Passage to $H^{*}(G/B)$

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

Last update: 21 February 2013

Passage to $H^{*}(G/B)$

Let us explain how our results in $K(G/B)$ are related to the cohomology $H^{*}(G/B)$ 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])

$$\text{ch}:K_{vb}\left(X\right)\otimes \mathbb{Q}⟶H^{*}(X;\mathbb{Q})$$

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

$$\text{ch}\left(f_{!}\left(x\right)\right)=f_{*}\left(\text{ch}\left(x\right)\text{td}\left({𝒯}_f\right)\right),$$

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 $\lambda \in H^2(X;\mathbb{Q})$ then the Chern character and the Todd class of $ℒ$ are the elements of $H^{*}\left(X\right)$ given by

$$\text{ch}\left(ℒ\right)=e^{\lambda }=\sum _{k\ge 0}\frac{{\lambda }^k}{k!}\text{and}\text{td}\left(ℒ\right)=\frac{\lambda }{1-e^{-\lambda }},\text{respectively.}$$

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

$H^{*}(G/B)$ 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 $\mathbb{Q}\text{).}$ This is a polynomial ring in the $n$ variables ${\alpha }_1,\dots ,{\alpha }_n$ (the simple roots). Let $\hat{\epsilon }:S\left({𝔥}^{*}\right)\to \mathbb{Q}$ be the homomorphism given by $\hat{\epsilon }\left(\lambda \right)=0$ for all $\lambda \in {𝔥}^{*}\text{.}$ If $f\in S\left({𝔥}^{*}\right)$ then $\hat{\epsilon }\left(f\right)$ is the constant term of $f\text{.}$ It is a classical theorem of Borel (see [BGG1973, Prop. 1.3]) that

$$\begin{array}{cc}{H}^{*}(G/B;\mathbb{Q})\cong \frac{S\left({𝔥}^{*}\right)}{\widehat{ℐ}},& \text{(5.1)}\end{array}$$

where $\widehat{ℐ}$ is the ideal of $S\left({𝔥}^{*}\right)$ generated by $\{f\in S{\left({𝔥}^{*}\right)}^W\hspace{0.17em}\mid \hspace{0.17em}f-\hat{\epsilon }\left(f\right)=0\}\text{.}$ 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 $c_1\left({ℒ}_{-\lambda }\right)\in H^2(G/B)$ be the first Chern class of the line bundle ${ℒ}_{-\lambda }=\varphi \left(e^{\lambda }\right)$ where $\varphi :R\left(T\right)\to K(G/B)$ is the map in (1.2). Because of the isomorphism in (5.1) we often abuse notation and write $\lambda =c_1\left({ℒ}_{-\lambda }\right)\in H^{*}(G/B)\text{.}$ All of the maps in the following commutative diagram are isomorphisms. (Recall that we can identify $K(G/B)$ and $K_{vb}(G/B)\text{.)}$

$$\begin{array}{cc}\begin{array}{ccc}R\left(T\right)/ℐ\otimes \mathbb{Q}& \stackrel{\varphi }{⟶}& K_{vb}(G/B)\otimes \mathbb{Q}\\ \downarrow \text{ch}& & \downarrow \text{ch}\\ S\left({𝔥}^{*}\right)/\widehat{ℐ}& \stackrel{\hat{\varphi }}{⟶}& H^{*}(G/B;\mathbb{Q})\end{array}\text{where}\begin{array}{ccc}{e}^{\lambda }& \stackrel{\varphi }{⟶}& \left[{ℒ}_{-\lambda }\right]\\ \downarrow \text{ch}& & \downarrow \text{ch}\\ e^{\lambda }& \stackrel{\hat{\varphi }}{⟶}& e^{c_1\left({ℒ}_{-\lambda }\right)}\end{array}& \text{(5.2)}\end{array}$$

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 $\widehat{H^{*}}\left({ℬ}_T\right),$ 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 ${\mathbb{P}}_1\text{-bundle.}$

