{-}& ⟼& W_{\text{af}}/W\\ w{t}_{-h}& ⟼& w{t}_{-h}/W\end{array}$$ is of course a bijection.

Now another model for $W_{\text{af}}/W$ is $\Gamma \simeq Q^{\vee }\text{:}$ $$\begin{array}{ccc}\Gamma & \stackrel{\sim }{⟶}& W_{\text{af}}/W\\ t_h& ⟼& t_h/W\text{.}\end{array}$$ In other words, each coset $W_{\text{af}}/W$ has a unique translation element $t_{-h}$ in it, namely $$w{t}_{-h}/W=w{t}_{-h}{w}^{-1}/W=t_{-w·h}/W\text{.}$$

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 $$w{t}_{-h}\in W_{\text{af}}^{-}$$ 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 $$x\in W_{\text{af}}^{-}\cap \Gamma \text{.}$$ 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 $$x=t_{-w·h}⟺w{t}_{-h}=t_{-w·h}⟺w=1$$ so $$\begin{array}{ccc}{W}_{\text{af}}^{-}\cap P& =& \{t_{-h}:h\hspace{0.17em}\text{is dominant}\}\text{.}\end{array}$$
(6)	Let's now calculate the length $\ell \left(t_{-h}\right)$ when $h$ is dominant. Recall that $\alpha +n\delta >0$ $\iff$ either $n>0$ or $n=0,\alpha >0\text{.}$ Now we need to see for $\alpha +n\delta >0,$ when do we have $$t_{-h}·(\alpha +n\delta )<0\text{.}$$ Now $$t_{-h}·(\alpha +n\delta )=\alpha +(n+⟨h,\alpha ⟩)\delta$$ If $n>0,\alpha <0,$ then $t_{-h}·(\alpha +n\delta )<0$ for $n=0,1,\dots ,⟨h,\alpha ⟩-1\text{.}$ If $n>0,\alpha =0,$ then $t_{-h}·(\alpha +n\delta )n\delta <0\text{.}$ If $n>0,\alpha >0,$ then $t_{-h}·(\alpha +n\delta )<0\text{.}$ If $n=0,\alpha >0,$ then $t_{-h}·(\alpha +n\delta )<0\text{.}$ Then the only case when $\alpha +n\delta >-$ and $t_{-h}·(\alpha +n\delta )<0$ 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 $$\ell \left(t_{-h}\right)=⟨h,2\rho ⟩=\sum _{\beta >0}⟨h,\beta ⟩$$ for $h$ dominant. Let's notice that $$\begin{array}{ccc}\text{the sum of all}\hspace{0.17em}\{\alpha +n\delta >0;t_{-h}(\alpha +n\delta )<0\}& =& \sum _{\beta >0}(-\beta -\beta +\delta +(-\beta +2\delta )+\dots +(-\beta +(⟨h,\beta ⟩-1)\delta ))\\ & =& \sum _{\beta >0}(-⟨h,\beta ⟩\beta +\frac{1}{2}⟨h,\beta ⟩(⟨h,\beta ⟩-1)\delta )\text{.}\end{array}$$
(7)	For any $x=w{t}_{-h}\in W_{\text{af}}^{-},$ $t=t_{-h_1}\in {\Gamma }^{-}=W_{\text{af}}^{-}\cap \Gamma$ we have $xt=w{t}_{-(h+h_1)}\in W_{\text{af}}^{-}$ 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. $x·(\alpha +n\delta )=w\alpha +(n+⟨h,\alpha ⟩)\delta <0$ $⟺$ 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 $$\begin{array}{ccc}\{\alpha +n\delta >0:w{t}_{-h}·(\alpha +n\delta )<0\}& =& \begin{array}{c}\{-\beta +n\delta :\beta >0,w\beta <0,n=1,\dots ,⟨\beta ,h⟩-1\}\\ \cup \{-\beta +n\delta :\beta >0,w\beta >0,n=1,2,\dots ⟨\beta ,h⟩\}\text{.}\end{array}\end{array}$$ Consequently, $$\ell \left(w{t}_{-h}\right)=⟨2\rho ,h⟩-\ell \left(w\right)\text{.}$$

Lecture 9: March 12, 1997

Recall the ${\underset{\_}{A}}_{\text{af}}\text{-action}$ on ${\text{Hom}}_S(H^T\left(\Omega K\right),S)\text{:}$ $$\begin{array}{ccc}{A}_x·{\sigma }_{\left(y\right)}^{\Omega }& =& \{\begin{array}{cc}{\sigma }_{\left(xy\right)}^{\Omega }& \text{if}\hspace{0.17em}xy\in W_{\text{af}}^{-}\hspace{0.17em}\ell \left(x\right)+\ell \left(y\right)=\ell \left(xy\right),\\ 0& \text{otherwise,}\end{array}\\ w·{\psi }_t& =& {\psi }_{wt}\\ t\prime ·{\psi }_t& =& {\psi }_{t\prime t}\end{array}$$ $t,t\prime \in \Gamma ,w\in W\text{.}$

