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: 19 December 2013

Lecture 16: April 22, 1997

Recall last time:

• Defined $q{H}^{*}(G/P)$ from $$\begin{array}{cccc}{J}_{n,\tau }:& {\otimes }^{n-1}{H}^{*}(G/P)& ⟶& H^{*}(G/P)\text{.}\end{array}$$
• $\sigma *\sigma \prime$ from $J_{3,\tau }\text{'s.}$
• Gave formula for ${\sigma }^{r_i}*$ in $q{H}^2(G/B)\text{.}$
• Comparison of $q{H}^{*}(G/B)$ and $q{H}^{*}(G/P)\text{.}$

Today: Compare $R_P$ and $q{H}^{*}(G/P)\text{:}$

Theorem 1: We have an isomorphism $$\begin{array}{ccc}\mathbb{Z}{\otimes }_S{R}_P& \cong & q{H}^{*}(G/P)\\ 1\otimes q_{\tau }{\sigma }_P^{\left(w\right)}& ⟼& q_{\tau }{\sigma }_P^w\text{.}\end{array}$$

		Proof.
	First the case of $G/B\text{:}$ The map $1\otimes q_{\tau }{\sigma }_B^{\left(w\right)}\to q_{\tau }{\sigma }_B^w$ is bijective. Since both sides are generated by $H^2$ and there is no torsion remains to check formula for multiplications by $H^2$ on each side. We write these formulas down in Lectures 14 and 15. For any $P,$ use the commutative diagram given at the end of Lecture 15. $\square$

Theorem 2: We have an isomorphism $$\begin{array}{ccc}{R}_P{\otimes }_{{\Lambda }_P}\mathbb{Z}& \stackrel{\sim }{⟶}& H^T(G/P)\\ s{\sigma }_P^{\left(w\right)}\otimes 1& ⟼& s{\sigma }_P^{\left(w\right)}\text{.}\end{array}$$

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

We have the commutative diagram $$\begin{array}{c}{R}_P\\ \begin{array}{cc}\swarrow & \searrow \\ q{H}^{*}(G/P)& H^T(G/P)\\ {}_{q=0}\searrow & {\swarrow }_{s=0}\end{array}\\ H^{*}(G/P)\end{array}$$

From now on, will denote $$\begin{array}{ccc}{R}_P& =& q{H}^T(G/P),\\ R_P^{\prime }& =& q{H}^T{(G/P)}_{\left(q\right)}\text{(quantum cohomology with the quantum parameter inverted).}\end{array}$$

The homomorphism ${\psi }_P\text{:}$

$$\begin{array}{ccc}{H}_T\left(\Omega K\right)& \stackrel{{\psi }_P}{⟶}& q{H}^T{(G/P)}_{\left(q\right)}\\ \downarrow & & \downarrow \\ H_{*}\left(\Omega K\right)& \stackrel{{\bar{\psi }}_P}{⟶}& qH{(G/P)}_{\left(q\right)}\end{array}$$ When $P=B,$ ${\bar{\psi }}_P$ is an isomorphism if we also invert the tranlational elements in $H_{*}\left(\Omega K\right)\text{.}$ Using ${\bar{\psi }}_P,$ we get structure constants for $H_{*}\left(\Omega K\right)$ as Gromov-Witten invariants, which, since they are numbers of certain curves, are non-negative integers.

Results of Bott:

For certain $h\in Q^{\vee },$ construct $K/T\stackrel{{\varphi }_h}{\hookrightarrow }\Omega K$ by $$\begin{array}{ccc}{\varphi }_h{\left(kT\right)}^{\left(t\right)}& =& t^{h}k{t}^{-h}{k}^{-1}\end{array}$$

• $\exists$ $h$ s.t. $\text{Im}\hspace{0.17em}{{\varphi }_h}_{*}\left(H_{•}(K/T)\right)$ generates $H_{*}\left(\Omega K\right)\text{.}$
• Can find $\text{Im}\hspace{0.17em}\left[\text{Prim}\hspace{0.17em}{H}^{*}\left(\Omega K\right)\right]$ in $H^{*}(K/T)\text{.}$
• Related to $H^T(K/T){\otimes }_S{H}_T\left(\Omega K\right)\to H_T\left(\Omega K\right)$ or $\text{Spec}\hspace{0.17em}{H}^T(K/T){\times }_{\underset{\_}{h}}\text{Spec}\hspace{0.17em}{H}_T\left(\Omega K\right)\leftarrow \text{Spec}\hspace{0.17em}{H}_T\left(\Omega K\right)\text{.}$