		A proof of Proposition 3.3:
	Since $K(G/B)$ 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 $$\begin{array}{cc}\text{ch}\left({\left(f_{\alpha }\right)}_{!}\left(x\right)\right)={\left(f_{\alpha }\right)}_{*}\left(\text{ch}\left(x\right)\text{td}\left({𝒯}_{f_{\alpha }}\right)\right),& \text{(5.3)}\end{array}$$ 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 $-\alpha ,$ $c_1\left({𝒯}_{f_{\alpha }}\right)=\alpha \in H^2(G/B:\mathbb{Z})$ and so (5.3) becomes $$\begin{array}{cc}\text{ch}\left({\left(f_{\alpha }\right)}_{!}\left(x\right)\right)={\left(f_{\alpha }\right)}_{*}\left(\text{ch}\left(x\right)\frac{\alpha }{1-e^{-\alpha }}\right)\text{.}& \text{(5.4)}\end{array}$$ The BGG operator ${\partial }_{\alpha }={\left(f_{\alpha }\right)}^{*}{\left(f_{\alpha }\right)}_{*}$ is explicitly given by the formula $$\begin{array}{cc}{\partial }_{\alpha }\left(z\right)=\frac{z-s_{\alpha }\left(z\right)}{\alpha },& \text{(5.5)}\end{array}$$ see [Dem1974]. Thus, by applying ${\left(f_{\alpha }\right)}^{*}$ to both sides of (5.4) we obtain $$\begin{array}{cc}\text{ch}\left({\left(f_{\alpha }\right)}^{!}{\left(f_{\alpha }\right)}_{!}\left(x\right)\right)={\left(f_{\alpha }\right)}^{*}{\left(f_{\alpha }\right)}_{*}\left(\text{ch}\left(x\right)\frac{\alpha }{1-e^{-\alpha }}\right)={\partial }_{\alpha }\left(\text{ch}\left(x\right)\frac{\alpha }{1-e^{-\alpha }}\right)\text{.}& \text{(5.6)}\end{array}$$ 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 $${\partial }_{\alpha }\left(y\frac{\alpha }{1-e^{-\alpha }}\right)=y{\partial }_{\alpha }\left(\frac{\alpha }{1-e^{-\alpha }}\right)+s_{\alpha }\left(\frac{\alpha }{1-e^{-\alpha }}\right){\partial }_{\alpha }\left(y\right)$$ and we claim $$\text{(a)}{s}_{\alpha }\left(\frac{\alpha }{1-e^{-\alpha }}\right)=\frac{\alpha }{e^{\alpha }-1},\text{(b)}{\partial }_{\alpha }\left(\frac{\alpha }{1-e^{-\alpha }}\right)=1\text{.}$$ 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 $${\partial }_{\alpha }\left(y\frac{\alpha }{1-e^{-\alpha }}\right)=y+\frac{\alpha }{e^{\alpha }-1}\left(\frac{y-s_{\alpha }\left(y\right)}{\alpha }\right)\text{.}$$ Now cancelling the $\alpha \text{'s'}$ in the second term on the right and recalling that $y=\text{ch}\left(x\right)$ we find $$\text{ch}\left({\left(f_{\alpha }\right)}^{!}{\left(f_{\alpha }\right)}_{!}\left(x\right)\right)=\text{ch}\left(T_{\alpha }\left(x\right)\right)\text{.}$$ $\square$

We see that

$$\begin{array}{cc}\text{ch}\left(T_{\alpha }\left(x\right)\right)={\partial }_{\alpha }\left(\text{ch}\left(x\right)\frac{\alpha }{1-e^{-\alpha }}\right)& \text{(5.7)}\end{array}$$

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 $f^{!}:K(G/P)\to K(G/B)$ is an injection.

