## Background

Last update: 21 February 2013

## Background

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 ${H}^{*}\left(G/B\right)$ and the combinatorial theory of Schubert polynomials) the K-theory of $G/B$ is isomorphic to a quotient of $R\left(T\right),$ which is a ring of Laurent polynomials. In section 5 we shall explain how our results about $K\left(G/B\right)$ relate to and generalize well known results about Schubert polynomials and ${H}^{*}\left(G/B\right)\text{.}$

The K-theory K(G/B)

If $X$ is a quasi-projective variety let

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

If $f:X\to Y$ is a morphism of projective varieties then we have maps

$f!: K(Y) ⟶ K(X) [ℱ] ⟼ [f*ℱ], f!: K(X) ⟶ K(Y) [ℱ] ⟼ ∑i(-1)i[Rif*ℱ].$

See [CGi1433132, §5.2] for K-theory background. If $X$ is smooth then the isomorphism between $K\left(X\right)$ and ${K}_{lf}\left(X\right)$ 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 ${K}_{lf}\left(X\right)\to {K}_{vb}\left(X\right)$ 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 $T\subseteq B\subseteq G\text{.}$ The Bruhat decomposition says that $G$ is a disjoint union of double cosets of $B$ indexed by the elements of the Weyl group $W,$ $G={\bigcup }_{w\in W}BwB\text{.}$ 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 $G/B\text{.}$ For each $w\in W$ the subset ${X}_{w}^{\circ }=BwB/B$ is the Schubert cell and its closure ${X}_{w}$ is the Schubert variety. The formulas

$dim(Xw∘)= ℓ(w)and Xw=⋃v≤w Xv∘ (1.1)$

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

$K(G/B) is a free ℤ-module with basis { [𝒪Xw] ∣ w∈W } ,$

where ${X}_{w}$ are the Schubert varieties in $G/B$ and ${𝒪}_{{X}_{w}}$ is the structure sheaf of ${X}_{w}$ extended to $G/B$ by defining it to be 0 outside ${X}_{w}\text{.}$

The isomorphism $K\left(G/B\right)\cong R\left(T\right)/ℐ$

For any group $H$ let $R\left(H\right)$ be the Grothendieck group of complex representations of $H\text{.}$ Let $\Lambda ={\sum }_{i=1}^{n}ℤ{\omega }_{i},$ where the ${\omega }_{i}$ are the fundamental weights of the Lie algebra $𝔤$ of $G\text{.}$ We shall use the “geometric” convention (see [CGi1433132, 6.1.9(ii)]) and let ${e}^{-\lambda }$ be the element of $R\left(T\right)$ corresponding to the character determined by $\lambda \in \Lambda \text{.}$ Then

$R(T) has ℤ -basis {eλ ∣ λ∈Λ}, with multiplication eλeμ =eλ+μ,$

and Weyl group action determined by $w{e}^{\lambda }={e}^{w\lambda },$ for $w\in W$ and $\lambda \in \Lambda \text{.}$ In this way $R\left(T\right)$ is a Laurent polynomial ring and $R\left(G\right)\cong R{\left(T\right)}^{W}$ is the subalgebra of "symmetric functions" in $R\left(T\right)\text{.}$

Suppose that $V$ is a $T\text{-module.}$ Since $T\cong B/U,$ where $U$ is the unipotent radical of $B,$ we can extend $V$ to be a $B\text{-module}$ 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 $G{×}_{B}V$ is the set of pairs $\left(g,v\right),$ $g\in G,$ $v\in V,$ modulo the equivalence relation $\left(g,v\right)\sim \left(gb,{b}^{-1}v\right)\text{.}$ This construction induces a ring homomorphism

$ϕ: R(T) ⟶ K(G/B) V ⟼ ( G⊗BV ⟶π G/B ) . (1.2)$

If $V$ is a $G\text{-module}$ then $\varphi \left(V\right)=\text{dim}\left(V\right)$ in $K\left(G/B\right)\text{.}$ This is because the map

$G×BV ⟶ G/B×V (g,x) ⟼ (gB,gx) (1.3)$

is an isomorphism between $G{×}_{B}V$ and the trivial bundle $G/B×V\text{.}$ Define $\epsilon :R\left(T\right)\to ℤ$ by $\epsilon \left({e}^{\lambda }\right)=1$ for $\lambda \in \Lambda \text{.}$ Then the map $\varphi$ in (1.2) gives an isomorphism (see Proposition 1.5 below)