Define $$H_T\left(\Omega K\right)=\sum _{x\in W_{\text{af}}^{-}}s{\sigma }_{\left(x\right)}^{\Omega }$$ as the ${\underset{\_}{A}}_{\text{af}}\text{-submodule}$ of ${\text{Hom}}_S(H^T\left(\Omega K\right),S)$ spanned over $S$ by $\{{\sigma }_{\left(x\right)}^{\Omega }:x\in W_{\text{af}}^{-}\}\text{.}$ For $x\in W_{\text{af}}^{-},$ set $$F_x=\sum _{\underset{y\le x}{y\in W_{\text{af}}^{-}}}s{\sigma }_{\left(y\right)}^{\Omega }\text{.}$$ Then $$\begin{array}{cccc}{i}_x^{\Omega }:& {\underset{\_}{\overset{‾}{X}}}_x^{\Omega }& ⟶& \Omega K\end{array}$$ gives $$\begin{array}{ccc}{\text{Hom}}_S(H^T\left({\underset{\_}{\overset{‾}{X}}}_x^{\Omega }\right),S)& \stackrel{\sim }{⟶}& F_x\\ {\text{Hom}}_S(F_x,S)& \stackrel{\sim }{⟶}& H^T\left({\underset{\_}{\overset{‾}{X}}}_x^{\Omega }\right)\text{.}\end{array}$$

Structure on $F_x$

(1)

$\{1\otimes {\psi }_t:t\in \Gamma ,t\le x{w}_0\}$ is a free $S\text{-basis}$ for $\text{Frac}\left(S\right){\otimes }_S{F}_x$ where $$\begin{array}{ccc}\text{Frac}\left(S\right)& =& \text{the fractional field of}\hspace{0.17em}S\\ x\in {\Gamma }_{-}& \iff & \text{the minimal rep. of}\hspace{0.17em}xW\hspace{0.17em}\text{coincides with the translational representative of}\hspace{0.17em}xW\text{.}\end{array}$$

(2)

Set $$\begin{array}{c}{\Gamma }_{-}=\Gamma \cap W_{\text{af}}^{-}=\{t_{-h}:h\in {\underset{\_}{h}}_{\mathbb{Z}}\hspace{0.17em}\text{dominant}\}\\ \text{see end of Lecture 8 on}\hspace{0.17em}{W}_{\text{af}}^{-}\simeq W_{\text{af}}/W\simeq \Gamma \text{.}\end{array}$$ 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 $$P_{0}t·P_0\subset {\underset{\_}{\overset{‾}{X}}}_t^{\Omega }⟺t^{-1}{B}_{-}t\in P_0$$ But for any $\alpha \in {\Delta }_{+}$ $$t_{-h}·\alpha =\alpha +⟨h,\alpha ⟩\delta \in \Delta (P_0/b_{\text{af}})\text{(ie. a root for}\hspace{0.17em}{P}_0\text{)}$$ $$\begin{array}{c}\Rightarrow t^{-1}{B}_{-}t\in P_0\\ \Rightarrow {\underset{\_}{\overset{‾}{X}}}_t^{\Omega }\hspace{0.17em}\text{is}\hspace{0.17em}J\text{-stable}\\ \Rightarrow F_t\hspace{0.17em}\text{is}\hspace{0.17em}\underset{\_}{A}\text{-submod. of}\hspace{0.17em}{H}_T\left(\Omega K\right)\text{.}\end{array}$$ 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 $j\in I$ $\Rightarrow$ $r_i{\alpha }_j>0\text{"}\text{.}$ 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 $t^{-1}<t^{-1}{r}_i$ $\Rightarrow$ $t^{-1}{\alpha }_i>0\text{.}$ But $t^{-1}·{\alpha }_i=t_h·{\alpha }_i={\alpha }_i-⟨h,{\alpha }_i⟩\delta ,$ Since $⟨h,{\alpha }_i⟩>0$ $\Rightarrow$ $t_h·{\alpha }_i<0\text{.}$ 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 $$\begin{array}{cccc}m:& \Omega K\times \Omega K& ⟶& \Omega K\end{array}$$ induces the product map: $$\begin{array}{cccc}\mu :& H_T\left(\Omega K\right)\otimes H_T\left(\Omega K\right)& ⟶& H_T\left(\Omega K\right)\text{.}\end{array}$$ Since $$m({\underset{\_}{\overset{‾}{X}}}_x^{\Omega }\times {\underset{\_}{\overset{‾}{X}}}_t^{\Omega })\subset {\underset{\_}{\overset{‾}{X}}}_{xt}^{\Omega }$$ we actually have $$\begin{array}{cccc}\mu :& F_x\otimes F_t& ⟶& F_{xt}\text{.}\end{array}$$ The diagonal imbedding $$\begin{array}{ccc}\Omega K& ⟶& \Omega K\times \Omega K\end{array}$$ induces the co-product: $$\begin{array}{cccc}\Delta :& H_T\left(\Omega K\right)& ⟶& H_T\left(\Omega K\right)\otimes H_T\left(\Omega K\right)\text{.}\end{array}$$ Clearly $$\Delta F_x\subset F_x\otimes F_x$$ 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 $\{1\otimes {\psi }_t:t\in \Gamma ,t\le x{w}_0\}$ 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 $$\omega ·{\alpha }_i=-{\alpha }_{\omega \left(i\right)},i\ne 0,\omega \left(0\right)=0$$ $\text{(}\Rightarrow$ $\omega \left(w\right)=w_{0}w{w}_0$ 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. $$\begin{array}{ccc}\epsilon \left({\psi }_t\right)& =& 1,\\ c\left({\psi }_t\right)& =& {\psi }_{t^{-1}},\\ \Delta {\psi }_t& =& {\psi }_t\otimes {\psi }_t,\\ {\psi }_t{\psi }_{t\prime }& =& {\psi }_{tt\prime },\\ {\psi }_{\text{id}}& =& 1\text{.}\end{array}$$ $\square$

