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% Author Package file for use with AMS-LaTeX 1.2
\controldates{16-APR-2001,16-APR-2001,16-APR-2001,16-APR-2001}
 
\documentclass{era-l}
\issueinfo{7}{06}{}{2001}
\dateposted{April 24, 2001}
\pagespan{37}{44}
\PII{S 1079-6762(01)00092-0}
\copyrightinfo{2001}{American Mathematical Society}
\copyrightinfo{2001}{American Mathematical Society}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}

\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}

\theoremstyle{remark}
\newtheorem*{acknowledgments}{Acknowledgments}

\newcommand{\bR}{\mathbb{R}}
\newcommand{\bC}{\mathbb{C}}
\newcommand{\mya}{\alpha}
\newcommand{\Ga}{\Gamma}
\newcommand{\cO}{\mathcal{O}}

\begin{document}

\title[Parabolic principal bundles]{Principal bundles with parabolic structure}

\author{V. Balaji}
\address{Institute of Mathematical Sciences,
C.I.T. Campus, Taramani Chennai 600113, India}
\email{vbalaji@imsc.ernet.in}

\author{I. Biswas}
\address{School of Mathematics, Tata Institute of Fundamental
Research, Homi Bhabha Road, Bombay 400005, India}
\email{indranil@math.tifr.res.in}

\author{D. S. Nagaraj}
\address{Institute of Mathematical Sciences,
C.I.T. Campus, Taramani Chennai 600113, India}
\email{dsn@imsc.ernet.in}

\commby{Frances C. Kirwan}
\date{February 1, 2001}
\subjclass[2000]{Primary 14F05; Secondary 32L05}

\begin{abstract}
We define a principal bundle analog of vector bundles with
parabolic structure over a normal crossing divisor. Various
results on parabolic vector bundles and usual principal bundles
are extended to the context of parabolic principal bundles.
\end{abstract}

\maketitle

\section{Introduction}

Although parabolic vector bundles have been existing for a long time, a
satisfactory definition of parabolic $G$-bundles is still
lacking. In this note we define a $G$-bundle analog of vector
bundles with parabolic structure over a normal crossing divisor
and rational parabolic weights.

Let $Y$ be an algebraic variety.
Nori showed that a principal $G$-bundle over $Y$ can equivalently be
thought of as a functor from the category of
finite-dimensional $G$-modules to the category of vector
bundles over $Y$
which is compatible with the tensor product, direct sum
and dualization operation. A $G$-bundle $P$ sends a $G$-module
$V$ to the associated vector bundle $P\times_G V$. The content
of the observation of Nori is that this functor
determines $P$ uniquely.

Like in the case of usual vector bundles, the parabolic vector
bundles also have {\it parabolic} tensor product, direct sum and
dualization operations. Therefore, the above definition of
$G$-bundles can be extended to parabolic bundles. This is done in
Section 2. However, it is desirable to have a parabolic
$G$-bundle as concrete object, a scheme, {\it representing} this
functor.

If we restrict ourselves to the situation where the parabolic
divisor is a normal crossing divisor and the parabolic weights
are all rational, then a parabolic $G$-bundle over $X$
can be realized as an equivariant $G$-bundle over a ramified Galois
cover $Y$ over $X$.

Let $\Gamma$ denote the Galois group of the cover $Y$ of $X$.
Given an equivariant $G$-bundle $P$ on $Y$, we may take the
quotient of $P$ by $\Gamma$, which is a principal
$G$-bundle over the complement of the parabolic divisor.
Using the properties of this quotient that we establish, it is possible to characterize all $G$-spaces that are of the form $P/\Gamma$.
This enables us to define a parabolic $G$-bundle as a $G$-space
with certain properties.


\begin{acknowledgments} We are very grateful to M. S. Narasimhan
for his comments on \cite{BBN1} which led us to the parabolic
$G$-objects defined in Section 5.
\end{acknowledgments}
\section{The parabolic analog of principal bundles}

Let $G$ be an affine algebraic group over $\bC$.
We will briefly recall a reformulation of the definition
of principal $G$-bundles constructed by Nori in \cite{No1}, \cite{No2}.

