Affine Schubert Calculus

Abstract

This chapter discusses how k-Schur and dual k-Schur functions can be defined for all types. This is done via some combinatorial problems that come from the geometry of a very large family of generalized flag varieties. They apply to the expansion of products of Schur functions, k-Schur functions and their dual basis, and Schubert polynomials. Despite the geometric origin of these problems, concrete algebraic models will be given for the relevant cohomology rings and their Schubert bases.

Appendices

A Appendix: Proof of Coalgebra Properties

Let M and N be left F _W -modules. Then M, N, M ⊗_F N, and Hom_F(M, N) are left F-modules. We define an F _W -module structure on M ⊗_F N and Hom_F(M, N).

Proof of Proposition 3.14.

We first check the well-definedness of the formula for a ⋅ m ⊗ n. Let a ∈ F _W with $\Delta (a) =\sum _{(a)}a_{(1)} \otimes a_{(2)}$. We further expand $a_{(k)} =\sum _{w}a_{(k)}^{w}w$ for k = 1, 2 where $a_{(k)}^{w} \in \mathrm{ F}$. The condition of membership in $\mathrm{Im}(\Delta )$ of the right hand side of (3.33), is that only terms of the form v ⊗ v survive:

$$\displaystyle{ \sum _{a}a_{(1)}^{v}a_{ (2)}^{w} = 0\qquad \mbox{ for all $v\neq w$.} }$$

(A.1)

We take a typical generator of the relations in M ⊗ _Q N: qm ⊗ n − m ⊗ qn. We compute the componentwise action of $\Delta (a)$ on qm ⊗ n and m ⊗ qn.

$$\displaystyle\begin{array}{rcl} \Delta (a) \cdot (qm \otimes n)& =& \sum _{a}a_{(1)} \cdot qm \otimes a_{(2)} \cdot n {}\\ & =& \sum _{a,v,w}a_{(1)}^{v}a_{ (2)}^{w}v \cdot qm \otimes w \cdot n {}\\ & =& \sum _{a,v,w}a_{(1)}^{v}a_{ (2)}^{w}(v \cdot q)(v \cdot m) \otimes w \cdot n {}\\ & =& \sum _{a,v,w}a_{(1)}^{v}a_{ (2)}^{w}(v \cdot q)(v \cdot m \otimes w \cdot n). {}\\ \end{array}$$

Similarly

$$\displaystyle\begin{array}{rcl} \Delta (a) \cdot (m \otimes qn) =\sum _{a,v,w}a_{(1)}^{v}a_{ (2)}^{w}(w \cdot q)(v \cdot m \otimes w \cdot n).& & {}\\ \end{array}$$

The difference of these two expressions is

$$\displaystyle\begin{array}{rcl} & & \quad \,\,\sum _{a}\sum _{v\neq w}a_{(1)}^{v}a_{ (2)}^{w}(v \cdot q - w \cdot q)(v \cdot m \otimes w \cdot n) {}\\ & & =\sum _{v\neq w}(v \cdot q - w \cdot q)(v \cdot m \otimes w \cdot n)\sum _{a}a_{(1)}^{v}a_{ (2)}^{w} {}\\ & & = 0. {}\\ \end{array}$$

Thus the formula for a ⋅ (m ⊗ n) is well-defined.

Applying this to the special case of the action of F _W on $\mathrm{F}_{W} \otimes _{\mathrm{F}}\mathrm{F}_{W}$, we recover part (1), including (3.32). For a, b ∈ F _W we have

$$\displaystyle\begin{array}{rcl} a \cdot (b \cdot (m \otimes n))& =& \sum _{(a)}\sum _{(b)}a_{(1)} \cdot (b_{(1)} \cdot m) \otimes a_{(2)} \cdot (b_{(2)} \cdot n) {}\\ & =& \sum _{(a)}\sum _{(b)}(a_{(1)}b_{(1)}) \cdot m \otimes (a_{(2)}b_{(2)}) \cdot n {}\\ & =& (ab) \cdot (m \otimes n) {}\\ \end{array}$$

where the last step holds because of (3.32). Hence we have an action of F _W on M ⊗_F N.