		Proof.
	Since $K(G/P)$ is torsion free there is a natural injection $K(G/P)\hookrightarrow K(G/P)\otimes \mathbb{Q}\to H^{*}(G/P;\mathbb{Q})$ 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 $H^{*}(FP;\mathbb{Z})\text{.}$ $\square$

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

In summary, the Chern character gives an isomorphism

$$\begin{array}{ccccccc}R\left(T\right)/ℐ& \cong & K(G/B)& \stackrel{\text{ch}}{⟶}& H^{*}(G/B)& \cong & S\left({𝔥}^{*}\right)/\widehat{ℐ}\\ & & e^{\lambda }& ⟼& e^{\lambda },\end{array}$$

where

$$\begin{array}{c}ℐ=\text{ideal generated by}\hspace{0.17em}\{f\in R{\left(T\right)}^W\hspace{0.17em}\mid \hspace{0.17em}f-\epsilon \left(f\right)=0\},\begin{array}{cccc}\epsilon :& R\left(T\right)& ⟶& \mathbb{Z}\\ & e^{\lambda }& ⟼& 1,\end{array}\\ \widehat{ℐ}=\text{ideal generated by}\hspace{0.17em}\{f\in S{\left({𝔥}^{*}\right)}^W\hspace{0.17em}\mid \hspace{0.17em}f-\hat{\epsilon }\left(f\right)=0\},\begin{array}{cccc}\hat{\epsilon }:& S\left({𝔥}^{*}\right)& ⟶& \mathbb{Z}\\ & \lambda & ⟼& 0\text{.}\end{array}\end{array}$$

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

$$K(G/B)\hspace{0.17em}\text{has basis}\hspace{0.17em}\{\left[{𝒪}_{X_w}\right]\hspace{0.17em}\mid \hspace{0.17em}w\in W\}\text{and}{H}^{*}(G/B)\hspace{0.17em}\text{has basis}\hspace{0.17em}\{\left[X_w\right]\hspace{0.17em}\mid \hspace{0.17em}w\in W\}\text{.}$$

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

$$\text{ch}\left(\left[{𝒪}_{X_w}\right]\right)=\left[X_w\right]+\text{higher degree terms.}$$

where $\text{deg}\left(\left[X_w\right]\right)=\text{dim}(G/B)-\text{dim}\left(X_w\right)=N-\ell \left(w\right),\hspace{0.17em}\text{where}\hspace{0.17em}N\hspace{0.17em}\text{is the number of positive roots for}\hspace{0.17em}𝔤\text{.}$

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

$$T_{\alpha }\left(x\right)={\left(f_{\alpha }\right)}^{!}{\left(f_{\alpha }\right)}_{!}\left(x\right)=\frac{e^{\alpha }x-s_{\alpha }\left(x\right)}{e^{\alpha }-1}\text{and}{\partial }_{\alpha }\left(x\right)={\left(f_{\alpha }\right)}^{*}{\left(f_{\alpha }\right)}_{*}\left(x\right)=\frac{x-s_{\alpha }\left(x\right)}{\alpha }$$

in $K(G/B)$ and $H^{*}(G/B),$ 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])

$$T_{\alpha }\left(\left[{𝒪}_{X_w}\right]\right)=\{\begin{array}{cc}\left[{𝒪}_{X_{w{s}_{\alpha }}}\right],& \text{if}\hspace{0.17em}w{s}_{\alpha }>w,\\ \left[{𝒪}_{X_w}\right],& \text{if}\hspace{0.17em}w{s}_{\alpha }<w,\end{array}\text{and}{\partial }_{\alpha }\left(\left[X_w\right]\right)=\{\begin{array}{cc}\left[X_{w{s}_{\alpha }}\right],& \text{if}\hspace{0.17em}w{s}_{\alpha }>w,\\ \left[X_w\right],& \text{if}\hspace{0.17em}w{s}_{\alpha }<w,\end{array}$$

are equivalent.

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

$$e^{\lambda }\left[{𝒪}_{X_w}\right]=\sum _{\eta \in {𝒯}_w^{\lambda }}\left[{𝒪}_{X_{v(\eta ,w)}}\right]\text{and}\lambda ·\left[X_w\right]=\sum _{\underset{v\to w}{\alpha }}⟨\lambda ,{\alpha }^{\vee }⟩\left[X_v\right],$$

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.

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