Let $X$ be a connected smooth projective variety over $\bC$.
Denote by ${\rm Vect}(X)$ the category of vector bundles over
$X$. The category
${\rm Vect}(X)$ is equipped with an algebra structure
defined by the tensor product operation
\[
{\rm Vect}(X)\times {\rm Vect}(X) \longrightarrow
{\rm Vect}(X) ,
\]
which sends any pair $(E,F)$ to $E\otimes F$, and the direct
sum operation $\bigoplus$, making it an additive tensor category in
the sense of \cite[Definition 1.15]{DM}.

Let ${\rm Rep}(G)$ denote the category of all finite-dimensional 
complex left representations of the group $G$,
or equivalently, left $G$-modules. By a $G$-module (or
a representation) we shall always mean a left $G$-module
(or a left representation).

Given a principal $G$-bundle $P$ over $X$ and a left $G$-module
$V$, the associated fiber bundle $P\times_G V$ has a natural
structure of a vector bundle over $X$. Consider the functor
\begin{equation*}
F(P) : {\rm Rep}(G) \longrightarrow
{\rm Vect}(X) , \tag{2.1}
\end{equation*}
which sends any $V$ to the vector bundle $P\times_G V$ and sends
any homomorphism between two $G$-modules to the naturally induced
homomorphism between the two corresponding vector bundles. The
functor $F(P)$ enjoys several natural abstract properties. For
example, it is compatible with the algebra structures of ${\rm
Rep}(G)$ and ${\rm Vect}(X)$ defined using direct sum and tensor
product operations. Furthermore, $F(P)$ takes an exact sequence
of $G$-modules to an exact sequence of vector bundles, it also
takes the trivial $G$-module $\bC$ to the trivial line bundle on
$X$, and the dimension of $V$ coincides with the rank of the
vector bundle $F(P)(V)$.

In Proposition 2.9 of \cite{No1} (also Proposition 2.9 of
\cite{No2}) it has been established that the collection of
principal $G$-bundles over $X$ is in bijective correspondence with
the collection of functors from ${\rm Rep}(G)$ to ${\rm Vect}(X)$
satisfying the abstract properties that the functor $F(P)$ in
(2.1) enjoys. The four abstract properties are described on page
31 of \cite{No1}, where they are marked F1--F4. The bijective
correspondence sends a principal bundle $P$ to the functor $F(P)$
defined in (2.1).

We will define parabolic $G$-bundles along
the above lines.

Let $D$ be an effective divisor on $X$. For a coherent sheaf $E$
on $X$, the image of $E\otimes_{{\cO}_X} {\cO}_X(-D)$ in $E$
will be denoted by $E(-D)$. The following definition of a
parabolic sheaf was introduced in \cite{MY}.
\begin{definition}\label{def2.2}
Let $E$ be a torsion free ${\cO}_X$-coherent sheaf on $X$. A
{\it quasi-parabolic} structure on $E$ over $D$ is a filtration
by ${\cO}_X$-coherent subsheaves
\[
E\, =\, F_1(E)\, \supset\, F_2(E)\, \supset\, \cdots
\,\supset\, F_l(E)\,\supset\, F_{l+1}(E)\,=\, E(-D)\, .
\]
The integer $l$ is called the {\it length of the filtration}.
A {\it parabolic structure} is a quasi-parabolic structure,
as above, together with a system of {\it weights}
$\{{\mya}_1,\dots ,{\mya}_l\}$ such that
\[
0\, \leq\,
{\mya}_1\, < \, {\mya}_2 < \, \cdots \, < \, {\mya}_{l-1} \, < \,
{\mya}_l \, < \, 1, 
\]
where the weight ${\mya}_i$ corresponds to the subsheaf $F_i(E)$.
\end{definition}

We shall denote the parabolic sheaf defined above
by $(E,F_*,{\mya}_*)$.
When there is no possibility of confusion, it will be denoted by $E_*$.