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 $$\begin{array}{cccc}\varphi :& \Omega K\times X& ⟶& X\end{array}$$ is a $T\text{-equivariant}$ map (with $T$ acting on $\Omega K$ by conjugations and on $\Omega K\times X$ by the diagonal action), then each $\sigma \in H_T\left(\Omega K\right)\subset {\text{Hom}}_S(H^T\left(\Omega K\right),S)$ defines the following composition map $$\begin{array}{ccccccc}{H}^T\left(X\right)& \stackrel{{\varphi }^{*}}{⟶}& H^T\left(\Omega K\right){\otimes }_S{H}^T\left(X\right)& \stackrel{c\left(\sigma \right)\otimes \text{id}}{⟶}& S{\otimes }_S{H}^T\left(X\right)& \simeq & H^T\left(X\right)\text{.}\end{array}$$ 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 $$\begin{array}{cccc}\varphi :& \Omega K\times X& ⟶& X\end{array}$$ 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 $$\begin{array}{cccc}j:& H_T\left(\Omega K\right)& ⟶& {\underset{\_}{\hat{A}}}_{\text{af}}\text{.}\end{array}$$ 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 $$j\left(H_T\left(\Omega K\right)\right)\subset Z_{{\underset{\_}{A}}_{\text{af}}}\left(s\right),\text{centralizer of}\hspace{0.17em}s\hspace{0.17em}\text{in}\hspace{0.17em}{\underset{\_}{A}}_{\text{af}}\hspace{0.17em}\text{(}t\in W_{\text{af}}\subset {\underset{\_}{A}}_{\text{af}}\hspace{0.17em}\text{commutes with}\hspace{0.17em}s\text{).}$$ Set $${\underset{\_}{A}}_{\Omega }=Z_{{\underset{\_}{A}}_{\text{af}}}\left(s\right)\text{.}$$ It is a commutative $S\text{-algebra.}$ Thus we have an $S\text{-algebra}$ homomorphism $$\begin{array}{cccccc}j:& H_T\left(\Omega K\right)& ⟶& {\underset{\_}{A}}_{\Omega }& =& Z_{{\underset{\_}{A}}_{\text{af}}}\left(S\right)\end{array}$$ Will show that it is in fact an isomorphism.

Connection between $j:H_T\left(\Omega K\right)\to {\underset{\_}{A}}_{\text{af}}$ and $j_{\Omega }:\Omega K\to G_{\text{af}}/B_{\text{af}}:k\mapsto k{B}_{\text{af}}\text{.}$

Have commutative diagram $$\begin{array}{ccc}{H}_T\left(\Omega K\right)& \stackrel{j}{⟶}& {\underset{\_}{A}}_{\text{af}}\\ \begin{array}{c} ↪ \end{array}& & \downarrow \\ {\text{Hom}}_S(H^T\left(\Omega K\right),S)& \underset{{\left(j_{\Omega }\right)}_X}{⟶}& {\text{Hom}}_S(H^T(G_{\text{af}}/B_{\text{af}}),S)\end{array}\begin{array}{c}a\\ ↧\\ \epsilon ·a_R& =& \epsilon ·c{\left(a\right)}_L\end{array}$$ 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?) $$\begin{array}{cccc}\text{ev}:& {\underset{\_}{A}}_{\text{af}}& ⟶& \underset{\_}{A}\end{array}$$ where, recall, $$\begin{array}{ccc}\text{ev}{|}_S& =& \text{id,}\\ \text{ev}{|}_{A_{\stackrel{\vee }{\beta }}}& =& A_{{\bar{\beta }}^{\vee }},\\ \text{ev}{|}_{w{t}_h}& =& w\text{.}\end{array}$$

Lemma 2: For $\sigma \in H^T\left(\Omega K\right),$ $$\begin{array}{cc}(\text{id}\otimes \text{ev})\Delta ·j\left(\sigma \right)=j\left(\sigma \right)\otimes 1\text{.}& \text{?}\end{array}$$

		Proof.
	This is roughly due to the fact that $$\begin{array}{cccccc}\Omega K& \hookrightarrow & K_{\text{af}}:& k& ⟼& (k,1)\text{.}\end{array}$$ $\square$

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, $$j\left(\sigma \right)·(m\otimes n)=j\left(\sigma \right)·m\otimes n\text{.}$$ Apply this to the action map $$\begin{array}{cccc}F:& H_T\left(\Omega K\right){\otimes }_S{H}^T\left(X\right)& ⟶& H^T\left(X\right)\text{.}\end{array}$$

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 $$\begin{array}{cc}F(\sigma \otimes z)=j\left(\sigma \right)·z& \text{?}\end{array}$$ so for $w\in W$ $$w·F(\sigma \otimes z)=w·j\left(\sigma \right)·z\text{.}$$ In particular $$w·F({\psi }_t\otimes z)=w·t·z=wt{w}^{-1}·w·z=(w·t)·(w·z)\text{.}$$ On the other hand $$F(w·({\psi }_t\otimes z))=F(w·{\psi }_t\otimes w·z)=F(w·({\psi }_t\otimes z))\text{.}$$ Also $$t\prime ·F({\psi }_t\otimes t)=t\prime t·z=F({\psi }_{t\prime t}\otimes z)=F(t\prime ·({\psi }_t\otimes z))\text{.}$$ $\square$

Proposition: The multiplication map $$\begin{array}{ccc}{H}_T\left(\Omega K\right){\otimes }_S{H}_T\left(\Omega K\right)& ⟶& H_T\left(\Omega K\right)\end{array}$$ is an ${\underset{\_}{A}}_{\text{af}}\text{-map.}$

		Proof.
	This is because $$\sigma \sigma \prime =j\left(\sigma \right)·\sigma \prime \text{.}$$ $\square$