For the left F _W -module M, the dual ${M}^{{\ast}} =\mathrm{ Hom}_{\mathrm{F}}(M,\mathrm{F})$ has a left F _W -module structure defined by w ⋅ m ^∗ = m ^∗∘ w ⁻¹ or more generally $a \cdot {m}^{{\ast}} = {m}^{{\ast}}\circ {a}^{t}$ for w ∈ W and a ∈ F _W . Consider the left F-linear isomorphism ${M}^{{\ast}}\otimes _{\mathrm{F}}N\mathop{\cong}\mathrm{Hom}_{\mathrm{F}}(M,N)$ given by ${m}^{{\ast}}\otimes n\mapsto (x\mapsto {m}^{{\ast}}(x)n)$. We define a left F _W -module structure on Hom_F(M, N) by declaring that the above map is an isomorphism of left F _W -modules. It is enough to consider the action of a = w: $w \cdot {m}^{{\ast}}\otimes n = w \cdot {m}^{{\ast}}\otimes w \cdot n = {m}^{{\ast}}\circ {w}^{-1} \otimes w \cdot n$. This corresponds to the function $x\mapsto {m}^{{\ast}}({w}^{-1}x)w \cdot n = w \cdot {m}^{{\ast}}({w}^{-1}x)n$. If f ∈ Hom_F(M, N) corresponds to m ^∗⊗ n then the above function corresponds to $x\mapsto w \cdot f({w}^{-1}x)$, which is the required action.

Proof of Proposition 3.16.

We first compute

$$\displaystyle\begin{array}{rcl} \Delta (A_{i})& =& \Delta (\alpha _{i}^{-1}(1 - s_{ i})) {}\\ & =& \alpha _{i}^{-1}(1 \otimes 1 - s_{ i} \otimes s_{i}) {}\\ & =& \alpha _{i}^{-1}(1 \otimes 1 - s_{ i} \otimes 1 + s_{i} \otimes 1 - s_{i} \otimes s_{i}) {}\\ & =& A_{i} \otimes 1 + s_{i} \otimes A_{i} {}\\ & =& \alpha _{i}^{-1}(1 \otimes 1 - 1 \otimes s + 1 \otimes s_{ i} - s_{i} \otimes s_{i}) {}\\ & =& 1 \otimes A_{i} + A_{i} \otimes s_{i}. {}\\ \end{array}$$

This yields (3.35). It follows that the restriction of $\Delta$ to $\mathbb{A}$ has image in $\mathbb{A} \otimes _{S}\mathbb{A}$ and inherits the required properties by Proposition 3.14. All other assertions follow directly.

B Appendix: Small Torus GKM Proofs

Proof.

We prove (1) as (2) follows from it. There is a commutative diagram of ring homomorphisms

(B.1)

The horizontal maps are restriction to torus-fixed points. $\mbox{ For}_{T}^{T_{\mathrm{af}}}$ is the map that forgets from T _af-equivariance to T-equivariance. The top map is an isomorphism by Theorem 3.45. It is not hard to show that For is surjective and that ${H}^{T}(\tilde{\mathrm{Fl}}_{\mathrm{af}})$ has an H ^T(pt)-basis given by the T-equivariant classes of Schubert varieties ${[{X}^{v}]}^{T} \in {H}^{T}(\tilde{\mathrm{Fl}}_{\mathrm{af}})$ for v ∈ W _af. By commutativity of the diagram,

$$\displaystyle{ \mathrm{Im}(\mathrm{{res}}^{{\prime}}) =\mathrm{ Im}{(\pi }^{{\ast}}) =\bigoplus _{ v\in W_{\mathrm{af}}}{S\bar{\xi }}^{v}. }$$

(B.2)

The functions $\bar{{\xi }}^{v}$ are independent since the matrix $(\pi (d_{v,w}))_{v,w\in W_{\mathrm{af}}}$ is triangular with nonvanishing diagonal entries.

It remains to show that

$$\displaystyle{ \mathrm{Im}{(\pi }^{{\ast}}) = \Lambda _{\mathrm{ af}}^{{\prime}}. }$$

(B.3)

Let v ∈ W _af. Certainly $\bar{{\xi }}^{v}$ satisfies (3.53) since ξ ^v does. We must check the conditions (4.1) and (4.2). Let w ∈ W _af, $\alpha \in \Phi$, and $d \in \mathbb{Z}_{>0}$. Let W ^′ ⊂ W _af be the subgroup generated by $t_{{\alpha }^{\vee }}$ and r _α ; it is isomorphic to the affine Weyl group of SL ₂. Define the function f: W ^′ → S by $f(x) {=\bar{\xi } }^{v}(xw)$. Since ξ ^v satisfies (3.53) for Fl_af, f satisfies (3.53) for the affine flag variety Fl^′ corresponding to α. It follows that f is an S-linear combination of Schubert classes in Fl^′. By Propositions B.1 and B.2 (proved below), π ∘ f satisfies (4.1) and (4.2), so that $\bar{{\xi }}^{v} \in \Lambda _{\mathrm{af}}^{{\prime}}$, as required.