For a parabolic sheaf $(E,F_*, {\mya}_*)$, define
the following filtration $\{E_t\}_{t\in \bR}$ of coherent
sheaves on $X$ parameterized by $\bR$:
\[
E_t := F_i(E)(-[t]D) ,
\]
where $[t]$ is the integral part of $t$
and ${\mya}_{i-1} < t - [t]
\leq {\mya}_i$, with the convention that ${\mya}_0 = {\mya}_l -1 $
and ${\mya}_{l+1} = 1$.

If the underlying sheaf $E$ is locally free, then
$E_*$ will be called a parabolic vector bundle. {\it Henceforth,
all parabolic sheaves will be assumed to be parabolic vector
bundles.}

The class of parabolic vector bundles that are dealt with in the
present work satisfies certain conditions which will be explained
now. The first condition is that all
parabolic divisors are assumed to be {\it divisors
with normal crossings}. In other words, any parabolic divisor
is assumed to be reduced, each of its irreducible components is
smooth, and furthermore the irreducible
components intersect transversally.
The second condition is that
all the parabolic weights are {\it rational
numbers}. Before stating the third condition,
we remark that quasi-parabolic filtrations on a vector bundle
can be defined by giving filtrations by subsheaves of the
restriction of the vector bundle to each component of the
parabolic divisor. The third and final
condition states that on each component of the parabolic divisor
the filtration is given by {\it subbundles}.
The precise formulation of the last condition is given
in \cite[Assumptions 3.2 (1)]{Bi2}. {\it Henceforth,
all parabolic vector bundles will be assumed to satisfy
the above three conditions.} Note that if $\dim X\, =\, 1$, then
all conditions except the rationality of
weights are automatically satisfied.

Let ${\rm PVect}(X,D)$ denote the category whose
objects are parabolic vector
bundles over $X$ with parabolic structure over the divisor
$D$ satisfying the above three conditions, and the
morphisms of the category are homomorphisms of parabolic
vector bundles.
For any two parabolic bundles $E_*, V_* \in {\rm PVect}(X,D)$, their parabolic
tensor product $E_*\otimes V_*$ is also an element of ${\rm
PVect}(X,D)$. (See \cite{Bi1}, \cite{Yo} for the definition of
parabolic tensor product.)
The trivial line bundle with the trivial parabolic structure
(this means that the length of the parabolic flag is zero) acts as
the identity element for the parabolic tensor multiplication.
The parabolic tensor product operation on ${\rm PVect}(X,D)$
has all the abstract properties
enjoyed by the usual tensor product operation of vector bundles.

The direct sum of two vector bundles with parabolic structures
has an obvious parabolic structure.
Evidently, ${\rm PVect}(X,D)$ is closed under the operation of
taking direct sum. The category ${\rm PVect}(X,D)$ is an additive
tensor category with the direct sum and the parabolic
tensor product operation. Also the notion of dual of a vector
bundle can be extended to the context of parabolic bundles.

For an integer $N\geq 2$, let ${\rm PVect}(X,D,N) \, \subseteq \,
{\rm PVect}(X,D)$ denote the subcategory consisting of all
parabolic vector bundles all of whose parabolic weights are
multiples of $1/N$. It is straightforward to check that ${\rm
PVect}(X,D,N)$ is closed under all the above operations, namely
parabolic tensor product, direct sum and taking the parabolic
dual.
\begin{definition}\label{def2.3} A {\it parabolic principal $G$-bundle}
with parabolic structure over $D$ is a {\it functor} $F$ from the
category ${\rm Rep}(G)$ to the category ${\rm PVect}(X,D)$
satisfying the four conditions of \cite{No1} mentioned earlier.
The functor is further required to satisfy the condition that
there is an integer $N$, which depends on the functor, such that
the image of the functor is contained in ${\rm PVect}(X,D,N)$.
\end{definition}

A justification of the above definition will be provided
by the following proposition:
\begin{proposition}\label{prop2.4}
The collection of parabolic $GL(n, \bC)$-bundles on $X$
with parabolic structure over $D$
is identified, in a bijective fashion, with the
subclass of ${\rm PVect}(X,D)$ consisting of parabolic vector
bundles of rank $n$. Under this identification, a parabolic
$GL(n,\bC)$-bundle is identified
with the parabolic vector bundle
associated to it for the standard representation of
$GL(n,\bC)$ on ${\bC}^n$.
\end{proposition}

The proof is given in \cite{BBN1}.