Geometrical Models

Will construct geometrical models for

• the groupoid scheme $𝒰=\text{Spec}\hspace{0.17em}{H}^T(G/B)$ (finite $G\text{);}$
• scheme ${𝒰}_{G/P}=\text{Spec}\hspace{0.17em}{H}^T(G/P)$ with a groupoid $𝒰\text{-action;}$
• group schemes: $$\begin{array}{ccc}\widehat{𝒜}& =& \text{Spec}\hspace{0.17em}{H}_T\left(\Omega K\right)\\ 𝒜& =& \text{Spec}\hspace{0.17em}{H}_T\left({\Omega }_{0}K\right)\end{array}$$ (do not assume $G$ is simply connected);
• $$\begin{array}{ccc}q𝒰& =& \text{Spec}\hspace{0.17em}q{H}^T(G/B),\\ q{𝒰}_{G/P}& =& \text{Spec}\hspace{0.17em}q{H}^T(G/P)\\ {\left(q{𝒰}_{G/P}\right)}_q& =& \text{Spec}\hspace{0.17em}q{H}^T{(G/P)}_q\text{.}\end{array}$$ All equipped with groupoid $𝒰\text{-actions;}$
• The variety $𝒴$ (used to be denoted by $Y\text{)}$
  • It is a projective scheme over $\underset{\_}{h}$ "with pieces $q{𝒰}_{G/P}\text{."}$
  • It has an "open piece" ${𝒴}_{\left(q\right)}$ where the $\ell$ distinguished line bundles have nonvanishing sections (?) (will explain later).
  • Has $\mathbb{Z}\text{-points}$ $Y_P\in 𝒴\left(\mathbb{Z}\right)$ for each parabolic $P\text{.}$
• ${𝔾}_m=\text{Spec}\hspace{0.17em}\mathbb{Z}[t,t^{-1}]$ acts on all and gives gradings;
• Have homomorphism $\widehat{𝒜}\to 𝒜$ as group schemes;
• $𝒜,$ as a groupoid scheme, acts on $𝒴,$ and can identify the $𝒜\text{-orbit}$ through $Y_G$ with ${𝒴}_{\left(q\right)}\text{;}$
• Have natural morphisms $$\begin{array}{cccc}{𝒰}_{G/P}& ⟶& q{𝒰}_{G/P}& \text{(corresponding to}\hspace{0.17em}q{H}^T(G/P)\stackrel{q=0}{⟶}{H}^T(G/P)\text{)}\\ q{𝒰}_{G/P}& ⟶& 𝒴& \text{imbeddings}\\ {\left(q{𝒰}_{G/P}\right)}_{\left(q\right)}& ⟶& {𝒴}_{\left(q\right)}\end{array}$$
• $$\begin{array}{ccc}q{𝒰}_{G/P}\cap q{𝒰}_{G/P\prime }& =& \varnothing \text{if}\hspace{0.17em}P\ne P\prime \end{array}$$ but $$\begin{array}{ccc}q{𝒰}_{G/P}\cap 𝒜& =& {\left(q{𝒰}_{G/P}\right)}_{\left(q\right)}\text{.}\end{array}$$ (In the Peterson lingo, "the quantum cohomology res do not see each other, but they all see the homology of $\Omega K\text{").}$

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

The first model $𝒰$ of $\text{Spec}\hspace{0.17em}{H}^T(G/B)\text{:}$

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

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

Theorem: $$\begin{array}{ccc}𝒰& =& \text{Spec}\hspace{0.17em}{H}^T(G/B)\simeq {(G/B)}^{e+\underset{\_}{h}}\\ & =& \{(e+h,x)\in (e+\underset{\_}{h})\times G/B:V_{e+h}\left(x\right)=0\}\\ & =& \{(e+h,u_{-}·B):u_{-}^{-1}·(e+h)\in \underset{\_}{b}\}\text{where the second}\hspace{0.17em}·\hspace{0.17em}\text{is the adjoint action.}\end{array}$$ But since $U_{-}$ stabilises $e+{\underset{\_}{b}}_{-},$ when $u_{-}^{-1}·(e+h)\in \underset{\_}{b},$ we have $$u_{-}^{-1}·(e+h)\in \underset{\_}{b}\cap (e+{\underset{\_}{b}}_{-})=e+\underset{\_}{h}\text{.}$$ ????? $$\begin{array}{ccc}𝒰& =& \{(e+h,u_{-}·B):u_{-}^{-1}·(e+h)\in e+\underset{\_}{h}\}\\ & =& \{(e+h,u_{-},e+h\prime ):u_{-}^{-1}·(e+h)=e+h\prime \}\text{.}\end{array}$$

The groupoid structure on $𝒰\text{:}$

• $𝒰\underset{t}{\overset{s}{\rightrightarrows }}\underset{\_}{h}:$ $(e+h,u_{-},e+h\prime )\genfrac{}{}{0ex}{}{\stackrel{s}{\mapsto }}{\underset{t}{\mapsto }}\genfrac{}{}{0ex}{}{e+h}{e+h\prime }$
• $𝒰{\times }_h𝒰\to 𝒰:$ $(e+h,u_{-},e+h\prime )·(e+h\prime ,u\prime ,e+h″)=(e+h,\text{?????})$
• $\underset{\_}{h}\to 𝒰:$ $h\mapsto (e+h,1,e+h)$ (identities)
• inverse: $𝒰\to 𝒰:$ $(e+h,u_{-},e+h\prime )\mapsto (e+h\prime ,u_{-}^{-1},e+h)$

As a model for $\text{Spec}\hspace{0.17em}{H}^T(G/B),$ we must have two $W\text{-actions}$ on $𝒰$ which gave $w_L$ and $w_R$ on $H^T(G/B)\text{.}$ We identify these two actions in the next lecture.