$K(G/B)≅ R(T)/ℐ,$

where $ℐ$ is the ideal generated by $\left\{f\in R{\left(T\right)}^{W} \mid f-\epsilon \left(f\right)=0\right\}\text{.}$ Equivalently, $K\left(G/B\right)\cong R\left(T\right){\otimes }_{R\left(G\right)}ℤ,$ where $R\left(G\right)$ acts on $ℤ$ by $\left[V\right]·1=\text{dim}\left(V\right),$ if $V$ is a $G\text{-module.}$

K(G/P) for a parabolic subgroup P

A similar setup works when $B$ is replaced by any parabolic subgroup $P$ containing $B\text{.}$ The coset space $G/P$ is a projective variety and the Bruhat decomposition takes the form $G={\bigcup }_{\stackrel{‾}{w}\in W/{W}_{P}}B\stackrel{‾}{w}P$ where ${W}_{P}$ is the subgroup of $W$ given by ${W}_{P}=⟨{s}_{i} \mid {𝔤}_{-{\alpha }_{i}}\in 𝔭⟩,$ where $𝔭$ is the Lie algebra of $P\text{.}$ The Schubert varieties ${X}_{\stackrel{‾}{w}}$ are the closures of the Schubert cells ${X}_{\stackrel{‾}{w}}^{\circ }=B\stackrel{‾}{w}P$ in $G/P\text{.}$

$K(G/P) is a free ℤ-module with basis { [𝒪Xw‾] ∣ w‾∈W/ WP } .$

Write $P=LU$ where $U$ is the unipotent radical of $P$ and $L$ is a Levi subgroup. The Weyl group of $L$ is ${W}_{P}$ and

$R(L)≅R (T)WP$

is the subring of ${W}_{P}\text{-symmetric}$ functions in $R\left(T\right)\text{.}$ 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 $G\text{.}$ Let $P$ be a parabolic subgroup of $G$ with Levi decomposition $P=LU$ and let ${W}_{P}$ be the Weyl group of $L\text{.}$ Then

$K(G/P)≅ R(T)WP ℐP ,$

where ${ℐ}_{P}$ is the ideal generated by $\left\{f\in R{\left(T\right)}^{{W}_{P}} \mid f-\epsilon \left(f\right)=0\right\}$ and $\epsilon :R\left(T\right)\to ℤ$ is the map given by $\epsilon \left({e}^{\lambda }\right)=1$ for $\lambda \in \Lambda \text{.}$ Equivalently, $K\left(G/P\right)=R\left(L\right){\otimes }_{R\left(G\right)}ℤ\text{.}$

 Proof. Let ${K}_{vb}\left(G/P\right)$ be the Grothendieck group of ${C}^{\infty }$ vector bundles on $G/P$ and let $\eta :K\left(G/P\right)\to {K}_{vb}\left(G/P\right)$ be the map which assigns to a locally free sheaf its underlying vector bundle. Let ${\varphi }_{P}$ be the composition $ϕ∼P:R(L) ⟶ϕPK(G/P) ⟶ηKvb (G/P).$ Since ${\pi }_{1}\left(G\right)=0,$ ${\pi }_{1}\left(P\right)\cong {\pi }_{2}\left(G/P\right)$ which is free abelian by the Bruhat decomposition. Since the unipotent radical $U$ of $P$ is contractible the projection $f:P\to P/U\cong L$ is a homotopy equivalence. Thus ${\pi }_{1}\left(L\right)$ is free abelian and we may apply the results of [Pit1972] to conclude that ${\stackrel{\sim }{\varphi }}_{P}:R\left(L\right)\to {K}_{vb}\left(G/P\right)$ is surjective, $R\left(L\right)$ is projective over $R\left(G\right)$ with rank $\mid W/{W}_{P}\mid$ and ${K}_{vb}\left(G/P\right)$ is a free $ℤ\text{-module}$ of the same rank. (Note: The results of [Pit1972] can be applied since $G$ and $L$ are the complexifications of compact groups.) Since ${\stackrel{\sim }{\varphi }}_{P}$ is surjective the map $\eta$ is also surjective. Then, since $K\left(G/P\right)$ and ${K}_{vb}\left(G/P\right)$ are both free $ℤ\text{-modules}$ of rank $\mid W/{W}_{P}\mid ,$ it follows that $\eta$ must be an isomorphism. This means two things: (1) that we can identify $K\left(G/P\right)$ and ${K}_{vb}\left(G/P\right),$ and (2) that ${\varphi }_{P}$ is surjective. The kernel ${ℐ}_{P}$ of ${\varphi }_{P}$ is identified by using (1.3). $\square$

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

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

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. This map is given explicitly by

$f!([𝒪Xw‾]) =[𝒪Xv],$

where $v\in W$ is the unique element of longest length in the coset $\stackrel{‾}{w}=v{W}_{P}\text{.}$

## 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.