\section{Semistability for parabolic principal bundles}

Fix an ample line bundle $L$ over $X$. For
a coherent sheaf $F$ over $X$, define the degree $\mbox{deg}(F)$
of $F$ using $L$.

Let $P$ be a principal $G$-bundle over $X$.
A reduction of the structure
group of $P$ to a subgroup $Q \subset G$ is defined by giving
a section of the fiber bundle $P/Q \to X$ with
fiber $G/Q$. Henceforth, we will assume $G$ to be semisimple.
\begin{definition}[\cite{RR}]\label{def3.1} Let $P(Q)$ denote
a reduction of the structure group of $P$ to a maximal parabolic
subgroup $Q \subset G$ over an open set $U \subseteq X$ with
$\mbox{codim}(X-U) \geq 2$. The principal $G$-bundle $P$ is
called {\it semistable} (respectively, {\it stable}) if for
every such situation, the line bundle over $U$ associated to
$P(Q)$ for any character of $Q$ dominant with respect to a Borel
subgroup contained in $Q$, is of nonpositive degree
(respectively, strictly negative degree). The principal bundle
$P$ is called {\it polystable} if there is a reduction of the
structure group of $P$ to $M$, namely $P(M) \subset P$,
where $M \subset G$ is a maximal reductive subgroup of a
parabolic subgroup of $G$, such that $P(M)$ is a stable
principal $M$-bundle and furthermore, for any character of $M$
trivial on the intersection with the center of $G$, the
corresponding line bundle associated to $P(M)$ is of degree
zero.
\end{definition}

Given a homomorphism $G \to H$ and a principal
$G$-bundle $P$, the space $P\times_G H$, where
$G$ acts as left translations of $H$,
has a natural structure of a principal $H$-bundle. This
construction of a principal $H$-bundle from the
principal $G$-bundle $P$ is called the {\it extension of the
structure group} of $P$ to $H$. From \cite{RR} we know
that if $P$ is a semistable (respectively, polystable) $G$-bundle,
then $P\times_G H$ is a semistable (respectively, polystable)
$H$-bundle.

If $H$ is reductive and
$P$ is a semistable $G$-bundle, then the extension
$P\times_G H$ is also semistable (\cite{RR}).
This indicates how we may define
semistability in the context of parabolic $G$-bundles.
The following proposition, which
is proved in \cite{BBN1}, will be needed for that purpose.
The definition of parabolic semistable and parabolic
polystable vector bundles
is given in \cite{MY} and \cite{MS}.

\begin{proposition}\label{prop3.2} Let $E_*\, , \,F_*
\, \in \, {\rm PVect}(X,D)$ be two parabolic semistable
{\rm (}respectively, parabolic polystable{\rm )} vector bundles
on $X$. Then
the parabolic tensor product $E_*\otimes F_*$ is also
parabolic semistable {\rm (}respectively, parabolic polystable{\rm )},
and furthermore the parabolic dual of $E_*$ is also parabolic
semistable {\rm (}respectively, parabolic polystable{\rm )}.
\end{proposition}
\begin{definition}
Let $P_*$ be a functor from
the category ${\rm Rep}(G)$ to the category ${\rm PVect}(X,D)$
defining a parabolic $G$-bundle as in Definition \ref{def2.3}. This
functor $P_*$ will be called a {\it parabolic semistable
{\rm (}respectively, parabolic polystable{\rm )} principal
$G$-bundle} if
and only if the image of the functor is contained in the
category of parabolic semistable (respectively, parabolic
polystable) vector bundles.
\end{definition}

We observe that Proposition \ref{prop3.2} implies that the
subcategory of ${\rm PVect}(X,D)$ consisting of parabolic
semistable (respectively, parabolic polystable) vector bundles
is closed under tensor product. Furthermore, to check parabolic
semistability (respectively, parabolic polystability) it is not
necessary to check the criterion for $V_1\otimes V_2 \in {\rm
Rep}(G)$ if it has been checked for $V_1$ and $V_2$
individually.