Lecture 17: April 23, 1997

The following works for the general Kac-Moody case:

Set $$\begin{array}{ccc}e& =& \sum _{i\in I}{e}_i\in {\underset{\_}{n}}_{+},\\ f_i^{\left(n\right)}& =& \frac{f_i^n}{n!}\in U\left({\underset{\_}{n}}_{-}\right)\text{.}\end{array}$$ Then $$\begin{array}{ccc}U{\left({\underset{\_}{n}}_{-}\right)}_{\mathbb{Z}}& =& {⟨f_i^{\left(n\right)}⟩}_{i\in I,n\ge 0}\end{array}$$ and using the action of $2{\rho }^{\vee }$ we can give $U{\left({\underset{\_}{n}}_{-}\right)}_{\mathbb{Z}}$ a $\mathbb{Z}\text{-grading}$ with $\text{deg}\hspace{0.17em}{f}_i=-2\text{.}$

Define $$\begin{array}{ccc}𝒪\left(U_{-}\right)& =& {\text{Hom}}_{\mathbb{Z}}(U{\left({\underset{\_}{n}}_{-}\right)}_{\mathbb{Z}},\mathbb{Z})\text{(graded dual)}\end{array}$$ and we use $U_{-}$ to denote the groupscheme defined by $𝒪\left(U_{-}\right)\text{.}$

Lemma: For any $w\in W,$ there exists a scheme morphism $$\begin{array}{cccc}{U}_w:& \underset{\_}{h}& ⟶& U_{-}\end{array}$$ s.t. $$\begin{array}{ccc}{U}_w\left(h\right)·(e+h)& =& e+w·h\text{.}\end{array}$$ We have $$\begin{array}{ccc}{U}_{vw}\left(h\right)& =& U_v(w·h)U_w\left(h\right)\end{array}$$ and $$U_{r_i}\left(h\right)=\text{exp}\left(⟨{\alpha }_i,h⟩f_i\right)=y_i\left(⟨{\alpha }_i,h⟩\right)$$ where, recall, ${\varphi }_i:SL(2,ℂ)\to G$ and for $u\in ℂ$ $\text{(}\mathbb{Z}\text{?)}$ $$\begin{array}{ccc}{x}_i\left(u\right)& =& {\varphi }_i\left(\begin{array}{cc}1& u\\ 0& 1\end{array}\right),\\ y_i\left(u\right)& =& {\varphi }_i\left(\begin{array}{cc}1& 0\\ u& 1\end{array}\right)\text{.}\end{array}$$

The finite case:

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

Example: For $𝔤=sl\left(3\right),$ $h=\text{diag}(x_1,x_2,x_3),$ have $$\begin{array}{ccc}\begin{array}{ccc}{U}_{r_1}\left(h\right)& =& \left(\begin{array}{ccc}1& 0& 0\\ x_1-x_2& 1& 0\\ 0& 0& 1\end{array}\right)\\ U_{r_1{r}_2{r}_1}\left(h\right)& =& \left(\begin{array}{ccc}1& 0& 0\\ x_1-x_3& 1& 0\\ (x_1-x_2)(x_1-x_3)& x_1-x_3& 1\end{array}\right)\end{array}& & \begin{array}{ccc}{u}_{r_2}\left(h\right)& =& \left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& x_2-x_3& 1\end{array}\right)\\ u_{r_2{r}_1}\left(h\right)& =& \left(\begin{array}{ccc}1& 0& 0\\ x_1-x_3& 1& 0\\ (x_1-x_2)(x_1-x_3)& x_1-x_3& 1\end{array}\right)\end{array}\end{array}$$

Fact: $U_{w_0}\left(t{\rho }^{\vee }\right)=\text{exp}\left(tf\right),$ $t\in ℂ,$ where $\{e,f,2{\rho }^{\vee }\}$ is a TDS.

The affine case:

In this case, define $U_{r_i}\left(h\right)=y_i\left(⟨{\alpha }_i,h⟩\right)$ for $i\in I_{\text{af}}$ and use $$\begin{array}{ccc}{U}_{vw}\left(h\right)& =& U_v(w·h)U_w\left(h\right)\end{array}$$ to extend to any $w\text{.}$ This is well-defined because of the braid relations: assume that $2<m_{ij}<\infty$ and $⟨{\alpha }_j,{\alpha }_i^{\vee }⟩=-1$ for $i\ne j\text{.}$ Set $a=⟨{\alpha }_i,h⟩,$ $b=⟨{\alpha }_j,h⟩\text{.}$ Then $$\begin{array}{ccc}{m}_{ij}=3\text{:}{U}_{iji}& =& U_{r_i}(r_j{r}_i·h)U_{r_j}(r_i·h)U_{r_i}\left(h\right)\\ & =& y_i\left(b\right)y_j(a+b)y_j\left(a\right)\\ U_{jij}& =& y_j\left(a\right)y_i(a+b)y_j\left(b\right)\\ \\ m_{ij}=4\text{:}{U}_{ijij}& =& y_i\left(a\right)y_j(a+b)y_i(a+2b)y_j\left(b\right)\\ U_{jiji}& =& y_j\left(b\right)y_i(a+2b)y_j(a+b)y_i\left(a\right)\\ \\ m_{ij}=6\text{:}{U}_{ijijij}& =& y_i\left(a\right)y_j(3a+b)y_i(2a+b)y_j(3a+2b)y_i(a+b)y_j\left(b\right)\\ U_{jijiji}& =& y_j\left(b\right)y_i(a+b)y_j(3a+2b)y_i(2a+b)y_j(3a+b)y_i\left(a\right)\end{array}$$ The fact that they are equal is due to Kostant's theorem $\text{(}{U}_w\left(\text{?????}\right)$ is unique). These are called universal exponential solutions to the Yang-Baxter Equations by Fomin and Kirilov in their paper in Lett. Math. Phys (1996) 273-284.

What about $m_{ij}=\infty \text{?}$ This is what is needed in the affine case?

