## Passage to ${H}^{*}\left(G/B\right)$

Last update: 21 February 2013

## Passage to ${H}^{*}\left(G/B\right)$

Let us explain how our results in $K\left(G/B\right)$ are related to the cohomology ${H}^{*}\left(G/B\right)$ and Schubert polynomials. The transfer is by way of the Chern character ch.

If $X$ is a finite CW complex then the Chern character (see [Mac1968, Ch. 10], [Hir1995, §23-24], [Har1977, App. A])

$ch:Kvb(X)⊗ ℚ⟶H*(X;ℚ)$

is a natural ring isomorphism, i.e. if $f:X\to Y$ is continuous then $\text{ch}\left({f}^{!}\left(x\right)\right)={f}_{*}\left(\text{ch}\left(x\right)\right)\text{.}$ If $f:X\to Y$ is a morphism of nonsingular projective varieties then the Grothendieck-Riemann-Roch theorem [Har1977, App. A Theorem 5.3], henceforth G-R-R, says

$ch(f!(x))= f*(ch(x)td(𝒯f)),$

where $\text{td}\left({𝒯}_{f}\right)$ is Todd class of the relative tangent sheaf of $f\text{.}$ If $ℒ$ is a line bundle on $X$ with first Chern class $\lambda \in {H}^{2}\left(X;ℚ\right)$ then the Chern character and the Todd class of $ℒ$ are the elements of ${H}^{*}\left(X\right)$ given by

$ch(ℒ)=eλ= ∑k≥0 λk k! andtd(ℒ)= λ1-e-λ, respectively.$

The expression ${e}^{\lambda }$ is finite sum since ${\lambda }^{k}=0$ in ${H}^{*}\left(X\right)$ whenever $k>\text{dim}\left(X\right)\text{.}$

${H}^{*}\left(G/B\right)$ as the quotient of a polynomial ring

Let $X=G/B\text{.}$ Let $𝔥$ be the Lie algebra of $T$ and let $S\left({𝔥}^{*}\right)$ be the ring of polynomials on ${𝔥}^{*}$ (over $ℚ\text{).}$ This is a polynomial ring in the $n$ variables ${\alpha }_{1},\dots ,{\alpha }_{n}$ (the simple roots). Let $\stackrel{^}{\epsilon }:S\left({𝔥}^{*}\right)\to ℚ$ be the homomorphism given by $\stackrel{^}{\epsilon }\left(\lambda \right)=0$ for all $\lambda \in {𝔥}^{*}\text{.}$ If $f\in S\left({𝔥}^{*}\right)$ then $\stackrel{^}{\epsilon }\left(f\right)$ is the constant term of $f\text{.}$ It is a classical theorem of Borel (see [BGG1973, Prop. 1.3]) that

$H*(G/B;ℚ) ≅S(𝔥*)ℐ^, (5.1)$

where $\stackrel{^}{ℐ}$ is the ideal of $S\left({𝔥}^{*}\right)$ generated by $\left\{f\in S{\left({𝔥}^{*}\right)}^{W} \mid f-\stackrel{^}{\epsilon }\left(f\right)=0\right\}\text{.}$ Proposition 1.5 is the K-theory analogue of Borel’s theorem.

Let $-\lambda \in \Lambda \text{.}$ The element $-\lambda$ determines a character of $T,$ denoted by ${e}^{\lambda }\in R\left(T\right)$ (see section 1). Let ${c}_{1}\left({ℒ}_{-\lambda }\right)\in {H}^{2}\left(G/B\right)$ be the first Chern class of the line bundle ${ℒ}_{-\lambda }=\varphi \left({e}^{\lambda }\right)$ where $\varphi :R\left(T\right)\to K\left(G/B\right)$ is the map in (1.2). Because of the isomorphism in (5.1) we often abuse notation and write $\lambda ={c}_{1}\left({ℒ}_{-\lambda }\right)\in {H}^{*}\left(G/B\right)\text{.}$ All of the maps in the following commutative diagram are isomorphisms. (Recall that we can identify $K\left(G/B\right)$ and ${K}_{vb}\left(G/B\right)\text{.)}$