Conversely, suppose $\xi \in \Lambda _{\mathrm{af}}^{{\prime}}$. We show that

$$\displaystyle{ \xi \in \bigoplus _{v\in W_{\mathrm{af}}}{S\,\bar{\xi }}^{v}. }$$

(B.4)

Let x = t _λ u be of minimal length in the support of ξ, with u ∈ W and λ ∈ Q ^∨. It suffices to show that

$$\displaystyle{ \xi (x) {\in \bar{\xi }}^{x}(x)S. }$$

(B.5)

Suppose (B.5) holds. Define ${\xi }^{{\prime}}: W_{\mathrm{af}} \rightarrow S$ by ${\xi }^{{\prime}} =\xi -{(\xi (x){/\bar{\xi }}^{x}(x))\bar{\xi }}^{x}$. Since $\Lambda _{\mathrm{af}}^{{\prime}}$ is an S-module, ${\xi }^{{\prime}} \in \Lambda _{\mathrm{af}}^{{\prime}}$. Moreover $\Omega {(\xi }^{{\prime}}) \supsetneq \Omega (\xi )$ where $\Omega (\xi )$ is defined by (3.64). By induction (B.4) holds for ξ ^′ and therefore it holds for ξ.

We now show (B.5). The elements $\{\alpha \mid \alpha \in {\Phi }^{+}\}$ are relatively prime in S. Letting $\alpha \in {\Phi }^{+}$, by (3.61) it suffices to show that $\xi (x) \in J:{=\alpha }^{d}S$ where $d = \vert \mathrm{Inv}_{\alpha }({x}^{-1})\vert$ and $\mathrm{Inv}_{\alpha }({x}^{-1})$ is the set of roots in Inv(x ⁻¹) (see (2.21)) of the form ±α + k δ for some $k \in \mathbb{Z}_{\geq 0}$.

Note that for $\beta \in \Phi _{\mathrm{af}}^{+\mathrm{re}}$, β ∈ Inv(x ⁻¹) if and only if ${x}^{-1}\cdot \beta \in -\Phi _{\mathrm{af}}^{+\mathrm{re}}$. We have

$$\displaystyle{ {x}^{-1} \cdot (\pm \alpha + k\delta ) = {u}^{-1}t_{ -\lambda }\cdot (\pm \alpha + k\delta ) = \pm {u}^{-1}\alpha + (k\pm \langle \lambda \,,\,\alpha \rangle )\delta. }$$

Letting $\chi =\chi (\alpha \in \mathrm{ Inv}({u}^{-1}))$ we have

$$\displaystyle{ \mathrm{Inv}_{\alpha }({x}^{-1}) = \left \{\begin{array}{@{}l@{\quad }l@{}} \{\alpha,\alpha +\delta,\mathop{\ldots },\alpha -(\langle \lambda \,,\,\alpha \rangle +1-\chi )\delta \} \quad &\mbox{ if $\langle \lambda \,,\,\alpha \rangle \leq 0$} \\ \{-\alpha +\delta,-\alpha + 2\delta,\mathop{\ldots },-\alpha + (\langle \lambda \,,\,\alpha \rangle -\chi )\delta \}\quad &\mbox{ if $\langle \lambda \,,\,\alpha \rangle > 0$.} \end{array} \right. }$$

(B.6)

Suppose first that $\langle \lambda \,,\,\alpha \rangle > 0$. Then $d =\langle \lambda \,,\,\alpha \rangle -\chi (\alpha \in \mathrm{ Inv}({u}^{-1}))$. Applying (4.2) to $y = t_{{(1-d)\alpha }^{\vee }}x$, we have Z ₁ ∈ J where

$$\displaystyle\begin{array}{rcl} Z_{1}& =& \xi ({(1 - t_{{\alpha }^{\vee }})}^{d-1}(1 - s_{\alpha })y) {}\\ & =& {(-1)}^{d-1}\xi ({(1 - t_{{ -\alpha }^{\vee }})}^{d-1}x) -\xi ({(1 - t_{{\alpha }^{\vee }})}^{d-1}s_{\alpha }y) {}\\ & =& {(-1)}^{d-1}\xi (x) -\xi ({(1 - t_{{\alpha }^{\vee }})}^{d-1}s_{\alpha }y) {}\\ \end{array}$$

