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 $R\left(T\right),$ 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 ${H}^{*}(G/B)\text{.}$

The K-theory K(G/B)

If $X$ is a quasi-projective variety let

$$\begin{array}{ccc}K\left(X\right)& =& \text{the Grothendieck group of coherent sheaves on}\hspace{0.17em}X,\\ {K}_{lf}\left(X\right)& =& \text{the Grothendieck group of locally free sheaves on}\hspace{0.17em}X,\\ {K}_{vb}\left(X\right)& =& \text{the Grothendieck group of vector bundles on}\hspace{0.17em}X\text{.}\end{array}$$If $f:X\to Y$ is a morphism of projective varieties then we have maps

$$\begin{array}{cccc}{f}^{!}:& K\left(Y\right)& \u27f6& K\left(X\right)\\ & \left[\mathcal{F}\right]& \u27fc& \left[{f}^{*}\mathcal{F}\right],\\ \text{}\\ {f}_{!}:& K\left(X\right)& \u27f6& K\left(Y\right)\\ & \left[\mathcal{F}\right]& \u27fc& \sum _{i}{(-1)}^{i}\left[{R}^{i}{f}_{*}\mathcal{F}\right]\text{.}\end{array}$$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

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

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

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

For any group $H$ let $R\left(H\right)$ be the Grothendieck group of complex representations of $H\text{.}$ Let $\Lambda ={\sum}_{i=1}^{n}\mathbb{Z}{\omega}_{i},$ where the ${\omega}_{i}$ are the fundamental weights of the Lie algebra $\U0001d524$ 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

$$R\left(T\right)\hspace{0.17em}\text{has}\hspace{0.17em}\mathbb{Z}\text{-basis}\hspace{0.17em}\{{e}^{\lambda}\hspace{0.17em}\mid \hspace{0.17em}\lambda \in \Lambda \},\phantom{\rule{2em}{0ex}}\text{with multiplication}\hspace{0.17em}{e}^{\lambda}{e}^{\mu}={e}^{\lambda +\mu},$$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

$$\begin{array}{cccc}\pi :& G{\times}_{B}V& \u27f6& G/B\\ & (g,v)& \u27fc& gB\end{array}\phantom{\rule{2em}{0ex}}\text{where}\phantom{\rule{2em}{0ex}}G{\times}_{B}V=\frac{G\times V}{\u27e8(g,v)\sim (gb,{b}^{-1}v)\u27e9},$$so that $G{\times}_{B}V$ is the set of pairs $(g,v),$ $g\in G,$ $v\in V,$ modulo the equivalence relation $(g,v)\sim (gb,{b}^{-1}v)\text{.}$ This construction induces a ring homomorphism

$$\begin{array}{cc}\begin{array}{cccc}\varphi :& R\left(T\right)& \u27f6& K(G/B)\\ & V& \u27fc& (G{\otimes}_{B}V\stackrel{\pi}{\u27f6}G/B)\text{.}\end{array}& \text{(1.2)}\end{array}$$If $V$ is a $G\text{-module}$ then $\varphi \left(V\right)=\text{dim}\left(V\right)$ in $K(G/B)\text{.}$ This is because the map

$$\begin{array}{cc}\begin{array}{ccc}G{\times}_{B}V& \u27f6& G/B\times V\\ (g,x)& \u27fc& (gB,gx)\end{array}& \text{(1.3)}\end{array}$$is an isomorphism between $G{\times}_{B}V$ and the trivial bundle $G/B\times V\text{.}$ Define $\epsilon :R\left(T\right)\to \mathbb{Z}$ 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)