The following proposition is proved in \cite{BBN1}.

\begin{proposition}\label{3.4} A parabolic
$G$-bundle $P_*$ is parabolic semistable {\rm (}respectively,
parabolic polystable{\rm )} if and only if there is a
faithful representation
\[
\rho : G 
\longrightarrow GL(V)
\]
such that the corresponding parabolic vector bundle
$P_*(\rho)$ is parabolic semistable {\rm (}respectively, parabolic
polystable{\rm )}. Consequently, if for
one faithful representation $\rho$ the parabolic
vector bundle $P_*(\rho)$ is parabolic semistable
{\rm (}respectively, parabolic polystable{\rm )}, then for any
representation ${\rho}'$, the parabolic
vector bundle $P_*({\rho}')$ is parabolic semistable
{\rm (}respectively, parabolic polystable{\rm )}.
\end{proposition}

The following proposition is immediate.

\begin{proposition}\label{prop3.5} If $G
\longrightarrow H$ is a homomorphism of groups and
if $P_*$ is a parabolic semistable {\rm (}respectively, parabolic
polystable{\rm )} $G$-bundle, then the parabolic $H$-bundle,
obtained by the extension of the structure group of $P_*$, is
also parabolic semistable {\rm (}respectively,
parabolic polystable{\rm )}.
\end{proposition}

The following theorem is proved in \cite{BBN1}. The definition of
Chern classes of parabolic bundles is given in \cite{Bi3}.

\begin{theorem}\label{thm3.6} A parabolic
$G$-bundle $P_*$ admits a unitary flat connection
if and only if the following two conditions hold{\rm :}
\begin{enumerate}

\item[(1)] $P_*$ is parabolic polystable{\rm ;}

\item[(2)] $c^*_2(P_*({\rm ad}))\, = \, 0$, where $c^*_2$
is the second parabolic Chern class.
\end{enumerate}

Furthermore, a parabolic $G$-bundle
satisfying the above two conditions admits a
unique unitary flat connection.
\end{theorem}

For parabolic vector bundles over a Riemann surface, this
theorem was proved in \cite{MS}.

\section{Orbifold bundles and parabolic bundles}

The following is the well-known ``covering lemma'' of
Y. Kawamata.

Given $X$ and a divisor $D =\sum_{i=1}^c D_i$ as above,
where $D_i$ are irreducible components, and an integer $N$,
there is a Galois cover
\[
p : Y \longrightarrow X,
\]
where $Y$ is a connected smooth
projective manifold, $p^{-1}(D)_{\rm red}$
is a normal crossing divisor and
$p^{-1}(D_i)=k_iN(p^{-1}(D_i))_{\rm red}$ (\cite{KMM}).
We fix such a cover. Let $\Gamma\,:={\rm Gal}(Y/X)$.

An {\it orbifold} $G$-bundle over $Y$ is a $G$-bundle
$P$ together with a lift of the action of $\Gamma$ on
the total space of $P$ as bundle automorphisms. An 
orbifold $G$-bundle will be called a $(\Ga ,G)$-bundle.

Let $P$ be a $(\Ga ,G)$-bundle on $Y$. It is
called {\it $\Ga$-semistable}
(respectively, {\it $\Ga$-polystable}) if and only if $P$
satisfies the condition of semistability (respectively,
polystability) in Definition \ref{def3.1} with all reductions of
the structure group being $\Ga$-equivariant.

\begin{proposition}\label{prop4.1} A
$(\Ga ,G)$-bundle $P$ is $\Ga$-semistable {\rm (}respectively,
$\Ga$-polystable{\rm )} if and only if $P$ is
semistable {\rm (}respectively, polystable{\rm )}
according to Definition \ref{def3.1}.
\end{proposition}

If we fix $N$, the space of all parabolic principal $G$-bundles
satisfying the condition that the image of the corresponding
functor in Definition \ref{def2.3}
is contained in ${\rm PVect}(X,D,N)$ will be denoted by
$PG(X,D,N)$.

