Passage to H*(G/B)

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

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])

ch:Kvb(X) H*(X;)

is a natural ring isomorphism, i.e. if f:XY is continuous then ch(f!(x))= f*(ch(x)). If f:XY 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 td(𝒯f) is Todd class of the relative tangent sheaf of f. If is a line bundle on X with first Chern class λH2(X;) then the Chern character and the Todd class of are the elements of H*(X) given by

ch()=eλ= k0 λk k! andtd()= λ1-e-λ, respectively.

The expression eλ is finite sum since λk=0 in H*(X) whenever k>dim(X).

H*(G/B) as the quotient of a polynomial ring

Let X=G/B. Let 𝔥 be the Lie algebra of T and let S(𝔥*) be the ring of polynomials on 𝔥* (over ). This is a polynomial ring in the n variables α1,,αn (the simple roots). Let ε^:S(𝔥*) be the homomorphism given by ε^(λ)=0 for all λ𝔥*. If fS(𝔥*) then ε^(f) is the constant term of f. It is a classical theorem of Borel (see [BGG1973, Prop. 1.3]) that

H*(G/B;) S(𝔥*)^, (5.1)

where ^ is the ideal of S(𝔥*) generated by { fS(𝔥*)W f-ε^ (f)=0 } . Proposition 1.5 is the K-theory analogue of Borel’s theorem.

Let -λΛ. The element -λ determines a character of T, denoted by eλR(T) (see section 1). Let c1(-λ)H2(G/B) be the first Chern class of the line bundle -λ=ϕ(eλ) where ϕ:R(T)K(G/B) is the map in (1.2). Because of the isomorphism in (5.1) we often abuse notation and write λ=c1(-λ)H*(G/B). All of the maps in the following commutative diagram are isomorphisms. (Recall that we can identify K(G/B) and Kvb(G/B).)

R(T)/ ϕ Kvb (G/B) ch ch S(𝔥*) /^ ϕ^ H* (G/B;) where eλ ϕ [-λ] ch ch eλ ϕ^ ec1(-λ) (5.2)

The left hand ch map is obtained by viewing R(T) and S(𝔥*) as subsets of K(T) and H*^(T), respectively, where T is the classifying space of T.

The relation between Tα and α

Let α be a simple root and let Pα be the parabolic subgroup with Lie algebra 𝔭α spanned by 𝔟 and the root space 𝔤-α. Let fα:G/B G/Pα corresponding 1-bundle.

A proof of Proposition 3.3:

Since K(G/B) is torsion free, we can check the identity (fα)! (fα)! (x)=Tα (x) by applying ch to both sides and checking the result in cohomology.

The G-R-R for the map fα says

ch((fα)!(x))= (fα)* (ch(x)td(𝒯fα)), (5.3)

where 𝒯fα is the bundle of tangents along the fibres, which is the line bundle associated to the ad(B)-module 𝔭α/𝔟. Since this module has weight -α, c1(𝒯fα)= αH2 (G/B:) and so (5.3) becomes

ch((fα)!(x)) =(fα)* ( ch(x) α1-e-α ) . (5.4)

The BGG operator α= (fα)* (fα)* is explicitly given by the formula

α(z)= z-sα(z)α, (5.5)

see [Dem1974]. Thus, by applying (fα)* 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 α to obtain ch(Tα(x)). For convenience, let y=ch(x). Then the right side of (5.6) is

α (yα1-e-α)= yα (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 α's' in the second term on the right and recalling that y=ch(x) we find

ch ( (fα)! (fα)! (x) ) =ch(Tα(x)).

We see that

ch(Tα(x))= α ( ch(x) α1-e-α ) (5.7)

which relates to Tα to α in an explicit way. In fact, if one wished one could reverse the argument and derive the formula (5.5) for α from the formula for Tα.

A proof of Proposition 1.6

Proposition 1.6. If f:G/BG/P is the natural projection then the induced map f!:K(G/P)K(G/B) is an injection.


Since K(G/P) is torsion free there is a natural injection K(G/P) K(G/P) H*(G/P;) 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/BG/BG/P shows that f* is injective even for H*(FP ;).

Dictionary between K(G/B) and H*(G/B)

In summary, the Chern character gives an isomorphism

R(T)/ K(G/B) ch H*(G/B) S(𝔥*)/^ eλ eλ,


=ideal generated by { fR(T)W f-ε(f)=0 } , ε: R(T) eλ 1, ^=ideal generated by { fS(𝔥*)W f-ε^(f)=0 } , ε^: S(𝔥*) λ 0.

Let [Xw]H*(G/B) be the element which is Poincaré dual to the fundamental cycle of Xw in H*(G/B). This element is called a Schubert polynomial. Then

K(G/B)has basis { [𝒪Xw] wW } andH* (G/B)has basis {[Xw]wW} .

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 deg([Xw])= dim(G/B)- dim(Xw)=N- (w),whereN is the number of positive roots for𝔤.

If α is a simple root and fα:G/BG/Pα is the corresponding 1-bundle, then

Tα(x)= (fα)! (fα)!(x)= eαx-sα(x) eα-1 andα(x)= (fα)* (fα)*(x)= x-sα(x)α

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])

Tα([𝒪Xw])= { [𝒪Xwsα], ifwsα>w, [𝒪Xw], ifwsα<w, andα ([Xw])= { [Xwsα], ifwsα>w, [Xw], ifwsα<w,

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

eλ[𝒪Xw]= η𝒯wλ [𝒪Xv(η,w)] andλ·[Xw]= αvw λ,α [Xv],

where the sum is over all vW such that (v)=(w)-1 and there is a root α such that v=sαw. Chevalley’s formula can be obtained from ours formula by subtracting [𝒪Xw] 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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