where the last equality holds by the assumption on Supp(ξ) and a calculation of $\mathrm{Inv}_{\alpha }({(s_{\alpha +d\delta }x)}^{-1})$, giving $x > s_{\alpha +d\delta }x > t_{-{k\alpha }^{\vee }}x$ for all 1 ≤ k ≤ d − 1. By Lemma 4.1 we have Z ₂ ∈ J where

$$\displaystyle{ Z_{2} =\xi ({(1 - t_{{\alpha }^{\vee }})}^{d-1}s_{\alpha }y) -\xi ({(1 - t_{{\alpha }^{\vee }})}^{d-1}t_{{ p\alpha }^{\vee }}s_{\alpha }y) }$$

for any $p \in \mathbb{Z}$. Thus

$$\displaystyle{ Z_{1} + Z_{2} = {(-1)}^{d-1}\xi (x) -\xi ({(1 - t_{{\alpha }^{\vee }})}^{d-1}t_{{ p\alpha }^{\vee }}s_{\alpha }y) \in J. }$$

By the assumption on Supp(x) and (B.6),

$$\displaystyle{ \xi ({(1 - t_{{\alpha }^{\vee }})}^{d-1}t_{{ p\alpha }^{\vee }}s_{\alpha }y) = 0 }$$

for p = 2 − d. It follows that ξ(x) ∈ J.

Otherwise $\langle \lambda \,,\,\alpha \rangle \leq 0$. By the previous case we may assume that $t_{{d\alpha }^{\vee }}x\not\in \mathrm{Supp}(\xi )$. Thus

$$\displaystyle{ \xi (x) =\xi ({(1 - t_{{\alpha }^{\vee }})}^{d}x) \in J }$$

by induction on Supp(ξ) and (4.1). This proves (B.5) and (B.4) as required.

2.1 B.1 Small Torus GKM Condition for $\widehat{\mathrm{sl}}_{2}$

In this section we consider the root datum for $\widehat{\mathrm{sl}}_{2}$. Let ${\Phi }^{+} =\{\alpha \}$ where α = α ₁. Also δ = α ₀ +α ₁ so that π(α ₀) = −π(α ₁) and the level zero action of W _af is given by s ₀ ⋅ α = s ₁ ⋅ α = −α. For $i \in \mathbb{Z}_{\geq 0}$ let

$$\displaystyle{ \begin{array}{rcl} \sigma _{2i}& =&{(s_{1}s_{0})}^{i}\qquad \sigma _{-2i} = {(s_{0}s_{1})}^{i},\\ \sigma _{ 2i+1} & =&s_{0}\sigma _{2i}\qquad \sigma _{-(2i+1)} = s_{1}\sigma _{-2i}. \end{array} }$$

(B.7)

Then we have ℓ(σ _j ) =  | j | for $j \in \mathbb{Z}$, $W_{\mathrm{af}}^{I} =\{\sigma _{j}\mid j \in \mathbb{Z}_{\geq 0}\}$, and

$$\displaystyle{ \sigma _{2i} = t_{-{i\alpha }^{\vee }}\qquad \mbox{ for $i \in \mathbb{Z}$.} }$$

Let $\xi _{j}^{i}:=\pi {(\xi }^{\sigma _{i}}(\sigma _{j}))$ for $i,j \in \mathbb{Z}$. For $m,a \in \mathbb{Z}_{\geq 0}$,

$$\displaystyle{ \xi _{j}^{m} = {(-1)}^{mj}\binom{m + a}{{m}\alpha }^{m} }$$

(B.8)

where j ∈ { m + 2a, m + 2a + 1, −m − 2a − 1, −m − 2a − 2}. This is easily proved by induction using (3.60). The rest of the values for ξ ^m are zero by (3.58). For m < 0 we may use

$$\displaystyle{ \xi _{j}^{m} = {(-1)}^{m}\xi _{ -j}^{-m}\qquad \mbox{ for $m,j \in \mathbb{Z}$} }$$

(B.9)

which follows from the Dynkin symmetry $0 \leftrightarrow 1$.

Proposition B.1.