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

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

Suppose that $U_{-}$ acts on a scheme $Z\text{.}$ Then $W$ acts on $(e+\underset{\_}{h})\times Z$ by $$\begin{array}{ccc}w\left((e+h,z)\right)& =& (e+wh,U_w\left(h\right)·z)\text{.}\end{array}$$ Assume that $Z$ is affine. Then $W$ acts on $𝒪((e+\underset{\_}{h})\times Z)\simeq S\otimes 𝒪\left(Z\right)\text{:}$ $$\begin{array}{ccc}(w·p_1)(e=h,z)& =& p_1(w^{-1}·(e+h,z)),p_1=s\otimes p\in S\otimes 𝒪\left(Z\right)\text{.}\end{array}$$

Lemma: For $s\otimes p\in S\otimes 𝒪\left(Z\right),$ $$\begin{array}{ccc}{r}_i·(s\otimes p)& =& \sum _{n\ge 0}(r_i·\left({\alpha }_i^{n}s\right))\otimes f_i^{\left(n\right)}·p\text{.}\end{array}$$

		Proof.
	By definition, $$\begin{array}{ccc}{r}_i·(s\otimes p)(e+h,z)& =& (s\otimes p)(r_i·(e+h,z))\\ & =& (s\otimes p)(e+r_i·h,U_{r_i}\left(h\right)·z)\\ & =& s(r_i·h)p(U_{r_i}\left(h\right)·z)\\ & =& (r_i·s)\left(h\right)(U_{r_i}{\left(h\right)}^{-1}·p)\left(z\right)\\ & =& (r_i·s)\left(h\right)\left(\text{exp}(-⟨{\alpha }_i,h⟩f_i·p)\left(z\right)\right)\\ & =& (r_i·s)\left(h\right)(\sum _{n\ge 0}\frac{-{⟨{\alpha }_i,h⟩}^n}{n!}{f}_i^n·p)\left(z\right)\\ & =& \sum _{n\ge 0}({(-{\alpha }_i)}^n{r}_i·s)\left(h\right)(f_i^{\left(n\right)}·p)\left(z\right)\\ & =& \sum _{n\ge 0}{r}_i·\left({\alpha }_i^{n}s\right)\left(h\right)(f_i^{\left(n\right)}·p)\left(z\right)\\ \Rightarrow \hspace{0.17em}{r}_i·(s\otimes p)& =& \sum _{n\ge 0}(r_i·\left({\alpha }_i^{n}s\right))\otimes f_i^{\left(n\right)}·p\text{.}\end{array}$$ $\square$

Consequently, we get an integrable $\underset{\_}{A}\text{-module}$ structure on $S\otimes 𝒪\left(Z\right)$ by $$\begin{array}{ccc}{A}_i·(s\otimes p)& =& \frac{1}{{\alpha }_i}(1-r_i)·(s\otimes p)\\ & =& (A_i·s)\otimes p+\sum _{n\ge 1}{r}_i·\left({\alpha }_i^{n-1}s\right)\otimes f_i^{\left(n\right)}·p\text{.}\end{array}$$ For each $p\in 𝒪\left(Z\right),$ this is a finite sum.

The groupoid scheme $𝒰\prime =(e+\underset{\_}{h})\times U_{-}\text{:}$

Define $$\begin{array}{ccccccc}{P}_L=p_1:& 𝒰\prime & ⟶& e+\underset{\_}{h}:& (e+h,u)& ⟼& e+h\end{array}$$ and $$\begin{array}{ccccccc}{P}_R:& 𝒰\prime & ⟶& e+\underset{\_}{h}:& (e+h,u)& ⟼& \text{proj. of}\hspace{0.17em}{u}^{-1}·(e+h)\\ & & & & & & \text{to}\hspace{0.17em}e+\underset{\_}{h}\hspace{0.17em}\text{in}\\ & & & & & & e+{\underset{\_}{b}}_{-}=e+\underset{\_}{h}+n_{-}\text{.}\end{array}$$ These are the source and target maps for the groupoid struture on $𝒰\prime \text{.}$ Other structure maps: $$\begin{array}{ccccccc}\text{identities}\hspace{0.17em}i\prime :& e+\underset{\_}{h}& \hookrightarrow & 𝒰\prime :& e+h& ⟼& (e+h,1)\\ \text{multiplication:}\hspace{0.17em}\mu \prime :& 𝒰\prime {\times }_{e+\underset{\_}{h}}𝒰\prime & ⟶& 𝒰\prime :& (e+h,u)·(e+h\prime ,u\prime )& =& (e+h,u\prime )\\ & & & & \multicolumn{3}{c}{\text{if}\hspace{0.17em}{P}_R(e+h,u)=e+h\prime =P_L(e+h\prime ,u\prime )\text{.}}\\ \text{inverse:}\hspace{0.17em}\iota :& 𝒰\prime & ⟶& 𝒰\prime :& (e+h,u)& ⟼& (P_R(e+h,u),u^{-1})\text{.}\end{array}$$ The idea now is to embed $𝒰$ as a subgroupoid scheme of $𝒰\prime \text{.}$ Here the groupoidscheme structure on $𝒰$ is the one defined in Lecture 5. To this end, we use the integrable $\underset{\_}{A}\text{-module}$ structure on $𝒰\prime \text{.}$

