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

Last update: 21 February 2013


In this section we set up notations and recall how one is able to work with the K-theory of G/B “purely combinatorially”. This is possible because (in analogy with the cohomology H*(G/B) and the combinatorial theory of Schubert polynomials) the K-theory of G/B is isomorphic to a quotient of R(T), which is a ring of Laurent polynomials. In section 5 we shall explain how our results about K(G/B) relate to and generalize well known results about Schubert polynomials and H*(G/B).

The K-theory K(G/B)

If X is a quasi-projective variety let

K(X) = the Grothendieck group of coherent sheaves onX, Klf(X) = the Grothendieck group of locally free sheaves onX, Kvb(X) = the Grothendieck group of vector bundles onX.

If f:XY is a morphism of projective varieties then we have maps

f!: K(Y) K(X) [] [f*], f!: K(X) K(Y) [] i(-1)i[Rif*].

See [CGi1433132, §5.2] for K-theory background. If X is smooth then the isomorphism between K(X) and Klf(X) is given by assigning to the class of a sheaf the alternating sum of the sheaves in a locally free resolution, see [BSe1958, §4] and [Har1977, III Ex. 6.9]. There is always a map Klf(X)Kvb(X) which assigns to a locally free sheaf the underlying vector bundle and the results of [Pit1972] imply that this map is an isomorphism when X=G/B, see Proposition 1.5 below. Thus, in the case which we wish to consider in this paper, X=G/B, all three K-theories are isomorphic.

Let G be a complex connected simply connected semisimple Lie group. Fix a maximal torus T and a Borel subgroup B such that TBG. The Bruhat decomposition says that G is a disjoint union of double cosets of B indexed by the elements of the Weyl group W, G=wWBwB. The flag variety is the projective variety formed by the coset space G/B and the Bruhat decomposition of G induces a cell decomposition of G/B. For each wW the subset Xw=BwB/B is the Schubert cell and its closure Xw is the Schubert variety. The formulas

dim(Xw)= (w)and Xw=vw Xv (1.1)

define the length (w) of wW and the Bruhat-Chevalley order on the Weyl group, respectively. It follows from the Bruhat decomposition (see lecture 4 by Grothendieck in [Che1958]) that

K(G/B)is a free -module with basis { [𝒪Xw] wW } ,

where Xw are the Schubert varieties in G/B and 𝒪Xw is the structure sheaf of Xw extended to G/B by defining it to be 0 outside Xw.

The isomorphism K(G/B)R(T)/

For any group H let R(H) be the Grothendieck group of complex representations of H. Let Λ=i=1n ωi, where the ωi are the fundamental weights of the Lie algebra 𝔤 of G. We shall use the “geometric” convention (see [CGi1433132, 6.1.9(ii)]) and let e-λ be the element of R(T) corresponding to the character determined by λΛ. Then

R(T)has -basis {eλλΛ}, with multiplicationeλeμ =eλ+μ,

and Weyl group action determined by weλ=ewλ, for wW and λΛ. In this way R(T) is a Laurent polynomial ring and R(G)R(T)W is the subalgebra of "symmetric functions" in R(T).

Suppose that V is a T-module. Since TB/U, where U is the unipotent radical of B, we can extend V to be a B-module by defining the action of U to be trivial. Define a vector bundle

π: G×BV G/B (g,v) gB whereG×BV= G×V (g,v) (gb,b-1v) ,

so that G×BV is the set of pairs (g,v), gG, vV, modulo the equivalence relation (g,v)(gb,b-1v). This construction induces a ring homomorphism

ϕ: R(T) K(G/B) V ( GBV π G/B ) . (1.2)

If V is a G-module then ϕ(V)=dim(V) in K(G/B). This is because the map

G×BV G/B×V (g,x) (gB,gx) (1.3)

is an isomorphism between G×BV and the trivial bundle G/B×V. Define ε:R(T) by ε(eλ)=1 for λΛ. Then the map ϕ in (1.2) gives an isomorphism (see Proposition 1.5 below)

K(G/B) R(T)/,

where is the ideal generated by { fR(T)W f-ε(f)=0 } . Equivalently, K(G/B)R(T) R(G), where R(G) acts on by [V]·1=dim(V), if V is a G-module.

K(G/P) for a parabolic subgroup P

A similar setup works when B is replaced by any parabolic subgroup P containing B. The coset space G/P is a projective variety and the Bruhat decomposition takes the form G=wW/WP BwP where WP is the subgroup of W given by WP= si 𝔤-αi𝔭 , where 𝔭 is the Lie algebra of P. The Schubert varieties Xw are the closures of the Schubert cells Xw=BwP in G/P.

K(G/P) is a free-module with basis { [𝒪Xw] wW/ WP } .

Write P=LU where U is the unipotent radical of P and L is a Levi subgroup. The Weyl group of L is WP and


is the subring of WP-symmetric functions in R(T). The same construction as in (1.2) with B replaced by P and T replaced by L gives a ring homomorphism

ϕP: R(L) K(G/P) V ( G×PV π G/P ) . (1.4)

Proposition 1.5. Let G be a connected simply connected semisimple Lie group and let T be a maximal torus of G. Let P be a parabolic subgroup of G with Levi decomposition P=LU and let WP be the Weyl group of L. Then

K(G/P) R(T)WP P ,

where P is the ideal generated by { fR(T)WP f-ε(f)=0 } and ε:R(T) is the map given by ε(eλ)=1 for λΛ. Equivalently, K(G/P)=R(L) R(G).


Let Kvb(G/P) be the Grothendieck group of C vector bundles on G/P and let η:K(G/P) Kvb(G/P) be the map which assigns to a locally free sheaf its underlying vector bundle. Let ϕP be the composition

ϕP:R(L) ϕPK(G/P) ηKvb (G/P).

Since π1(G)=0, π1(P)π2(G/P) which is free abelian by the Bruhat decomposition. Since the unipotent radical U of P is contractible the projection f:PP/UL is a homotopy equivalence. Thus π1(L) is free abelian and we may apply the results of [Pit1972] to conclude that ϕP:R(L) Kvb(G/P) is surjective, R(L) is projective over R(G) with rank W/WP and Kvb(G/P) is a free -module of the same rank. (Note: The results of [Pit1972] can be applied since G and L are the complexifications of compact groups.)

Since ϕP is surjective the map η is also surjective. Then, since K(G/P) and Kvb(G/P) are both free -modules of rank W/WP, it follows that η must be an isomorphism. This means two things: (1) that we can identify K(G/P) and Kvb(G/P), and (2) that ϕP is surjective.

The kernel P of ϕP is identified by using (1.3).

Transfer from K(G/B) to K(G/P)

Although we will work primarily with K(G/B) it is standard to transfer results from K(G/B) to results on K(G/P). This can be accomplished with the following proposition. The proof will be given in section 5.

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. This map is given explicitly by

f!([𝒪Xw]) =[𝒪Xv],

where vW is the unique element of longest length in the coset w=vWP.

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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