For all d ≥ 1, $m \in \mathbb{Z}$ , and w ∈ W _af we have

$$\displaystyle{{ \xi }^{m}({(1 - t_{{\alpha }^{\vee }})}^{d}w) {\in \alpha }^{d}\mathbb{Z}[\alpha ]. }$$

(B.10)

Proof.

One may reduce to m ≥ 0 using (B.9). Equation (B.10) is proved for m = 2i, $w = t_{{(-i-a)\alpha }^{\vee }}$ and d = (i + a) + (i + 1 + b) = 2i + a + b + 1 for $a,b \in \mathbb{Z}_{\geq 0}$, as the other possibilities are similar or easier. Equation (B.10) can be rewritten as

$$\displaystyle{ Z:=\sum _{ k=0}^{d}{(-1)}^{k}\binom{d}{k}\xi _{ -2i-2-2b+2k}^{2i} {\in \alpha }^{d}\mathbb{Z}[\alpha ]. }$$

(B.11)

By (3.58) ξ _2p ²ⁱ = 0 for − 2i − 2 < 2p < 2i. Using (B.8),

$$\displaystyle\begin{array}{rcl} Z& =& \left (\sum _{k=0}^{b} +\sum _{ k=2i+1+b}^{d}\right ){(-1)}^{k}\binom{d}{k}\xi _{ -2i-2-2b+2k}^{2i} {}\\ & =& \sum _{k=0}^{b}{(-1)}^{k}\binom{d}{k}\xi _{ -2i-2-2b+2k}^{2i} +\sum _{ k=0}^{a}{(-1)}^{k+2i+1+b}\binom{d}{2i + 1 + b + k}\xi _{ 2i+2k}^{2i} {}\\ & =& {(-1)}^{b}\sum _{ k=0}^{b}{(-1)}^{k}\binom{d}{b - k}\xi _{ -2i-2-2k}^{2i} - {(-1)}^{b}\sum _{ k=0}^{a}{(-1)}^{k}\binom{d}{a - k}\xi _{ 2i+2k}^{2i} {}\\ & =& {(-1)}^{b}\sum _{ k=0}^{b}{(-1)}^{k}\binom{d}{b - k}\binom{2i + k}{{2i}\alpha }^{2i} - {(-1)}^{b}\sum _{ k=0}^{a}{(-1)}^{k}\binom{d}{a - k}\binom{2i + k}{{2i}\alpha }^{2i}. {}\\ \end{array}$$

Taking the coefficient of (−x)^b in (1 − x)^d∕(1 − x)²ⁱ⁺¹ = (1 − x)^a+b we have

$$\displaystyle{ \sum _{k=0}^{b}{(-1)}^{k}\binom{d}{b - k}\binom{2i + k}{2i} = \binom{a + b}{b}. }$$

Exchanging the roles of a and b we see that Z = 0 which implies (B.11).

Proposition B.2.

For all d ≥ 1, $m \in \mathbb{Z}$ , and w ∈ W _af we have

$$\displaystyle{{ \xi }^{m}({(1 - t_{{\alpha }^{\vee }})}^{d-1}(1 - s_{\alpha })w) {\in \alpha }^{d}\mathbb{Z}[\alpha ]. }$$

Proof.

Let $y = {(1 - t_{{\alpha }^{\vee }})}^{d-1}$. Without loss of generality we may assume $w = t_{{p\alpha }^{\vee }}$ for some $p \in \mathbb{Z}$. By Proposition B.1, ξ ^m satisfies (4.1). We have

$$ \displaystyle\begin{array}{rcl} {\xi }^{m}(y(1 - s_{\alpha })t_{{ p\alpha }^{\vee }})& =& {\xi }^{m}(yt_{{ p\alpha }^{\vee }}) {-\xi }^{m}(yt_{ -{p\alpha }^{\vee }}s_{\alpha }) {}\\ & \equiv & {\xi }^{m}(y) {-\xi }^{m}{(ys_{\alpha })\mod \alpha }^{d}\mathbb{Z}[\alpha ], {}\\ \end{array}$$

using Lemma 4.1 twice. However

$$ \displaystyle\begin{array}{rcl} {\xi }^{m}(y) {-\xi }^{m}(ys_{\alpha })& =& {\xi }^{m}(y\alpha A_{\alpha }) {}\\ & =& (y\cdot \alpha )(A{_{\alpha }\bullet \xi }^{m})(y). {}\\ \end{array}$$

Now y ⋅ α = ±α and A _α •ξ ^m is 0 if m ≥ 0 and is equal to ξ ^m+1 if m < 0. Assuming the latter, by (4.1) for d − 1, ξ ^m+1(y) ∈ α ^d−1 S, so that ${\xi }^{m}(y) {-\xi }^{m}(ys_{\alpha }) {\in \alpha }^{d}S$ as required.