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

Consider the $W_L$ action on $(e+\underset{\_}{h})\times U_{-}\text{:}$ $$\begin{array}{ccc}{w}_L·(e+h,u)& =& (e+wh,U_w\left(h\right)u)\text{.}\end{array}$$ It satisfies $$\begin{array}{ccc}{P}_R·w_L& =& P_R\end{array}$$ by the definition of $U_w\text{.}$ By the discussion in Lecture 17, we have an integrable ${\underset{\_}{A}}_L\text{-module}$ structure on $𝒪\left(𝒰\prime \right)\text{.}$ In other words we have a groupoid action $$\begin{array}{cccc}\varphi :& 𝒰{\times }_{\underset{\_}{h}}𝒰\prime & ⟶& 𝒰\prime \\ & P_R\circ p_2\downarrow \phantom{P_R\circ p_2}& & \phantom{P_R}\downarrow P_R\\ & e+\underset{\_}{h}& \simeq & e+\underset{\_}{h}\end{array}$$ Also have $$\begin{array}{ccc}𝒰{\times }_{\underset{\_}{h}}𝒰\prime {\times }_{\underset{\_}{h}}𝒰\prime & \stackrel{\text{id}\times \mu \prime }{⟶}& 𝒰{\times }_{\underset{\_}{h}}𝒰\prime \\ \varphi \times \text{id}\downarrow \phantom{\varphi \times \text{id}}& & \phantom{\varphi }\downarrow \varphi \\ 𝒰\prime {\times }_{\underset{\_}{h}}𝒰\prime & \stackrel{\mu \prime }{⟶}& 𝒰\prime \end{array}$$ where $\mu \prime :𝒰\prime {\times }_h𝒰\prime \to 𝒰\prime$ is the multiplication morphism for $𝒰\prime \text{.}$ These imply that the following composition is a morphism of groupoid schemes over $\underset{\_}{h}\text{:}$ $$\begin{array}{ccccccc}𝒰& =& 𝒰{\times }_{\underset{\_}{h}}\underset{\_}{h}& \stackrel{\text{id}\times i\prime }{⟶}& 𝒰{\times }_{\underset{\_}{h}}𝒰\prime & \stackrel{\varphi }{⟶}& 𝒰\prime \end{array}$$ where $i\prime :\underset{\_}{h}& \hookrightarrow & 𝒰\prime$ is the identity morphism for $𝒰\prime \text{.}$

The groupoid isomoprhism $𝒰\prime \simeq 𝒰\prime {\times }_{e+{\underset{\_}{b}}_{-}}(e+\underset{\_}{h})$

Define $$\begin{array}{ccccccccc}{P}_R^{\prime }:& 𝒰\prime & ⟶& e+{\underset{\_}{b}}_{-}:& (e+h,u)& ⟼& u^{-1}·(e+h)& \in & e+{\underset{\_}{b}}_{-}\text{.}\end{array}$$ Form $$\begin{array}{ccc}𝒰\prime {\times }_{e+{\underset{\_}{b}}_{-}}(e+\underset{\_}{h})& ≔& 𝒰″\end{array}$$ using $P_R^{\prime }$ and $e+\underset{\_}{h}\hookrightarrow e+{\underset{\_}{b}}_{-}$ (the inclusion). We think of $𝒰\prime {\times }_{e+{\underset{\_}{b}}_{-}(e+\underset{\_}{h})}=𝒰″$ as the subset of $𝒰\prime \text{:}$ $$\{(e+h,u):u^{-1}·(e+h)\in e+h\}\text{.}$$ We claim that the morphism $𝒰\to 𝒰\prime$ factors through $𝒰″\text{.}$ To prove this, we look at $$\begin{array}{ccccc}𝒪\left(𝒰\prime \right)& ⟶& 𝒪\left(𝒰\right)& \simeq & H^T(G/B)\text{.}\end{array}$$ For each $w\in W,$ recall that we have ${\psi }_w:H^T(G/B)\to S\text{.}$

The map $$\begin{array}{ccccccc}𝒪\left(𝒰\prime \right)& ⟶& 𝒪\left(𝒰\right)& \simeq & H^T(G/B)& \stackrel{{\psi }_w}{⟶}& S\end{array}$$ corresponds to the scheme morphism $$\begin{array}{cccccc}\underset{\_}{h}& ⟶& 𝒰\prime :& h& ⟼& (e+h,U_{w^{-1}}{\left(h\right)}^{-1})\text{.}\end{array}$$ Since $${\left(U_{w^{-1}}{\left(h\right)}^{-1}\right)}^{-1}·(e+h)=U_{w^{-1}}\left(h\right)·(e+h)=e+w^{-1}·h$$ we see that $$(e+h,U_{w^{-1}}{\left(h\right)}^{-1})\in 𝒰″\text{.}$$ Since $\{{\psi }_w:w\in W\}$ is a basis for ${\text{Hom}}_S(H^T(K/T),S),$ we conclude that the morphism $𝒰\to 𝒰\prime$ factors through $𝒰″$ to give $$\begin{array}{ccc}𝒰& ⟶& 𝒰″\text{.}\end{array}$$

Theorem: $$\begin{array}{ccc}𝒰& \stackrel{\sim }{⟶}& 𝒰″\end{array}$$ as groupoid schemes over $\underset{\_}{h}\text{.}$

Lecture 18: April 29, 1997

Last time we had morphisms of groupoid schemes over $\underset{\_}{h}$ $$\begin{array}{c}\text{Spec}\hspace{0.17em}{H}^T(K/T)=𝒰⟶(e+\underset{\_}{h})\times {𝒰}_{-}≕𝒰\prime \\ \begin{array}{c} ⤏ \end{array}\nearrow \\ [(e+\underset{\_}{h})\times {𝒰}_{-}]{\times }_{e+{\underset{\_}{b}}_{-}}(e+\underset{\_}{h})=𝒰″\end{array}$$

