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\begin{document}


\title{A Pieri-Chevalley formula in the K-theory of a $G/B$-bundle}

\author{Harsh Pittie}
\address{Department of Mathematics, Graduate Center,
City University of New York,
New York, NY 10036}
\thanks{Research supported in part by National
Science Foundation grant DMS-9622985.}

\author{Arun Ram}
\address{Department of Mathematics,
Princeton University,
Princeton, NJ 08544}
\email{rama@math.princeton.edu}



\issueinfo{5}{14}{}{1999}
\dateposted{July 14, 1999}
\pagespan{102}{107}
\PII{S 1079-6762(99)00067-0}
\def\copyrightyear{1999}
\copyrightinfo{1999}{American Mathematical Society}

\subjclass{Primary 14M15; Secondary 14C35, 19E08}

\date{February 9, 1999}
\revdate{April 29, 1999}
\commby{Efim Zelmanov}

%**************** ABSTRACT *************************

\begin{abstract}
Let $G$ be a semisimple complex Lie group, $B$ a
Borel subgroup, and $T\subseteq B$ a maximal torus of $G$. The
projective variety $G/B$ is a generalization of the classical flag
variety.  The structure sheaves of the Schubert subvarieties form a
basis of the K-theory $K(G/B)$ and every character of $T$ gives rise
to a line bundle on $G/B$.  This note gives a formula for the product
of a dominant line bundle and a Schubert class in $K(G/B)$. This
result generalizes a formula of Chevalley which computes an analogous
product in cohomology.  The new formula applies to the relative case,
the K-theory of a $G/B$-bundle over a smooth base $X$, and is
presented in this generality. In this setting the new formula is a
generalization of recent $G=GL_n({\mathbb C})$ results of Fulton and
Lascoux.
\end{abstract}

\maketitle

Let $G$ be a complex, semisimple, simply connected
algebraic group and $B \subseteq G$ a Borel subgroup.
We fix a smooth closed complex projective variety $X$ and a principal
algebraic $B$-bundle over it:
$B \longrightarrow E \mapright{\pi} X$.  For
any complex algebraic variety $F$ with a left algebraic $B$-action, we
denote by $E (F)$ the total space of the associated fibre bundle
with fibre $F$.  Thus $E(F) = E \times_B F$
and the projection to
$X$ is obtained from projection on the first factor.

Fix a maximal torus $T\subseteq B$ and let $W$ be its Weyl group.
For each $w\in W$ the Bruhat cell $Y_w^\circ=BwB\subseteq G/B$ and
the Schubert variety $Y_w=\overline{BwB}\subseteq G/B$ are $B$-stable subsets of
$G/B$ so we have inclusions of bundles $E(Y_w^\circ)\subseteq 
E(Y_w)\subseteq E(G/B)$.  The closed subvarieties $\Omega_w=E(Y_w)$
determine classes $[\cO_{\Omega_w}]$ in 
$K(E(G/B))$. \footnote{For any smooth variety $V$, 
$K(V)$ is the Grothendieck ring of coherent $\cO_V$-modules.}
In fact, by a well-known result of Grothendieck, these classes
form a $K(X)$-basis for $K(E(G/B))$.  On $E(G/B))$ we also have ``homogeneous''
line bundles associated to irreducible representations of $B$ (see below).
The main result of this announcement is a formula for the tensor
product of the class of a homogeneous line bundle with a 
Schubert class, expressed as a $K(X)$-linear combination of Schubert
classes.  

We believe that this formula is the most general uniform result in the
intersection theory of Schubert classes: it is related to a recent result
of Fulton and Lascoux \cite{FL}, who presented a similar formula for a
$GL_n(\CC)/B$-bundle.  Indeed, in this case, their formula and ours
coincide once one knows how to translate between their 
combinatorics with tableaux and ours with Littelmann paths.  O. Mathieu has also
proved the positivity which is implied by our formula; see \cite[p. 101]{FP}. 
Applying the Chern character to our formula, and equating the lowest order terms
we obtain a relative version of the result of Chevalley \cite{Ch} alluded to in the
title of this paper.
%, which is for the case $X=\mathrm{pt}$. 
%Complete proofs of our results will be given in a forthcoming paper.

