  Let $X$ be a smooth projective curve over the complex numbers. To every
representation $\rho\colon \GL(r)\lra \GL(V)$ of the complex general linear
group on the finite dimensional complex vector space $V$ which satisfies the
assumption that there be an integer $\alpha$ with $\rho(z \id_{\C^r})=z^\alpha
\id_V$ for all $z\in\C^*$ we associate the problem of classifying triples
$(E,L,\phi)$ where $E$ is a vector bundle of rank $r$ on $X$, $L$ is a line
bundle on $X$, and $\phi\colon E_\rho\lra L$ is a non trivial homomorphism.
Here, $E_\rho$ is the vector bundle of rank $\dim V$ associated to $E$ via
$\rho$. If we take, for example, the standard representation of $\GL(r)$ on
$\C^r$ we have to classify triples $(E,L,\phi)$ consisting of $E$ as before and
a non-zero homomorphism $\phi\colon E\lra L$ which includes the so-called
Bradlow pairs. For the representation of $\GL(r)$ on $S^2\C^3$ we find the
conic bundles of Gomez and Sols. In the present paper, we will formulate a
general semistability concept for the above triples which depends on a rational
parameter $\delta$ and establish the existence of moduli spaces of
$\delta$-(semi)stable triples of fixed topological type. The notion of
semistability mimics the Hilbert-Mumford criterion for $SL(r)$ which is the
main reason that such a general approach becomes feasible. In the known
examples (the above, Higgs bundles, extension pairs, oriented framed bundles)
we show how to recover the "usual" semistability concept. This process of
simplification can also be formalized. Altogether, our results provide a
unifying construction for the moduli spaces of most decorated vector bundle
problems together with an automatism for finding the right notion of
semistability and should therefore be of some interest.