More generally, for any ${\underset{\_}{A}}_{\text{af}}\text{-module}$ $M,$ the map $$\begin{array}{cccc}\varphi :& H_T\left(\Omega K\right){\otimes }_{S}M& ⟶& M\\ & \sigma \otimes m& ⟼& j\left(\sigma \right)·m\end{array}$$ 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) $$I=\sum _{\underset{w\ne \text{id}}{w\in W}}$ {\underset{\_}{A}}_{\text{af}}$A_w\text{.}$$ 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 $x\in W_{\text{af}}^{-}$ $$j\left({\sigma }_{\left(x\right)}^{\Omega }\right)=A_x\hspace{0.17em}\text{mod}\hspace{0.17em}I\text{.}$$

		Proof.
	$$\begin{array}{c}j\left({\sigma }_{\left(x\right)}^{\Omega }\right)·1={\sigma }_{\left(x\right)}^{\Omega }1={\sigma }_{\left(x\right)}^{\Omega }=A_x·{\sigma }_{\left(\text{id}\right)}^{\Omega }=A_x·1\text{.}\\ \Rightarrow j\left({\sigma }_{\left(x\right)}^{\Omega }\right)-A_x\in I\text{.}\end{array}$$ $\square$

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.
	$$j\left({\sigma }_{\left(x\right)}^{\Omega }\right)A_{w_0}=(A_x+a)A_{w_0}=A_x{A}_{w_0}=A_{x{w}_0}(a\in I)\text{.}$$ $\square$

Corollary 2: For any $x\in W_{\text{af}}^{-},$ $t\in {\Gamma }_{-}$ $$\begin{array}{ccc}{\sigma }_{\left(x\right)}^{\Omega }{\sigma }_{\left(t\right)}^{\Omega }& =& {\sigma }_{\left(xt\right)}^{\Omega },\\ F_x{F}_t& =& F_{xt}\text{.}\end{array}$$ $\text{(}\ell \left(x\right)+\ell \left(w_0\right)=\ell \left(x{w}_0\right)$ 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 $$\begin{array}{ccc}{\sigma }_{\left(x\right)}^{\Omega }{\sigma }_{\left(t\right)}^{\Omega }& =& j\left({\sigma }_{\left(x\right)}^{\Omega }\right)·{\sigma }_{\left(t\right)}^{\Omega }\\ & =& (A_x+a)·{\sigma }_{\left(t\right)}^{\Omega }a\in I\\ & =& A_x·{\sigma }_{\left(t\right)}^{\Omega }(a·({\sigma }_{\left(t\right)}^{\Omega }=0))\\ & =& {\sigma }_{\left(xt\right)}^{\Omega }\text{.}\end{array}$$ (We are saying $\ell \left(x\right)+\ell \left(t\right)=\ell \left(xt\right)\text{? automatically?)}$ $\square$

Proposition: $$\begin{array}{cccccc}{H}_T\left(\Omega K\right){\otimes }_S\underset{\_}{A}& ⟶& {\underset{\_}{A}}_{\text{af}}:& \sigma \otimes a& ⟼& j\left(\sigma \right)a\end{array}$$ 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 $${\underset{\_}{A}}_{\text{af}}\simeq {\underset{\_}{A}}_{\Omega }+I$$ as an ${\underset{\_}{A}}_{\Omega }\text{-module.}$

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

• First, by identifying $${\underset{\_}{A}}_{\Omega }\cong {\underset{\_}{A}}_{\text{af}}/I$$ 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 $$a·a\prime -aa\prime \in I\text{.}$$
• 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 $$j\left({\sigma }_{\left(x\right)}^{\Omega }\right)\in A_x+I\text{.}$$ In other words, $$j\left({\sigma }_{\left(x\right)}^{\Omega }\right)=A_x·1$$ 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 $\beta \in {\Delta }^{\text{re}}$ $$\begin{array}{ccc}s·a& =& sa=as\\ wt·a& =& wta{w}^{-1}\\ A_{{\beta }^{\vee }}·a& =& A_{{\beta }^{\vee }}a-r_{\beta }a{A}_{{\bar{\beta }}^{\vee }}(\beta =\alpha +n\delta ,\bar{\beta }=\alpha )\\ & =& A_{{\beta }^{\vee }}a{r}_{\bar{\beta }}+a{A}_{{\bar{\beta }}^{\vee }}\\ \text{(}{A}_{{\alpha }_0^{\vee }}·{\bar{a}}_0^{\vee }& =& -\stackrel{\vee }{\theta }\text{).}\end{array}$$ The proof of this proposition is not trivial. Need calculat?????

Introduce Hopf algebra (over $S\text{)}$ structure on ${\underset{\_}{A}}_{\Omega }\text{:}$ $$\begin{array}{ccc}\pi \left(s\right)& =& s,\\ \epsilon \left(t\right)& =& 1,\\ c\left(t\right)& =& t^{-1},\\ \Delta \left(t\right)& =& t\otimes t\text{.}\end{array}$$

Theorem: The map $$\begin{array}{cccc}j:& H_T\left(\Omega K\right)& ⟶& {\underset{\_}{A}}_{\Omega }\end{array}$$ 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 $\underset{\_}{h}=\text{Spec}\hspace{0.17em}S$ 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)	$m:𝒪\left(X\right){\otimes }_S𝒪\left(X\right)\to 𝒪\left(X\right)$ 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 $\underset{\_}{h}=\text{Spec}\hspace{0.17em}S,$ 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)	$m:𝒪\left(X\right){*}_S𝒪\left(X\right)\to 𝒪\left(X\right)$ 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 $m:𝒪\left(X\right){\otimes }_S𝒪\left(X\right)\to 𝒪\left(X\right)$ 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 $𝒜=\text{Spec}\hspace{0.17em}{H}_T\left(\Omega K\right)\text{.}$ 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 $$\begin{array}{ccccccc}g:& H_T\left(\Omega K\right)& ⟶& 𝒪\left(X\right):& g\left(\sigma \right)& =& j\left(\sigma \right)·1\text{.}\end{array}$$ Conversely, given (1) and (2), the ${\underset{\_}{A}}_{\text{af}}\text{-module}$ structure on $𝒪\left(X\right)$ is defined by $$\left(j\left(\sigma \right)a\right)·p=g\left(\sigma \right)(a·p)\text{.}$$