$$K(G/B)\cong R\left(T\right)/\mathcal{I},$$where $\mathcal{I}$ is the ideal generated by $\{f\in R{\left(T\right)}^{W}\hspace{0.17em}\mid \hspace{0.17em}f-\epsilon \left(f\right)=0\}\text{.}$ Equivalently, $K(G/B)\cong R\left(T\right){\otimes}_{R\left(G\right)}\mathbb{Z},$ where $R\left(G\right)$ acts on $\mathbb{Z}$ by $\left[V\right]\xb71=\text{dim}\left(V\right),$ 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 $G={\bigcup}_{\stackrel{\u203e}{w}\in W/{W}_{P}}B\stackrel{\u203e}{w}P$ where ${W}_{P}$ is the subgroup of $W$ given by ${W}_{P}=\u27e8{s}_{i}\hspace{0.17em}\mid \hspace{0.17em}{\U0001d524}_{-{\alpha}_{i}}\in \U0001d52d\u27e9,$ where $\U0001d52d$ is the Lie algebra of $P\text{.}$ The Schubert varieties ${X}_{\stackrel{\u203e}{w}}$ are the closures of the Schubert cells ${X}_{\stackrel{\u203e}{w}}^{\circ}=B\stackrel{\u203e}{w}P$ in $G/P\text{.}$

$$K(G/P)\hspace{0.17em}\text{is a free}\hspace{0.17em}\mathbb{Z}\text{-module with basis}\hspace{0.17em}\{\left[{\mathcal{O}}_{{X}_{\stackrel{\u203e}{w}}}\right]\hspace{0.17em}\mid \hspace{0.17em}\stackrel{\u203e}{w}\in W/{W}_{P}\}\text{.}$$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

$$R\left(L\right)\cong R{\left(T\right)}^{{W}_{P}}$$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

$$\begin{array}{cc}\begin{array}{cccc}{\varphi}_{P}:& R\left(L\right)& \u27f6& K(G/P)\\ & V& \u27fc& (G{\times}_{P}V\stackrel{\pi}{\u27f6}G/P)\text{.}\end{array}& \text{(1.4)}\end{array}$$
**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

where ${\mathcal{I}}_{P}$ is the ideal generated by $\{f\in R{\left(T\right)}^{{W}_{P}}\hspace{0.17em}\mid \hspace{0.17em}f-\epsilon \left(f\right)=0\}$ and $\epsilon :R\left(T\right)\to \mathbb{Z}$ is the map given by $\epsilon \left({e}^{\lambda}\right)=1$ for $\lambda \in \Lambda \text{.}$ Equivalently, $K(G/P)=R\left(L\right){\otimes}_{R\left(G\right)}\mathbb{Z}\text{.}$

Proof. | |

Let ${K}_{vb}(G/P)$ be the Grothendieck group of ${C}^{\infty}$ vector bundles on $G/P$ and let $\eta :K(G/P)\to {K}_{vb}(G/P)$ be the map which assigns to a locally free sheaf its underlying vector bundle. Let ${\varphi}_{P}$ be the composition $${\stackrel{\sim}{\varphi}}_{P}:R\left(L\right)\stackrel{{\varphi}_{P}}{\u27f6}K(G/P)\stackrel{\eta}{\u27f6}{K}_{vb}(G/P)\text{.}$$Since ${\pi}_{1}\left(G\right)=0,$ ${\pi}_{1}\left(P\right)\cong {\pi}_{2}(G/P)$ 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 ${\stackrel{\sim}{\varphi}}_{P}:R\left(L\right)\to {K}_{vb}(G/P)$ is surjective, $R\left(L\right)$ is projective over $R\left(G\right)$ with rank $\mid W/{W}_{P}\mid $ and ${K}_{vb}(G/P)$ is a free $\mathbb{Z}\text{-module}$ 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 $K(G/P)$ and ${K}_{vb}(G/P)$ are both free $\mathbb{Z}\text{-modules}$ of rank $\mid W/{W}_{P}\mid ,$ it follows that $\eta $ must be an isomorphism. This means two things: (1) that we can identify $K(G/P)$ and ${K}_{vb}(G/P),$ and (2) that ${\varphi}_{P}$ is surjective. The kernel ${\mathcal{I}}_{P}$ 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 $K(G/B)$ it is standard to transfer results from $K(G/B)$ to results on $K(G/P)\text{.}$ 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 ${f}^{!}:K(G/P)\to K(G/B)$
is an injection. This map is given explicitly by

where $v\in W$ is the unique element of longest length in the coset $\stackrel{\u203e}{w}=v{W}_{P}\text{.}$

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.