Let $[\Ga ,G,N]$ denote the collection of
$(\Ga ,G)$-bundles on $Y$ satisfying the following
two conditions:

\begin{enumerate}

\item[(1)] for a general point $y$ of an irreducible
component of $(p^*D_i)_{\rm red}$,
the action of ${\Ga}_y$ on $P_y$ is of order a divisor of $N$;

\item[(2)] for a general point $y$ of an irreducible
component of a ramification divisor for $p$
not contained in $(p^*D)_{\rm red}$,
the action of ${\Ga}_y$ on $P_y$ is the trivial action.

\end{enumerate}

In \cite{BBN1} we construct a map from $[\Ga ,G,N]$
to $PG(X,D,N)$ and also a map from $PG(X,D,N)$
to $[\Ga ,G,N]$. Using the
result of Nori, the constructions reduce to a construction
of maps between parabolic vector bundles and orbifold
vector bundles. Such constructions are given in \cite{Bi2}.
In \cite{BBN1} we prove

\begin{theorem}\label{thm4.2} The above-mentioned
maps between $PG(X,D,N)$ and $[\Ga ,G,N]$ are inverses
of each other. Furthermore, $P_*\, \in\, PG(X,D,N)$ is
parabolic semistable {\rm (}respectively, parabolic
polystable{\rm )} if and only if the corresponding
bundle $P\, \in\, [\Ga ,G,N]$
is $\Ga$-semistable {\rm (}respectively,
$\Ga$-polystable{\rm )}.
\end{theorem}

The constructions in Theorem 4.2 suggest a more concrete
definition of a parabolic $G$-bundle.

\section{Alternative definition of parabolic $G$-bundles}

Henceforth we will assume $\dim X \,= \, 1$.  This is only to simplify
the exposition. Everything in this section can be extended to higher
dimensions.

Suppose $P\, \in\, [\Ga ,G,N]$. Then both $G$ and $\Gamma$
act on the total space of $P$. It is easy to see that these
two actions commute. Therefore, the quotient space $P/\Gamma$,
which we will denote by $P'$, has an action of $G$. Let
\begin{equation*}
\phi : P'  \longrightarrow P'/G = X \tag{5.1}
\end{equation*}
be the quotient map. For any $y\,\in\, Y$, let $\Gamma_y\, \subset\,
\Gamma$ denote the isotropy of $y$.
If $\Gamma_y$ is trivial, then the action of $G$ on
$\phi^{-1}(p(y))$ is free and transitive.
If $x\,\in\, X\setminus D$ and $y\in p^{-1}(x)$, then $\Gamma_y$ may
be nontrivial, but by assumption, the action of $\Gamma_y$ on the
fiber $P_y$ is trivial. Therefore, the map $\phi$ in (5.1) defines
a principal $G$-bundle over $X\setminus D$.

For any $x\in D$, the action of $\Gamma_y$ on $P_y$ for any $y\in p^{-1}(x)$
is of order a divisor of
$N$. Note that $\Gamma_y$ is a cyclic group whose
order is a multiple of $N$.

It can be checked that
the variety $P'$ is smooth, but the map $\phi$ is not smooth
over any point $x\in D$ that has the property that the action
of $\Gamma_y$ on $P_y$ is nontrivial, where $y\in p^{-1}(x)$
(\cite{BBN2}).

The $G$-bundle $P$ can be constructed back from $P'$. Indeed,
$P$ is the normalization of the fiber product $P'\times_X Y$
(\cite{BBN2}). This shows that $[\Ga ,G,N]$ can be identified
with $G$-spaces of certain type.