		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 $$\begin{array}{ccc}g(a·\sigma )& =& a·g\left(\sigma \right)\text{.}\end{array}$$ Now $$\begin{array}{ccc}g(a·\sigma )& =& j(a·\sigma )·1,\\ a·g\left(\sigma \right)& =& a·j\left(\sigma \right)·1=\left(aj\left(\sigma \right)\right)·1\text{.}\end{array}$$ Thus we need to show $$(j(a·\sigma )-aj\left(\sigma \right))·1=0\in 𝒪\left(X\right)\text{.}$$ But we know that the action of $\underset{\_}{A}$ on $H_T\left(\Omega K\right)$ is characterised by the fact that $$j(a·\sigma )-aj\left(\sigma \right)\in I=\sum _{\underset{w\ne \text{id}}{w\in W}}{\underset{\_}{A}}_{\text{af}}{A}_w\text{.}$$ Since for any $i\in I,$ $$A_i·1=A_i·{\pi }_X\left(1\right)={\pi }_X(A_i·1)=0\in 𝒪\left(X\right)$$ we see that $b·1=0$ for any $b\in I\text{.}$ Thus $$(j(a·\sigma )-aj\left(\sigma \right))·1=0$$ 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 $a\in \underset{\_}{A},$ $p\in 𝒪\left(X\right)$ $$\left(j\left(\sigma \right)a\right)·p=g\left(\sigma \right)(a·p)\text{.}$$ 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 $$\begin{array}{cccccc}{H}_T\left(\Omega K\right){*}_S\underset{\_}{A}& ⟶& {\underset{\_}{A}}_{\text{af}}:& \sigma \otimes a& ⟼& j\left(\sigma \right)a\end{array}$$ is an ${\underset{\_}{A}}_{\text{af}}\text{-module}$ map. (?) In order to show $$\begin{array}{cccc}m:& 𝒪\left(X\right){\otimes }_S𝒪\left(X\right)& ⟶& 𝒪\left(X\right)\end{array}$$ is an ${\underset{\_}{A}}_{\text{af}}\text{-module}$ map, only need to show $$m(j\left(\sigma \right)·(p_1\otimes p_2))=j\left(\sigma \right)·\left(p_1{p}_2\right)\text{.}$$ But $$j\left(\sigma \right)·\left(p_1{p}_2\right)=g\left(\sigma \right)p_1{p}_2$$ and (Remark after Lemma 2 in Lecture 9 on page 9-7) $$\begin{array}{ccc}m(j\left(\sigma \right)·(p_1\otimes p_2))& =& m(j\left(\sigma \right)·p_1\otimes p_2)=m(g\left(\sigma \right)p_1\otimes p_2)\\ & =& g\left(\sigma \right)p_1{p}_2\end{array}$$ so $$m(j\left(\sigma \right)·(p_1\otimes p_2))=j\left(\sigma \right)·\left(p_1{p}_2\right)\text{.}$$ $\square$

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

In more geometrical terms, let $$𝒰=\text{Spec}\hspace{0.17em}{H}^T(K/T)\text{.}$$ We said in Lecture 6 that an integrable $\underset{\_}{A}\text{-module}$ should be thought of as an action $\varphi :𝒰{\times }_{\underset{\_}{A}}X\to X\text{.}$ In this language, an $\Omega \text{-integrable}$ ${\underset{\_}{A}}_{\text{af}}\text{-module}$ structure on $𝒪\left(X\right)$ $⟺$ pairs $(\varphi ,f)$ 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 $$j\left({\sigma }_{\left(x\right)}^{\Omega }\right)=\sum _{y\in W_{\text{af}}}{j}_x^y{A}_y\text{.}$$ In terms of the map $$\begin{array}{cccc}{j}_{\Omega }:& \Omega x& ⟶& G_{\text{af}}/B_{\text{af}}\end{array}$$ we have $$j_{\Omega }^{*}{\sigma }_{G_{\text{af}}/B_{\text{af}}}^{\left(y\right)}=\sum _{x\in W_{\text{af}}^{-}}{j}_x^y{\sigma }_{\Omega }^{\left(x\right)}\text{.}$$

Immediate properties of the polynomial $j_x^y\text{'s,}$ $x\in W_{\text{af}}^{-},$ $y\in W_{\text{af}}\text{:}$

Property 1: $$\text{deg}\hspace{0.17em}{j}_x^y=2(\ell \left(y\right)-\ell \left(x\right))\text{.}$$

This is because $$\begin{array}{ccc}\text{deg}\hspace{0.17em}\left({\sigma }_{\left(x\right)}^{\Omega }\right)& =& -2\ell \left(x\right),\\ \text{deg}\hspace{0.17em}{A}_y& =& -2\ell \left(y\right)\text{.}\end{array}$$

Property 2: $$j_x^y={\delta }_{xy}\text{if}\hspace{0.17em}y\in W_{\text{af}}^{-}\text{.}$$