The ring $K(E(G/B))$ is a $K(X)$-module via the map 
$\pi^*\colon K(X)\to K(E(G/B))$.  Since $G/B$ has a unique fixed point 
for the $B$-action, there is a canonical
section $\sigma\colon X\to E(G/B)$ of the bundle $E(G/B)$.
Consider the diagram
\[
\begin{matrix}
G& \rightarrow &E(G)& \rightarrow &X \\
\downarrow&& \downarrow {\scriptstyle \rho}&& \Vert \\
G/B & \rightarrow &E(G/B)& \rightarrow &X 
\end{matrix}
\]
where the vertical maps are quotients by the right action of
$B$ on $G$; precisely, $E(G/B) \simeq (E\times _B G) / B$.  Thus
$\rho$ is the projection map of a principal $B$-bundle over 
$E(G/B)$.

There are two vector bundles naturally associated to each $B$-module
$V$:
\[
E(V) \longrightarrow X, \qquad\hbox{and}\qquad
E_G(V)=E(G)\times_B V \longrightarrow E(G/B),\]
where the projection map for the latter of these is via $\rho$.
This assignment $V \mapsto E_G (V)$ of $B$-modules to vector
bundles over $E(G/B)$ preserves direct sums and tensor products,
and hence induces a ring homomorphism 
$R(B) \mapright{\phi} K (E(G/B))$,
where $R(B)$ is the representation ring of $B$. 
By construction $\sigma^*(E_G(V))=E(V)$ as vector bundles on $X$.
One also checks that if $V$ is the restriction
of a $G$-module, then $E_G(V)=\pi^*(\sigma^*(E_G(V)))$. 
Thus we have a commutative diagram
\[
\begin{matrix}
R(G) &-\ -\ \to &K(X) \\
\downarrow {\scriptstyle\mathrm{res}} &&\downarrow {\scriptstyle \pi^*} \\
R(B) &\longrightarrow &K(E(G/B))
\end{matrix}\]
and a map
\[K(X)\otimes_{R(G)} R(B)\quad \mapright{\pi^*\otimes\phi}\quad K(E(G/B)),\]
where $R(G)$ is the representation ring of $G$ and the $R(G)$-action
on $K(X)$ is given by the map $V\mapsto E(V)$.

Let $P$ be the weight lattice of 
$\mygg=\Lie(G)$. Then $R(B)=R(T)\cong \ZZ[P]$, the group algebra of $P$,
and $R(G)=R(T)^W$.  If $\lambda\in P$, let $e^\lambda$ be the corresponding
element of $R(T)$ and define
\begin{equation}
x^\lambda = E(e^\lambda)\in K(X)
\qquad\hbox{and}\qquad
y^\lambda = E_G(e^\lambda)\in K(E(G/B)).\end{equation}
The statement that
$E_G(V)=\pi^*(\sigma^*(E_G(V)))$ if $V$ is a $G$-module is equivalent
to the statement that, in $K(E(G/B))$, 
\[\chi(x) = E(\chi)
\quad\hbox{is equal to}\quad
\chi(y)=E_G(\chi),
\quad\hbox{for all $\chi\in R(T)^W$.}\]
We recall from \cite{P} that $R(T)$ is a free $R(G)$-module of rank
$|W|$, and $R(T)\otimes_{R(G)}\mathbb Z\longrightarrow K(G/B)$
is an isomorphism.\footnote{The discussion in [P] is entirely in
terms of compact groups and the $K$-theory of $C^\infty$ vector
bundles;  with trivial modifications the results hold
in the present context also.} 
According to Steinberg, \cite{S} there is an $R(G)$-basis of
$R(T)$ of the form $\{e^{\varepsilon_w}\ |\ w\in W\}$, where the $\varepsilon_w$
are certain specific elements of $P$.
Since the set $\{y^{\varepsilon_w}\ |\ w\in W\}$
is a set of globally defined elements in $K(E(G/B))$ which behaves 
properly under restriction, and which forms a basis locally,
it follows from standard yoga that it is also 
a $K(X)$-basis for $K(E(G/B))$.  Thus the map
$K(X)\otimes_{R(G)} R(T) \longrightarrow K(E(G/B))$ is an isomorphism and
\begin{equation}
K(E(G/B))\cong \frac{K(X)\otimes R(T)}{\cI},\end{equation}
where $\cI$ is the ideal in $K(X)\otimes R(T)$ generated by the
set $\{ \chi(x)\otimes 1 -1\otimes \chi\ |\ \chi\in R(T)^W\}$.

