Background

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

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

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

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

$$\begin{array}{cc}\text{dim}\left(X_w^{\circ }\right)=\ell \left(w\right)\text{and}{X}_w=\bigcup _{v\le w}{X}_v^{\circ }& \text{(1.1)}\end{array}$$

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

$$K(G/B)\hspace{0.17em}\text{is a free}\hspace{0.17em}\mathbb{Z}\text{-module with basis}\hspace{0.17em}\{\left[{𝒪}_{X_w}\right]\hspace{0.17em}\mid \hspace{0.17em}w\in W\},$$

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 $K(G/B)\cong R\left(T\right)/ℐ$

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

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

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

$$\begin{array}{cccc}\pi :& G{\times }_{B}V& ⟶& G/B\\ & (g,v)& ⟼& gB\end{array}\text{where}G{\times }_{B}V=\frac{G\times V}{⟨(g,v)\sim (gb,b^{-1}v)⟩},$$

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

$$\begin{array}{cc}\begin{array}{cccc}\varphi :& R\left(T\right)& ⟶& K(G/B)\\ & V& ⟼& (G{\otimes }_{B}V\stackrel{\pi }{⟶}G/B)\text{.}\end{array}& \text{(1.2)}\end{array}$$

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

$$\begin{array}{cc}\begin{array}{ccc}G{\times }_{B}V& ⟶& G/B\times V\\ (g,x)& ⟼& (gB,gx)\end{array}& \text{(1.3)}\end{array}$$

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)

$$K(G/B)\cong R\left(T\right)/ℐ,$$

where $ℐ$ 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 $\left[V\right]·1=\text{dim}\left(V\right),$ 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 $G={\bigcup }_{\bar{w}\in W/W_P}B\bar{w}P$ where $W_P$ is the subgroup of $W$ given by $W_P=⟨s_i\hspace{0.17em}\mid \hspace{0.17em}{𝔤}_{-{\alpha }_i}\in 𝔭⟩,$ where $𝔭$ is the Lie algebra of $P\text{.}$ The Schubert varieties $X_{\bar{w}}$ are the closures of the Schubert cells $X_{\bar{w}}^{\circ }=B\bar{w}P$ in $G/P\text{.}$

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

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

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

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

$$K(G/P)\cong \frac{R{\left(T\right)}^{W_P}}{{ℐ}_P},$$

where ${ℐ}_P$ 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 $${\stackrel{\sim }{\varphi }}_P:R\left(L\right)\stackrel{{\varphi }_P}{⟶}K(G/P)\stackrel{\eta }{⟶}{K}_{vb}(G/P)\text{.}$$ 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 ${ℐ}_P$ 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

$$f^{!}\left(\left[{𝒪}_{X_{\bar{w}}}\right]\right)=\left[{𝒪}_{X_v}\right],$$

where $v\in W$ is the unique element of longest length in the coset $\bar{w}=v{W}_P\text{.}$

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.

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