Property 3: $$j_x^y=0\text{unless}y\le t\le x{w}_0\text{for some}t\in \Gamma \text{.}$$ $$\left(\begin{array}{ccc}\Rightarrow \text{deg}\hspace{0.17em}{j}_x^y& =& 2(\ell \left(y\right)-\ell \left(x\right))\le 2(\ell \left(x{w}_0\right)-\ell \left(x\right))\\ & =& 2(\ell \left(x\right)+\ell \left(w_0\right)-\ell \left(x\right))=2\ell \left(w_0\right)\end{array}\right)\text{.}$$

		Proof.
	Since $$j_{\Omega }\left({\underset{\_}{\overset{‾}{X}}}_x^{\Omega }\right)\subset {\pi }_p^{-1}\left({\underset{\_}{\overset{‾}{X}}}_x^{G_{\text{af}}/B_{\text{af}}}\right)={\underset{\_}{\overset{‾}{X}}}_{x{w}_0}^{G_{\text{af}}/B_{\text{af}}}$$ 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 $x,z\in W_{\text{af}}^{-}$ $${\sigma }_{\left(x\right)}^{\Omega }{\sigma }_{\left(z\right)}^{\Omega }=\sum _{\underset{\underset{\ell \left(y\right)+\ell \left(z\right)=\ell \left(yz\right)}{yz\in W_{\text{af}}^{-}}}{y\in W_{\text{af}}}}{j}_x^y{\sigma }_{\left(xz\right)}^{\Omega }\text{.}$$

		Proof.
	$$\begin{array}{ccc}{\sigma }_{\left(x\right)}^{\Omega }{\sigma }_{\left(z\right)}^{\Omega }& =& j\left({\sigma }_{\left(x\right)}^{\Omega }\right)·{\sigma }_{\left(z\right)}^{\Omega }\\ & =& \sum _{y\in W_{\text{af}}}{j}_x^y{A}_y·{\sigma }_{\left(z\right)}^{\Omega }\\ & =& \sum _{\underset{\underset{\ell \left(y\right)+\ell \left(z\right)=\ell \left(yz\right)}{yz\in W_{\text{af}}^{-}}}{y\in W_{\text{af}}}}{j}_x^y{\sigma }_{\left(xz\right)}^{\Omega }\text{.}\end{array}$$ $\square$

Conjecture: The $j_x^y\text{'s}$ are polynomials in the ${\alpha }_i\text{'s}$ with coefficients in ${\mathbb{Z}}_{+}=\{0,1,2,\dots \}\text{.}$

Remark 1: Can show $j_x^y\in {\mathbb{Z}}_{+}$ 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 $t\in {\Gamma }^{-}=W_{\text{af}}^{-}\cap \Gamma ,$ $${\sigma }_{\left(x\right)}^{\Omega }{\sigma }_{\left(t\right)}^{\Omega }={\sigma }_{\left(xt\right)}^{\Omega }\text{.}$$ On the other hand, since $H_T\left(\Omega K\right)$ is commutative, we have $${\sigma }_{\left(x\right)}^{\Omega }{\sigma }_{\left(t\right)}^{\Omega }={\sigma }_{\left(t\right)}^{\Omega }{\sigma }_{\left(x\right)}^{\Omega }=j\left({\sigma }_{\left(t\right)}^{\Omega }\right)·{\sigma }_{\left(x\right)}^{\Omega }\text{.}$$ It follows that, for $h$ dominant $$j\left({\sigma }_{\left(t_{-h}\right)}^{\Omega }\right)=\sum _{w\in W}{A}_{t_{-w·h}}$$ 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 $$\begin{array}{cccc}{\text{ev}}_1:& K_{\text{af}}/T& ⟶& K/T\\ & {\text{ev}}_1\left(kT\right)& =& k_{\left(1\right)}T\text{.}\end{array}$$

Proposition: For $x,y\in W_{\text{af}}^{-}$ and $w\in W,$ $$\begin{array}{ccc}{j}_x^{y\omega \left(w^{-1}\right)}& =& ⟨{\sigma }_{G_{\text{af}}/B_{\text{af}}}^{\left(y{w}_0\right)}{\text{ev}}_1^{*}({w_0}_L·{\sigma }_{G/B}^{\left(w\right)}),{\sigma }_{\left(x{w}_0\right)}^{G_{\text{af}}/B_{\text{af}}}⟩\\ & =& {\int }_{[{\underset{\_}{\overset{‾}{X}}}_{\Omega }^{y{w}_0}\cap {\underset{\_}{\overset{‾}{X}}}_{x{w}_0}^{\Omega }]}{\text{ev}}_1^{*}({w_0}_L·{\sigma }_{G/B}^{\left(w\right)})\end{array}$$ where $${\underset{\_}{\overset{‾}{X}}}_{\Omega }^{y{w}_0}=\overline{B_{\text{af}}^{-}y{w}_0·B_{\text{af}}},{\underset{\_}{\overset{‾}{X}}}_{x{w}_0}^{\Omega }=\overline{B_{\text{af}}x{w}_0·B_{\text{af}}}$$ and $$\omega \left(w\right)=w_{0}w{w}_0^{-1}$$ 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 $$\begin{array}{ccc}{H}^T(K/T)& ⟶& H(K/T)\text{.}\end{array}$$
2.	The formula for $\ell \left(w\right)=1$ will be used later to show that $$\begin{array}{ccc}{H}_{*}\left(\Omega K\right)& \simeq & q{H}^{*}(G/B)\text{.}\end{array}$$