$R(T)/ℐ⊗ℚ ⟶ϕ Kvb (G/B) ⊗ℚ ↓ ch ↓ ch S(𝔥*) /ℐ^ ⟶ϕ^ H* (G/B;ℚ) where eλ ⟶ϕ [ℒ-λ] ↓ch ↓ch eλ ⟶ϕ^ ec1(ℒ-λ) (5.2)$

The left hand ch map is obtained by viewing $R\left(T\right)$ and $S\left({𝔥}^{*}\right)$ as subsets of $K\left({ℬ}_{T}\right)$ and $\stackrel{^}{{H}^{*}}\left({ℬ}_{T}\right),$ respectively, where ${ℬ}_{T}$ is the classifying space of $T\text{.}$

The relation between ${T}_{\alpha }$ and ${\partial }_{\alpha }$

Let $\alpha$ be a simple root and let ${P}_{\alpha }$ be the parabolic subgroup with Lie algebra ${𝔭}_{\alpha }$ spanned by $𝔟$ and the root space ${𝔤}_{-\alpha }\text{.}$ Let ${f}_{\alpha }:G/B\to G/{P}_{\alpha }$ corresponding ${ℙ}_{1}\text{-bundle.}$

 A proof of Proposition 3.3: Since $K\left(G/B\right)$ is torsion free, we can check the identity ${\left({f}_{\alpha }\right)}^{!}{\left({f}_{\alpha }\right)}_{!}\left(x\right)={T}_{\alpha }\left(x\right)$ by applying ch to both sides and checking the result in cohomology. The G-R-R for the map ${f}_{\alpha }$ says $ch((fα)!(x))= (fα)* (ch(x)td(𝒯fα)), (5.3)$ where ${𝒯}_{{f}_{\alpha }}$ is the bundle of tangents along the fibres, which is the line bundle associated to the $\text{ad}\left(B\right)\text{-module}$ ${𝔭}_{\alpha }/𝔟\text{.}$ Since this module has weight $-\alpha ,$ ${c}_{1}\left({𝒯}_{{f}_{\alpha }}\right)=\alpha \in {H}^{2}\left(G/B:ℤ\right)$ and so (5.3) becomes $ch((fα)!(x)) =(fα)* ( ch(x) α1-e-α ) . (5.4)$ The BGG operator ${\partial }_{\alpha }={\left({f}_{\alpha }\right)}^{*}{\left({f}_{\alpha }\right)}_{*}$ is explicitly given by the formula $∂α(z)= z-sα(z)α, (5.5)$ see [Dem1974]. Thus, by applying ${\left({f}_{\alpha }\right)}^{*}$ to both sides of (5.4) we obtain $ch ( (fα)! (fα)! (x) ) = (fα)* (fα)* ( ch(x) α1-e-α ) =∂α ( ch(x) α1-e-α ) . (5.6)$ The strategy now is to manipulate the right hand side of (5.6) using the “skew-Leibniz” rule satisfied by ${\partial }_{\alpha }$ to obtain $\text{ch}\left({T}_{\alpha }\left(x\right)\right)\text{.}$ For convenience, let $y=\text{ch}\left(x\right)\text{.}$ Then the right side of (5.6) is $∂α (yα1-e-α)= y∂α (α1-e-α) +sα (α1-e-α) ∂α(y)$ and we claim $(a)sα (α1-e-α)= αeα-1, (b)∂α (α1-e-α)= 1.$ The first equality is trivial and the second can be proved by formal computation (carefully done!) or by applying the G-R-R again. Using (a) and (b) we obtain $∂α (yα1-e-α)= y+αeα-1 (y-sα(y)α).$ Now cancelling the $\alpha \text{'s'}$ in the second term on the right and recalling that $y=\text{ch}\left(x\right)$ we find $ch ( (fα)! (fα)! (x) ) =ch(Tα(x)).$ $\square$

We see that

$ch(Tα(x))= ∂α ( ch(x) α1-e-α ) (5.7)$

which relates to ${T}_{\alpha }$ to ${\partial }_{\alpha }$ in an explicit way. In fact, if one wished one could reverse the argument and derive the formula (5.5) for ${\partial }_{\alpha }$ from the formula for ${T}_{\alpha }\text{.}$

A proof of Proposition 1.6

Proposition 1.6. If $f:G/B\to G/P$ is the natural projection then the induced map ${f}^{!}:K\left(G/P\right)\to K\left(G/B\right)$ is an injection.