Consider the corresponding ring homomorphism $$\begin{array}{cc}\begin{array}{ccccccc}𝒪\left(𝒰\prime \right)& =& S\otimes 𝒪\left({𝒰}_{-}\right)& ⟶& 𝒪\left(U\right)& =& H^T(K/T)\text{.}\end{array}& (*)\end{array}$$

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

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

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

Consider the quotient of $𝒰{\left({\underset{\_}{n}}_{-}\right)}_{\mathbb{Z}}$ by the 2-sided ideal generated by $\left\{f_i^{\left(m\right)}\hspace{0.17em}\right|\hspace{0.17em}i\in I,\hspace{0.17em}m\ge 2\}\text{.}$ The resulting ring with identity $𝒰{\left({\underset{\_}{n}}_{-}\right)}_{\mathbb{Z}}/⟨f_i^{\left(2\right)};i\in I⟩$ is given by generators $\left\{f_i\hspace{0.17em}\right|\hspace{0.17em}i\in I\}$ and relations: $$f_i{f}_i=0,f_i{f}_j{f}_i=0\text{if}\hspace{0.17em}{a}_{ij}=-1,\text{and}{f}_i{f}_j=f_j{f}_i\text{if}\hspace{0.17em}{a}_{ij}=0\text{.}$$ For $G^{\vee }\text{-abelian}$ $w$ with reduced expression $r_{i_1}\cdots r_{i_N},$ put $$\begin{array}{ccc}{f}_w& =& f_{i_1}\cdots f_{i_N}\text{.}\end{array}$$ These $f_w$ define a basis of $𝒰{\left({\underset{\_}{n}}_{-}\right)}_{\mathbb{Z}}/⟨f_i^{\left(2\right)};i\in I⟩\text{.}$ The dual basis gives us elements in $𝒪\left({𝒰}_{-}\right)\text{:}$ $$f_w^{*}\in \text{Hom}(𝒰{\left({\underset{\_}{n}}_{-}\right)}_{\mathbb{Z}}/⟨f_i^{\left(2\right)};i\in I⟩,\mathbb{Z})\subset \text{Hom}(𝒰{\left({\underset{\_}{n}}_{-}\right)}_{\mathbb{Z}},\mathbb{Z})=𝒪\left({𝒰}_{-}\right)\text{.}$$

Claim: Under the homomorphism $(*)$ $$\begin{array}{ccc}S\otimes 𝒪\left({𝒰}_{-}\right)& ⟶& H^T(G/B),\\ 1\otimes f_w^{*}& ⟼& {\sigma }_{G/B}^{\left(w\right)}\text{.}\end{array}$$

		Proof.
	Write $f_w^{*}$ for $1\otimes f_w^{*}\text{.}$ The statement is clear for the identity elements: $f_1^{*}\mapsto {\sigma }_{G/B}^{\left(1\right)}\text{.}$ Suppose $r_{i}w\le w\text{.}$ Then $r_{i}w$ is again $G^{\vee }\text{-abelian,}$ and we have $$(r_i·f_w^{*})(h,u)=f_w^{*}\left(u_{r_i}\left(h\right)u\right)={\alpha }_i\left(h\right)f_{r_{i}w}^{*}\left(u\right)+f_w^{*}\left(u\right)\text{.}$$ Therefore $$\begin{array}{ccc}{r}_i{f}_w^{*}& =& {\alpha }_i{f}_{r_{i}w}^{*}+f_w^{*},\\ A_i·f_w^{*}& =& -f_{r_{i}w}^{*}\text{.}\end{array}$$ Similarly, $A_j·f_w^{*}=0$ if $w\le r_{j}w\text{.}$ Define $x\in H^T(G/B)$ by $$\begin{array}{ccc}{f}_w^{*}& ⟼& {\sigma }_{G/B}^{\left(w\right)}+x\text{.}\end{array}$$ Then $$\begin{array}{ccc}{A}_i·f_w^{*}& ⟼& A_i{\sigma }_{G/B}^{\left(w\right)}+A_{i}x\text{.}\end{array}$$ We can assume by induction that $f_{r_{i}w}\mapsto {\sigma }_{G/B}^{\left(r_{i}w\right)}$ whenever $r_{i}w\le w\text{.}$ Therefore $A_i·x=0$ in this case. Also $A_j·x=0$ for $r_{j}w\ge w,$ by the above. So $x=0\text{.}$ $\square$

Miniscule representations

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

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

Let $V=V\left(\lambda \right)$ be a miniscule representation of $G$ with highest weight $v^{+}\in V\left(\lambda \right)\text{.}$ The stabilizer of the $\lambda$ weight space is the parabolic subgroup $P=P_{\lambda }=B{W}_{\lambda }B$ (where $W_{\lambda }$ is the stabilizer of $\lambda$ in $W\text{).}$ The weights of $V\left(\lambda \right)$ are precisely $\{w·\lambda \hspace{0.17em}|\hspace{0.17em}w\in W^P\}\text{.}$

Lemma: All $w\in W^P,$ for $P$ as above, are $G^{\vee }\text{-abelian,}$ and $\{v_w=f_w·v^{+}\hspace{0.17em}|\hspace{0.17em}w\in W^P\}$ gives a basis of $V\left(\lambda \right)\text{.}$