		Proof.
	The proof uses various formulas we have proved so far. $$\begin{array}{ccc}\multicolumn{3}{c}{⟨{\sigma }_{G_{\text{af}}/B_{\text{af}}}^{\left(y{w}_0\right)}{\text{ev}}_1^{*}({w_0}_L·{\sigma }_{G/B}^{\left(w\right)}),{\sigma }_{\left(x{w}_0\right)}^{G_{\text{af}}/B_{\text{af}}}⟩}\\ & =& \epsilon ({\left(A_{x{w}_0}\right)}_R·\left({\sigma }_{G_{\text{af}}/B_{\text{af}}}^{\left(y{w}_0\right)}{\text{ev}}_1^{*}({w_0}_L·{\sigma }_{G/B}^{\left(w\right)})\right))\\ & & \text{(definition of}\hspace{0.17em}⟨\hspace{0.17em}⟩\text{)}\\ & =& \epsilon (j{\left({\sigma }_{\left(x\right)}^{\Omega }\right)}_R·A_{w_{0}R}·\left({\sigma }_{G_{\text{af}}/B_{\text{af}}}^{\left(y{w}_0\right)}{\text{ev}}_1^{*}({w_0}_L·{\sigma }_{G/B}^{\left(w\right)})\right))\\ & & \text{(}{A}_{x{w}_0}=j\left({\sigma }_{\left(x\right)}^{\Omega }\right)A_{w_0}\hspace{0.17em}\text{from Lecture}\hspace{0.17em}\text{?????}\text{)}\\ & =& \epsilon (j{\left({\sigma }_{\left(x\right)}^{\Omega }\right)}_R·\left(\sum _{v\in W}({\left(A_{w,v}\right)}_R·{\sigma }_{G_{\text{af}}/B_{\text{af}}}^{\left(y{w}_0\right)})({\left(w_0{A}_v\right)}_R·{\text{ev}}_1^{*}\text{?????})\right))\\ & & \text{(}\Delta {\lambda }_{w_0}=\sum _{v\in W}{A}_{w_{0}v}\otimes w_0{A}_v\hspace{0.17em}\text{from Lecture}\hspace{0.17em}\text{?????}\text{)}\\ & =& \epsilon (j{\left({\sigma }_{\left(x\right)}^{\Omega }\right)}_R·\left(\sum _{v\in W}{\sigma }_{G_{\text{af}}/B_{\text{af}}}^{\left(y{w}_0{v}^{-1}{w}_0\right)}{\text{ev}}_1^{*}({w_0}_R{A_v}_R{w_0}_L·{\sigma }_{G/B}^{\left(w\right)})\right))\\ & & \begin{array}{c}\ell \left(y{w}_0{v}^{-1}{w}_0\right)+\ell \left(w_{0}v\right)=\ell \left(y{w}_0\right)\\ \Updownarrow \\ \ell (w-v^{-1}{w}_0)+\ell \left(w_{0}v\right)=\ell \left(w_0\right)\\ \text{automatically satisfied.}\end{array}\begin{array}{c}\text{(Formula for}\hspace{0.17em}{A_{w_{0}v}}_R·\hspace{0.17em}\text{from Lecture 6 and}\\ \text{beginning of Lecture 7),}\hspace{0.17em}{\text{ev}}_1^{*}\hspace{0.17em}\text{comm}\text{?????}\end{array}\\ & =& \epsilon (j{\left({\sigma }_{\left(x\right)}^{\Omega }\right)}_R·\sum _{\underset{\ell \left(w{v}^{-1}\right)+\ell \left(v\right)=\ell \left(w\right)}{v\in W}}{\sigma }_{G_{\text{af}}/B_{\text{af}}}^{\left(y\omega {\left(v\right)}^{-1}\right)}{\text{ev}}_1^{*}({w_0}_R{w_0}_L·{\sigma }_{G/B}^{\left(w{v}^{-1}\right)}))\\ & =& \epsilon \left(\sum _{\underset{\ell \left(w{v}^{-1}\right)+\ell \left(v\right)=\ell \left(w\right)}{v\in W}}(j{\left({\sigma }_{\left(x\right)}^{\Omega }\right)}_R·{\sigma }_{G_{\text{af}}/B_{\text{af}}}^{\left(y\omega {\left(v\right)}^{-1}\right)}){\text{ev}}_1^{*}({w_0}_R{w_0}_L·{\sigma }_{G/B}^{\left(w{v}^{-1}\right)})\right)\\ & & \text{(}(\text{id}\otimes \text{ev})\Delta ·j\left(\sigma \right)=j\left(\sigma \right)\otimes 1\hspace{0.17em}\text{in Lecture 9)}\\ & =& \sum _{\underset{\ell \left(w{v}^{-1}\right)+\ell \left(v\right)=\ell \left(w\right)}{v\in W}}\epsilon (j{{\sigma }_{\left(x\right)}^{\Omega }}_R·{\sigma }_{G_{\text{af}}/B_{\text{af}}}^{\left(y\omega {\left(v\right)}^{-1}\right)})\epsilon \left({\text{ev}}_1^{*}({w_0}_R{w_0}_L·{\sigma }_{G/B}^{\left(w{v}^{-1}\right)})\right)\\ & & \text{(}\epsilon \hspace{0.17em}\text{is a homomorphism).}\\ & =& \sum _{\underset{\ell \left(w{v}^{-1}\right)+\ell \left(v\right)=\ell \left(w\right)}{v\in W}}\epsilon (j{{\sigma }_{\left(x\right)}^{\Omega }}_R·{\sigma }_{G_{\text{af}}/B_{\text{af}}}^{\left(y\omega {\left(v\right)}^{-1}\right)}){\delta }_{v,w}\\ & & \text{(}\epsilon \left({\text{ev}}_1^{*}({w_0}_R{w_0}_K·{\sigma }_{G/B}^{\left(w{v}^{-1}\right)})\right)=\epsilon \left({\sigma }_{G/B}^{\left(w{v}^{-1}\right)}\right)={\delta }_{v,w}\hspace{0.17em}\text{Why?)}\\ & =& \epsilon (j{\left({\sigma }_{\left(x\right)}^{\Omega }\right)}_R·{\sigma }_{G_{\text{af}}/B_{\text{af}}}^{\left(y\omega {\left(w\right)}^{-1}\right)})\\ & =& ⟨j\left({\sigma }_{\left(x\right)}^{\Omega }\right),{\sigma }_{G_{\text{af}}/B_{\text{af}}}^{\left(y\omega {\left(w\right)}^{-1}\right)}⟩\\ & =& j_x^{y\omega {\left(w\right)}^{-1}}\text{.}\end{array}$$ The fact that this is then equal to the integral is almost by definition of the Schubert basis and of the pairing $⟨\hspace{0.17em}⟩\text{.}$ $\square$