Define a $W$-action on $K(X)\otimes R(T)$ as the $K(X)$-linear extension of the
action given by
\[wy^\lambda= y^{w\lambda},\qquad\hbox{for $w\in W$, $\lambda\in P$.}\]
This action descends to an action on $K(E(G/B))$, since the generators of 
the ideal $\cI$ are $W$-invariants for this action. 
Using this $W$-action on $K(E(G/B))$, we can
define the analogues of BGG-operators in this context.
Such operators were defined in the ``absolute case''
($X=\pt$) by Demazure, in $K_T (G/B)$ by Kostant and Kumar
\cite{KK}, and finally by Fulton and Lascoux \cite{FL} when $G=  SL(n, \mathbb C)$.
To make the definition, let $\alpha$ be a positive root with respect
to the pair $(B,T)$ and let $s_\alpha \in W$ be the corresponding
reflection.
Define $T_\alpha\colon R(T)\to R(T)$ by setting
\[T_\alpha (e^\lambda ) 
= (e^{\lambda + \alpha} - s_\alpha (e^\lambda) ) / (e^\alpha -1)\] 
and extending $\ZZ$-linearly.  Since $T_\alpha$ fixes elements of
$R(T)^W$, this operation can be extended $K(X)$-linearly to a well-defined
operator on $K(E(G/B))$.


Now fix a simple system of roots $\alpha_1 , \dots , \alpha _\ell$
for $(B,T)$ and let $P_j$ be the minimal parabolic subgroup 
corresponding to $\alpha_j$; this is the closed connected
subgroup of $G$ whose Lie algebra $\frak{p}_j$ is spanned by the
Lie algebra $\frak b$ of $B$ and the root space $\mygg_{-\alpha_j}$.
Let $f_j : E(G/B) \longrightarrow E (G/P_j)$ be the 
projection induced from the $B$-equivariant ${\mathbb P}^1$-bundle
$G/B \longrightarrow G/P_j$ (the canonical projection). 
The following result explains the geometric significance of the
operators $T_{\alpha_j}$ (henceforth abbreviated as $T_j$).
P.~Deligne pointed out an error in the proof of (a) below in an earlier
version of this preprint.  We are grateful to him for pointing this
out and have corrected the argument.

\begin{prop}
With the notation as above,
\begin{enumerate}
\item[(a)] $(f_j)^! \circ (f_j)_! ([\cO_{\Omega_w}]) =
\begin{cases}
[\cO_{\Omega_{ w s_j}}] & \mbox{if $\ell (w s_j ) > \ell (w)$,}\\
\mbox{$[\cO_{\Omega_w}]$} & \mbox{if $\ell(w s_j) < \ell (w)$.} 
\end{cases}$

%\end{displaymath}
%\end{flushleft}
%\noindent
\item[(b)] For any element $x \in K(E(G/B)),\ \
(f_j)^! \circ (f_j)_!(x) = T_j (x)$.
\end{enumerate}
\end{prop}

\begin{proof}
(a) Let $\bar w = \{w,ws_j\}$ be the coset of $w$ relative to 
$\langle s_j\rangle$.  The essential point is to prove the following
two equations:
\[f_! [\cO_{\Omega_v}] = [\cO_{\Omega_{\bar w}}],
\qquad\hbox{for $v\in \bar w$,}\]
where $\Omega_{\bar w}\subseteq E(G/P_j)$ is the relative Schubert variety
constructed from $Y_{\bar w}\subseteq E(G/P_j)$.  In turn, these equations
will follow from the isomorphisms
\[\hbox{(i)}\quad f_*(\cO_{\Omega_v})=\cO_{\Omega_{\bar w}},
\qquad\quad
\hbox{(ii)}\quad R^qf_*(\cO_{\Omega_v})=0,
\quad\hbox{for $q>0$, $v\in \bar w$.}\]