		Proof.
	$W^P$ is characterised as $$\begin{array}{ccc}{W}^P& =& \{w\in W\hspace{0.17em}|\hspace{0.17em}\alpha \in {\varphi }^{+},\hspace{0.17em}w,{\alpha }^{\vee }<0\hspace{0.17em}\Rightarrow \hspace{0.17em}⟨\lambda ,{\alpha }^{\vee }⟩=1\}\text{.}\end{array}$$ Therefore ${𝒰}_{-}^{\vee }\cap w{B}^{\vee }{w}^{-1}$ (for $w\in W^P\text{)}$ is generated by $1\text{-parameter}$ subgroups ${𝒰}_{-{\alpha }^{\vee }}^{\vee }=\text{exp}\hspace{0.17em}{𝔤}_{-{\alpha }^{\vee }}^{\vee }$ for which $⟨\lambda ,{\alpha }^{\vee }⟩=1\text{.}$ Any two such subgroups ${𝒰}_{-\alpha }^{\vee },{𝒰}_{-\beta }^{\vee }$ commute, since $⟨\lambda ,{\alpha }^{\vee }+{\beta }^{\vee }⟩=2$ and thus ${\alpha }^{\vee }+{\beta }^{\vee }$ is not a root of ${𝔤}^{\vee }$ (by condition (2) for miniscule $\lambda \text{).}$ So $w$ is $G^{\vee }\text{-abelian.}$ That $f_w·v^{+}\in V_{w·\lambda }$ is proved inductively. Let $w=r_{i}w\prime$ with $\ell \left(w\right)=\ell \left(w\prime \right)+1\text{.}$ Then $w\prime \in W^P$ and $$f_w·v^{+}=f_i{f}_{w\prime }·v^{+}\in V_{w\prime ·\lambda -{\alpha }_i}\text{.}$$ On the other hand $r_{i}w\prime ·\lambda =w\prime ·\lambda -⟨{\alpha }_i^{\vee },w\prime ·\lambda ⟩{\alpha }_i\text{.}$ We have $⟨{\alpha }_i^{\vee },w\prime ·\lambda ⟩=⟨{\left(w\prime \right)}^{-1}{\alpha }_i^{\vee },\lambda ⟩=1,$ since $r_{i}w\prime =w$ lies in $W^P$ and takes the positive weight ${\left(w\prime \right)}^{-1}{\alpha }_i^{\vee }$ to $-{\alpha }_i^{\vee }\text{.}$ Thus $w·\lambda =w\prime ·\lambda -{\alpha }_i$ and $f_w·v^{+}\in V_{w·\lambda }$ (and $f_w·v^{+}$ is nonzero). $\square$

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

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

Example: Consider the standard representation $V\left({\rho }_1\right)$ of ${SL}_3\text{.}$ It is clearly miniscule. The homomorphism $𝒪\left({𝒰}_{-}\right)\to H^T(G/B)$ gives rise to the "tautological" element $$u=\left(\begin{array}{ccc}1& & \\ {\sigma }_{G/B}^{\left(r_1\right)}& 1& \\ {\sigma }_{G/B}^{\left(r_2{r}_1\right)}& {\sigma }_{G/B}^{\left(r_2\right)}& 1\end{array}\right)\in {𝒰}_{-}\left(H^T(G/B)\right)\text{.}$$ Similarly the structure maps ${\pi }_L$ and ${\pi }_R:𝒪\left(\underset{\_}{h}\right)\to H^T(G/B)$ correspond to $$h_L=\left(\begin{array}{ccc}{\pi }_L\left({\rho }_1\right)& & \\ & {\pi }_L({\rho }_2-{\rho }_1)& \\ & & {\pi }_L(-{\rho }_2)\end{array}\right),h_R=\left(\begin{array}{ccc}{\pi }_R\left({\rho }_1\right)& & \\ & {\pi }_R({\rho }_2-{\rho }_1)& \\ & & {\pi }_R(-{\rho }_2)\end{array}\right)$$ in $\underset{\_}{h}\left(H^T(G/B)\right)\text{.}$

Then the following relation holds. $$\begin{array}{ccc}\left(\begin{array}{ccc}1& & \\ {\sigma }_{G/B}^{\left(r_1\right)}& 1& \\ {\sigma }_{G/B}^{\left(r_2{r}_1\right)}& {\sigma }_{G/B}^{\left(r_2\right)}& 1\end{array}\right)·(e+h_L)& =& e+h_R\text{.}\end{array}$$ This implies the factorization $$\begin{array}{c}𝒪((e+\underset{\_}{h})\times {𝒰}_{-})⟶H^T\left(K_T\right)\\ \begin{array}{c} ⤏ \end{array}\nearrow \\ 𝒪\left(𝒰″\right)\end{array}$$ from before expliticly.

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

There is also a similar construction for $H^{*}\left(\Omega K\right)\text{.}$ It will be shown later that $H^{*}\left(\Omega K\right)\cong 𝒰\left({\underset{\_}{n}}_{+}^e\right)\text{.}$ Therefore one can apply it to the lowest weight vector $v^{-}$ of a representation $V$ to obtain a subspace of that representation. If $V$ is miniscule we again recover all of $V$ (in types ADE). This is seen as follows.