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

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

For $x\in W_{\text{af}}^{-},$ set $${\sigma }_{\left[x\right]}=ϵ\left(x\right)\hspace{0.17em}c\left({\sigma }_{\left(x\right)}^{\Omega }\right)\in H_T\left(\Omega K\right)\text{.}$$ 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: $$\nu {|}_{\Delta }=\text{id}{|}_{\Delta },\nu {|}_{{\underset{\_}{A}}_{\Omega }}=c\text{.}$$ Can check that $$\nu \left(a\right)={(-1)}^{\frac{1}{2}\text{deg}\hspace{0.17em}a}{w}_0\omega \left(a\right)w_0,a\in W_{\text{af}}$$ 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 $$\nu \left(a\right)·c\left(\sigma \right)=c(a·\sigma )\text{.}$$

Fact 1: $\forall x\in W_{\text{af}}^{-}$ $${\sigma }_{\left[x\right]}=w_0·{\sigma }_{\left(\omega \left(x\right)\right)}^{\Omega }\text{.}$$

		Proof.
	$$\begin{array}{ccc}{\sigma }_{\left[x\right]}& =& ϵ\left(x\right)c\left({\sigma }_{\left(x\right)}^{\Omega }\right)\\ & =& ϵ\left(x\right)c(A_x·1)\\ & =& ϵ\left(x\right)\nu \left(A_x\right)·1\\ & =& ϵ\left(x\right){(-1)}^{\ell \left(x\right)}{w}_0\omega \left(A_x\right)w_0·1\\ & =& ϵ\left(x\right){(-1)}^{\ell \left(x\right)}{w}_0{A}_{\omega \left(x\right)}·1\\ & =& w_0·\left({\sigma }_{\left(\omega \left(x\right)\right)}^{\Omega }\right)\text{.}\end{array}$$ $\square$

Fact 2: For $x\in W_{\text{af}},$ $y\in W_{\text{af}}^{-}$ $$\nu \left(A_x\right)·{\sigma }_{\left[y\right]}=\{\begin{array}{cc}ϵ\left(x\right){\sigma }_{\left[xy\right]}& \text{if}\hspace{0.17em}xy\in W_{\text{af}}^{-},\ell \left(x\right)+\ell \left(y\right)=\ell \left(xy\right),\\ 0& \text{otherwise.}\end{array}$$

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

Fact 3: For $t\in {\Gamma }^{-},$ $x,z\in W_{\text{af}}^{-}$ $${\sigma }_{\left[t\right]}={\sigma }_{\left(\omega \left(t\right)\right)}^{\Omega }\text{.}$$

Fact 4: For $x,z\in W_{\text{af}}^{-}$ $${\sigma }_{\left[x\right]}{\sigma }_{\left[z\right]}\sum _{\underset{\underset{\ell \left(y\right)+\ell \left(z\right)=\ell \left(yz\right)}{yz\in W_{\text{af}}^{-}}}{y\in W_{\text{af}}}}ϵ\left(xy\right)j_x^y{\sigma }_{\left[yz\right]}\text{.}$$

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 $$\sigma m=j\left(\sigma \right)·m\text{.}$$ Since $M$ is ${\underset{\_}{A}}_{\text{af}}\text{-stable,}$ $\Rightarrow$ $j\left(\sigma \right)·m\in M$ $\Rightarrow$ $\sigma m\in M\text{.}$ Hence $M\subset H_T\left(\Omega K\right)$ is an ideal. Now for $i\in I$ and $m\in M,$ $$\begin{array}{ccc}{A}_{i}j\left(m\right)& =& j\left(m\right)A_i+j(A_i·m)r_i\\ \Rightarrow \underset{\_}{A}j\left(m\right)& \subset & j\left(M\right)\underset{\_}{A}\text{.}\end{array}$$ Also have $$\begin{array}{ccc}j\left(m\right)A_i& =& A_{i}j\left(m\right)-r_{i}j(A_i·m)\\ \Rightarrow j\left(M\right)\underset{\_}{A}& \subseteq & \underset{\_}{A}j\left(M\right)\\ \Rightarrow j\left(M\right)\underset{\_}{A}& =& \underset{\_}{A}j\left(M\right).\end{array}$$ 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 $$K\left(\beta \right)=\sum _{\underset{x·\beta <0}{x\in W_{\text{af}}^{-}}}s{\sigma }_{\left[x\right]}\text{.}$$ Since $\ell \left(zx\right)=\ell \left(z\right)+\ell \left(x\right)$ and $x·\beta <0$ $\Rightarrow$ $\left(zx\right)·\beta <0,$ 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 $qH(G/B)\text{.}$

Future Lectures:

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

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.

page history