To prove (i) and (ii) relabel the elements of $\bar w$ as $w'$
and $w''$ where $w'\tau_2>\cdots >\tau_r),\hfill
&\tau_i\in W/W_\lambda, \qquad \hbox{and}\hfill\\
\hfill\vec a&=(0=a_0\cdots>t_r$ and each $t_i$ is maximal in Bruhat order such that
$t_{i-1}>t_i$.   The \textit{ final direction} of $\pi$ with respect to $w$ is
\[v(\pi,w)=t_r,\]
where
$w\ge t_1>\cdots>t_r$ is a maximal lift of $\tau_1>\dots>\tau_r$
with respect to $w$.

\begin{thm}  Let $\lambda$ be a dominant integral weight and let $w\in W$.  
%Let $T_w$ and $Y^\mu$ be the operators on $K(E(G/B))$ 
%defined by ??? and ???, respectively. 
Then
\[Y^\lambda T_{w^{-1}} = 
\sum_{\eta\in \cT^\lambda_{\le w}} T_{v(\eta,w)^{-1}} Y^{\eta(1)}
\]
as operators on $K(E(G/B))$.
\end{thm}
\begin{proof}[Sketch of proof]
Fix a simple root $\alpha_i$.  Every path is in
a unique \textit{ $\alpha_i$-string} of paths
\[S_{\alpha_i}(\pi)=\{f_i^m\pi,\dots, f_i^2\pi,f_i\pi,\pi\},\]
where $f_i^m\pi=0$ and there does not exist any path $\eta$ such that
$f_i\eta=\pi$.  In a manner similar to that of \cite[Lemma 5.3]{Li} one
shows that, for any $\alpha_i$-string $S_{\alpha_i}(\pi)$,
\[\sum_{\eta\in S_{\alpha_i}(\pi)} T_{v(\eta,w)^{-1}}Y^{\eta(1)}
=T_{v(\pi,w)^{-1}}Y^{\pi(1)}T_i.\]
Given these facts, the proof of the Theorem follows the same lines as the proof
of the Demazure character formula given in \cite[5.5]{Li}. 
\end{proof}


By applying the formula in the Theorem to the element $[\cO_{\Omega_1}]\in
K(E(G/B))$ and using (3) and (4) we obtain the following.

\begin{cor}  Let $\lambda$ be a dominant integral weight
%, let $y^\lambda$ be the line bundle on $E(G/B)$ defined by ???, 
and let $w\in W$.  In $K(E(G/B))$,
\[y^\lambda [\cO_{\Omega_w}] = 
\sum_{\eta\in \cT^\lambda_w} [\cO_{\Omega_{v(\eta,w)}}]x^{\eta(1)}.
\]
%where $x^{\eta(1)}$ are the elements of $K(X)$ defined by ???.
\end{cor}

\section*{Acknowledgements}
We are grateful to many people for comments,
suggestions and encouragement: we would particularly like to thank
Jim Carlson, Mark Green, Shrawan Kumar, Bob MacPherson, Ted Shifrin 
and Al Vasquez.



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\MR{95e:14041}

\bibitem[FL]{FL} W. Fulton and A. Lascoux,
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\MR{96j:14036}

\bibitem[FP]{FP} W. Fulton and P. Pragacz,
\textit{Schubert varieties and degeneracy loci},
Lecture Notes in Math. \textbf{ 1689}, Springer-Verlag, Berlin 1998.
\CMP{98:17}

\bibitem[KK]{KK} B. Kostant and S. Kumar, 
\textit{$T$-equivariant K-theory of generalized flag varieties}, J. Differential
Geom. \textbf{ 32} (1990), 549--603. 
\MR{92c:19006}

\bibitem[Li]{Li} P. Littelmann,
\textit{ A Littlewood-Richardson rule for symmetrizable Kac-Moody algebras},
Invent. Math. \textbf{ 116} (1994), 329--346.
\MR{95f:17023}
\pagebreak
\bibitem[P]{P} H. Pittie, 
\textit{Homogeneous vector bundles over homogeneous spaces},
Topology \textbf{ 11} (1972), 199--203.
\MR{44:7583}

\bibitem[S]{S} R. Steinberg, 
\textit{On a theorem of Pittie},
Topology \textbf{ 14} (1975), 173--177. \MR{51:9101}
\end{thebibliography}
\end{document}