C Appendix: Homology of Gr

Let K be the maximal compact form of G and $T_{\mathbb{R}} = K \cap T$. The based loop group $\Omega K$ of continuous maps (S ¹, 1) → (K, 1), has a $T_{\mathbb{R}}$-equivariant multiplication map $\Omega K \times \Omega K \rightarrow \Omega K$ given by pointwise multiplication on K, and this induces the structure of a commutative and co-commutative Hopf-algebra on ${H}^{T_{\mathbb{R}}}(\Omega K)$. The co-commutativity of ${H}^{T_{\mathbb{R}}}(\Omega K)$ follows from the fact that $\Omega K$ is a homotopy double-loop space.

Gr = Gr _G and $\Omega K$ are weakly homotopy equivalent [135]. By “fattening loops”, it follows that H _T (Gr) and ${H}^{T}(\tilde{\mathrm{Gr}})$ are dual Hopf algebras over S = H ^T(pt) with duality given by an intersection pairing

$$\displaystyle{ \langle \cdot \,,\,\cdot \rangle: H_{T}(\mathrm{Gr}) \times {H}^{T}(\tilde{\mathrm{Gr}}) \rightarrow S. }$$

(C.1)

We may regard an element ξ ∈ H _T (Gr) as an S-module homomorphism ${H}^{T}(\tilde{\mathrm{Gr}}) \rightarrow S$ by $\xi (f) =\langle \xi \,,\,f\rangle$.

Lemma C.1.

Let λ,μ ∈ Q ^∨ , and consider the maps $i_{\lambda }^{{\ast}},i_{\mu }^{{\ast}}: {H}^{T}(\tilde{\mathrm{Gr}}) \rightarrow S$ as elements of H _T (Gr ). Then in H _T (Gr ), we have

$$\displaystyle{i_{\lambda }^{{\ast}}\;i_{\mu }^{{\ast}} = i_{\lambda +\mu }^{{\ast}}.}$$

Proof.

It suffices to work in ${H}^{T_{\mathbb{R}}}(\Omega K)$. The map $i_{\lambda }^{{\ast}}\;i_{\mu }^{{\ast}}$ is induced by the map $\mathrm{pt} \rightarrow \Omega K \times \Omega K \rightarrow \Omega K$ where the image of the first map is the pair $(t_{\lambda },t_{\mu }) \in \Omega K \times \Omega K$ of fixed points, and the latter map is multiplication. Treating $t_{\lambda },t_{\mu }: {S}^{1} \rightarrow K$ as homomorphisms into K, the pointwise multiplication of t _λ and t _μ gives t _λ+μ . Thus $i_{\lambda }^{{\ast}}\;i_{\mu }^{{\ast}} = i_{\lambda +\mu }^{{\ast}}$.

The antipode of H _T (Gr _G ) is given by $i_{\lambda }^{{\ast}}\mapsto i_{-\lambda }^{{\ast}}$, since the fixed points satisfy $t_{\lambda }^{-1} = t_{-\lambda }$ in $\Omega K$.

For $w \in W_{\mathrm{af}}^{0}$ denote by $[X_{w}]_{T} \in H_{T}(\mathrm{Gr})$ the equivariant fundamental homology class of the Schubert variety $X_{w}:= \overline{B_{\mathrm{af}}\dot{w}P_{\mathrm{af}}/P_{\mathrm{af}}} \subset G_{\mathrm{af}}/P_{\mathrm{af}} =\mathrm{ Gr}$.

Using the intersection pairing we have that $\{[X_{w}]\mid w \in W_{\mathrm{af}}^{0}\}$ is the basis of H _T (Gr _G ) that is dual to the basis $\{{[{X}^{w}]}^{T}\mid w \in W_{\mathrm{af}}^{0}\}$ of ${H}^{T}(\tilde{\mathrm{Gr}})$, which corresponds to the basis $\{\bar{{\xi }}^{w}\mid w \in W_{\mathrm{af}}^{0}\}$ of $\Lambda _{\mathrm{Fl}}^{{\prime}}$ under the isomorphism of Theorem 4.3.

Proof of Proposition 4.6.

Follows from the above discussion.