 Proof. Since $K\left(G/P\right)$ is torsion free there is a natural injection $K\left(G/P\right)↪K\left(G/P\right)\otimes ℚ\to {H}^{*}\left(G/P;ℚ\right)$ and so it suffices to check that the pull-back ${f}^{*}$ in rational cohomology is injective. Since the odd cohomology groups of the base and fiber are zero the Serre spectral sequence of the bundle $P/B\to G/B\to G/P$ shows that ${f}^{*}$ is injective even for ${H}^{*}\left(FP;ℤ\right)\text{.}$ $\square$

Dictionary between $K\left(G/B\right)$ and ${H}^{*}\left(G/B\right)$

In summary, the Chern character gives an isomorphism

$R(T)/ℐ ≅ K(G/B) ⟶ch H*(G/B) ≅ S(𝔥*)/ℐ^ eλ ⟼ eλ,$

where

$ℐ=ideal generated by { f∈R(T)W ∣ f-ε(f)=0 } , ε: R(T) ⟶ ℤ eλ ⟼ 1, ℐ^=ideal generated by { f∈S(𝔥*)W ∣ f-ε^(f)=0 } , ε^: S(𝔥*) ⟶ ℤ λ ⟼ 0.$

Let $\left[{X}_{w}\right]\in {H}^{*}\left(G/B\right)$ be the element which is Poincaré dual to the fundamental cycle of ${X}_{w}$ in ${H}^{*}\left(G/B\right)\text{.}$ This element is called a Schubert polynomial. Then

$K(G/B) has basis { [𝒪Xw] ∣ w∈W } andH* (G/B) has basis {[Xw] ∣ w∈W} .$

From a general fact (see [Ful1984, Ex. 15.2.16] or [CGi1433132, 5.8.13(i) and p. 289])

$ch([𝒪Xw])= [Xw]+higher degree terms.$

where $\text{deg}\left(\left[{X}_{w}\right]\right)=\text{dim}\left(G/B\right)-\text{dim}\left({X}_{w}\right)=N-\ell \left(w\right), \text{where} N \text{is the number of positive roots for} 𝔤\text{.}$

If $\alpha$ is a simple root and ${f}_{\alpha }:G/B\to G/{P}_{\alpha }$ is the corresponding ${ℙ}^{1}\text{-bundle,}$ then

$Tα(x)= (fα)! (fα)!(x)= eαx-sα(x) eα-1 and∂α(x)= (fα)* (fα)*(x)= x-sα(x)α$

in $K\left(G/B\right)$ and ${H}^{*}\left(G/B\right),$ respectively. As illustrated in (5.7) each of these two formulas can be derived from the other via the use of the Grothendieck-Riemann-Roch Theorem. This means that the following formulas (Proposition 3.4 and [BGG1973, Th. 3.14])

$Tα([𝒪Xw])= { [𝒪Xwsα], if wsα>w, [𝒪Xw], if wsαw, [Xw], if wsα

are equivalent.

Our new Pieri-Chevalley formula in $K\left(G/B\right),$ Theorem 4.3, and Chevalley’s classical Pieri formula in ${H}^{*}\left(G/B\right),$ [Che1994, Prop. 10], are

$eλ[𝒪Xw]= ∑η∈𝒯wλ [𝒪Xv(η,w)] andλ·[Xw]= ∑αv→w ⟨λ,α∨⟩ [Xv],$

where the sum is over all $v\in W$ such that $\ell \left(v\right)=\ell \left(w\right)-1$ and there is a root $\alpha$ such that $v={s}_{\alpha }w\text{.}$ Chevalley’s formula can be obtained from ours formula by subtracting $\left[{𝒪}_{{X}_{w}}\right]$ from each side, applying the Chern character ch, and comparing the lowest degree terms on each side.

## Notes and References

This is an excerpt of the paper entitled A Pieri-Chevalley formula for K(G/B) authored by H. Pittie and Arun Ram (preprint May 19, 1998).

Research supported in part by National Science Foundation grant DMS-9622985.