Let ${\underset{\_}{s}}_{+}$ be the centralizer in ${\underset{\_}{n}}_{+}^{\text{af}}$ of $e+e_0=e+t{e}_{-\theta }\text{.}$ Any representation of $G_{\text{af}}$ with miniscule highest weight ${\rho }_i$ is isomorphic to $V\left({\rho }_0^{\text{af}}\right),$ the representation with highest weight ${\rho }_0$ (since there is an admissible graph automorphism of the extended Dynkin diagram taking the vertex $i$ to the $0$ vertex). We have the following commutative diagram $$\begin{array}{ccccc}{V}^{*}\left({\rho }_i\right)& \twoheadleftarrow & V^{*}\left({\rho }_2^{\text{af}}\right)& \simeq & V^{*}\left({\rho }_0^{\text{af}}\right)\\ \uparrow & & & & \uparrow ·v^{-}\\ 𝒰\left({\underset{\_}{n}}_{+}^e\right)& \multicolumn{3}{c}{\stackrel{{\text{ev}}_0}{⟵}}& 𝒰\left({\underset{\_}{s}}_{+}\right)\end{array}$$ By a theorem in the Kac-Moody case, the map $𝒰\left({\underset{\_}{s}}_{+}\right)\to V^{*}\left({\rho }_0^{\text{af}}\right)$ on the right hand side is bijective. Hence the composition is surjective and so is $𝒰\left({\underset{\_}{n}}_{+}^e\right)\to V^{*}\left({\rho }_i\right)\text{.}$

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

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

		Proof.
	Suppose $g\in Z_G\left(e\right)\text{.}$ Then, by the Bruhat decomposition, $g=b_1\cap b_2$ for $b_1,b_2\in B$ and $n\in N_G\left(T\right)\text{.}$ We have $b_1\cap b_2·e=e,$ hence $$\begin{array}{ccc}n{b}_2·e& =& b_1^{-1}·e\text{.}\end{array}$$ Let $w\in W$ be the Weyl group element represented by $n\text{.}$ Then the left hand side of the above equation lies in the sum of weight spaces $\underset{\alpha \in w·{\Delta }_{+}}{⨁}{g}_{\alpha },$ while the right hand side has nonzero components in all the ${𝔤}_{{\alpha }_i},$ for ${\alpha }_i\in \Pi \text{.}$ Thus $\Pi \subseteq w·{\Delta }_{+},$ which implies that $w=\text{id.}$ $\square$

Consider the morphism $$\begin{array}{cccc}\varphi :& (e+\underset{\_}{h})\times {𝒰}_{-}& ⟶& e+{\underset{\_}{b}}_{-}\\ & (e+h,u)& ⟼& u^{-1}·(e+h)\text{.}\end{array}$$ Let $X≔(e+\underset{\_}{h})\times {𝒰}_{-}$ and $Y≔e+{\underset{\_}{b}}_{-}\text{.}$ Then $𝒪\left(X\right)$ and $𝒪\left(Y\right)$ are graded polynomial rings over $\mathbb{Z}$ in $N=\text{\#}{\Delta }_{+}$ generators, where the grading is given as follows. For $𝒪\left(X\right)=S\otimes 𝒪\left({𝒰}_{-}\right)$ let $S$ be graded as usual by $\text{deg}\hspace{0.17em}{\underset{\_}{h}}^{*}=2,$ and $𝒪\left({𝒰}_{-}\right)$ by $\text{deg}\hspace{0.17em}{𝔤}_{-\alpha }^{*}=2\text{ht}\left(\alpha \right)\text{.}$ The grading on $𝒪\left(Y\right)=𝒪\left({\underset{\_}{b}}_{-}\right)$ is given by $\text{deg}\hspace{0.17em}{𝔤}_{-\alpha }^{*}=2(\text{ht}\left(\alpha \right)+1)\text{.}$ Then we get that $$\begin{array}{cccc}{\varphi }^{*}:& 𝒪\left(Y\right)& ⟶& 𝒪\left(X\right)\end{array}$$ is a homomorphism of graded polynomial rings. Choose homogeneous generators of $𝒪\left(Y\right)$ and $𝒪\left(X\right)\text{.}$ So $𝒪\left(Y\right)=\mathbb{Z}[y_1,\dots ,y_N]$ and $𝒪\left(X\right)=\mathbb{Z}[x_1,\dots ,x_N]\text{.}$

Lemma: ${\varphi }^{*}\left(y_1\right),\dots ,{\varphi }^{*}\left(y_N\right)$ form a regular sequence in $𝒪\left(X\right)\otimes 𝔽\text{.}$

		Proof.
	Let $𝒮=⟨{\varphi }^{*}\left(y_1\right),\dots ,{\varphi }^{*}\left(y_N\right)⟩\text{.}$ Since the ${\varphi }^{*}\left(y_i\right)$ are homogeneous elements in a graded ring it suffices to show that the depth of $𝒮$ (or equivalently $\sqrt{𝒮}\text{)}$ equals $N\text{.}$ The following claim will imply that $\sqrt{𝒮}=⟨x_1,\dots ,x_N⟩$ and hence this lemma. Claim: Let $h\in \underset{\_}{h}\left(𝔽\right)$ and $u\in {𝒰}_{-}\left(𝔽\right),$ then $$\begin{array}{ccc}{u}^{-1}·(e+h)=e& \Rightarrow & u=1\text{.}\end{array}$$ Proof. Consider the semisimple part of $u·e=e+h\text{.}$ Since the semisimple part of $e$ is zero it must be zero. On the other hand it must be conjugate to $h\text{.}$ Hence $h=0_{*}$ and $u·e=e\text{.}$ So $u\in Z_G\left(e\right)\left(𝔽\right)$ which by a previous lemma is contained in $B\left(𝔽\right)\text{.}$ Therefore $u=1\text{.}$ $\square$ $\square$

We aim to prove the following

Theorem: The map $$\begin{array}{ccc}𝒪\left(((e+\underset{\_}{h})\times {𝒰}_{-}){\times }_{e+\underset{\_}{b}}(e+\underset{\_}{h})\right)& ⟶& H^T(G/B)\end{array}$$ is an isomorphism.

Notes and references

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

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