To explain this, let $Q$ be a smooth variety on which $G$ acts with
the property that there are only finitely many orbits on which the
isotropy is nontrivial. Suppose, moreover that at each of these finite
orbits, the isotropy is a cyclic subgroup of order which divides
$N$. Furthermore, assume that $Q/G \, =\, X$ and the image of the
orbits with nontrivial isotropy is contained in $D$. The quotient is
in the sense of geometric invariant theory.

From the earlier remarks on $P'$ it follows that the space of all
such $G$-spaces, $Q \longrightarrow Q/G = X$, is in bijective
correspondence with $[\Ga ,G,N]$.
\begin{definition}\label{def5.2}
A parabolic $G$-object is
a smooth variety $Q$ with an action of $G$ satisfying the
above properties.
\end{definition}

Using Theorem \ref{thm4.2} it follows that
parabolic $G$-objects are in bijective correspondence with
parabolic $G$-bundles.

For a finite-dimensional $G$-module $V$, consider
the diagonal action of $G$ on $Q\times V$. The invariant sections
on $G$-saturated open sets define a vector bundle over $X$
with a parabolic structure. The parabolic structure is obtained
the same way a parabolic bundle is obtained in \cite{Bi2} from
an orbifold bundle. Therefore, we get a functor from ${\rm Rep}(G)$
to ${\rm PVect}(X,D,N)$ that sends any $V$ to
the parabolic vector bundle constructed this way. In other words,
we get a parabolic $G$-bundle according to Definition \ref{def2.3}. For the
parabolic $G$-object $P'$, the parabolic $G$-bundle obtained this way
coincides with the one associated to $P$ by Theorem \ref{thm4.2}.

Fix a maximal torus $T$ and a Borel subgroup $B$ of $G$, with
$T\, \subset\, B$.
Let $H$ be a closed subgroup of $G$. A reduction of the structure group
of $Q$ to $H$ is a section $\sigma:
X\to Q/H$. So, in particular, $H$ acts on
$f^{-1}(\sigma (X))$, where $f : Q\to
Q/H$ is the projection. If $W$ is a finite-dimensional 
$H$-module, then consider $f^{-1}(\sigma (X))\times_H W$,
which is a parabolic vector bundle over $X$. This parabolic
vector bundle will be denoted by $Q(W)_*$.

\begin{definition}\label{def5.3} A parabolic $G$-object
$Q$ is semistable (respectively, stable) if for
every reduction of structure group of $Q$ to $H$, where $H$ is any
parabolic subgroup containing $B$, and any nontrivial dominant
character $\chi$ of $P$, the parabolic line bundle $Q(\chi)_*$
has nonpositive (respectively, negative) parabolic degree.
\end{definition}

Polystability of parabolic $G$-objects can be defined similarly.

A parabolic $G$-object $Q$ is semistable (respectively,
stable) if and only if the
corresponding parabolic $G$-bundle in the sense of Definition \ref{def2.3}
is semistable (respectively, stable) \cite{BBN2}. Definition \ref{def5.3}
can be reformulated in terms of the parabolic degree of the
parabolic vector bundle associated to the pullback of the relative
tangent bundle.

Using Proposition \ref{prop4.1}, the Harder-Narasimhan reduction of
the structure group of an orbifold $G$-bundle is simply the
Harder-Narasimhan reduction of the underlying $G$-bundle. Now
using the identification of orbifold $G$-bundles with parabolic
$G$-objects the notion of Harder-Narasimhan reduction extends
to the context of parabolic $G$-objects.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bibliographystyle{amsplain}
\begin{thebibliography}{22}

\bibitem{BBN1} V. Balaji, I. Biswas and D. S. Nagaraj, Principal
bundles over projective manifolds with parabolic structure
over a divisor, \textit{Tohoku Math. J.} (to appear).
\bibitem{BBN2} V. Balaji, I. Biswas and D. S. Nagaraj, in
preparation.
\bibitem{Bi1} I. Biswas, Parabolic ample
bundles, \textit{ Math. Ann.} \textbf{ 307} (1997), 511--529.
\MR{98e:14041}
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\end{document}