Let X be an irreducible smooth projective algebraic curve of genus g ≥ 2 over the ground field C, and let G be a semisimple simply connected algebraic group. The aim of this paper is to introduce the notion of semistable and stable parahoric torsors under a certain Bruhat-Tits group scheme G and to construct the moduli space of semistable parahoric G-torsors; we also identify the underlying topological space of this moduli space with certain spaces of homomorphisms of Fuchsian groups into a maximal compact subgroup of G. The results give a generalization of the earlier results of Mehta and Seshadri on parabolic vector bundles.which generalize the results in Narasimhan and Seshadri [27] and Seshadri [35].The purpose of this paper is to generalize the above results when the structure group G is no longer the full-linear group. Let us suppose hereafter that the group G is semisimple and simply connected (over C) unless otherwise stated.One can again give an equivalent description of (π, G)-bundles on H as certain intrinsically defined objects on X. However, the picture is more subtle than the case when G is the full-linear group. For instance, it is not possible, in general, to associate in a natural manner a principal G-bundle on X to a (π, G)-bundle on H. The new objects on X, which give an equivalent description of (π, G)-bundles on H, will be called parahoric bundles or parahoric torsors. These parahoric torsors are defined as pairs (E , θ), where E is a torsor (i.e., principal homogeneous space) on X under a parahoric Bruhat-Tits group scheme G, together with a prescription of weights θ, which are elements of the set of rational one-parameter subgroups of G (see the discussion below and Definition 6.1.1). We define notions of semistability and stability of such parahoric torsors and construct moduli spaces of these objects.The torsors under parahoric group schemes that we consider here have been studied earlier by Pappas and Rapoport, without however the notion of weights (see [29] and [30]); in [30] they made some precise conjectures on the moduli stack of such torsors. Heinloth has since settled many of their conjectures (see [21]; we note that Heinloth works over arbitrary ground fields not just C). We were led to the study of parahoric torsors in trying to interpret (π, G)-bundles on H as objects on X (inspired by A. Weil's work [44], as was the case in [24] and [36]). In Section 2 we link explicitly the ideas from the paper of Weil and Bruhat-Tits theory. This relationship plays a key role in the rest of the paper. We need to define a few technical terms before we can state the main results of our paper.Let A x i be the completion of the local ring at x i , and let K x i (or simply as K) be its quotient field, x i ∈ R. Let T be a maximal torus of G, and let Y (T ) := Hom(G m , T ) be the group of one-parameter subgroups of T . Let E Y (T )⊗R and E Q Y (T )⊗Q. By the general theory of Bruhat and Tits (see [9, Definition 5.2.6], and 2.1.2 below), one has certain collection of subsets {Θ i } ⊂ E m Q , where m...

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We define the notion of holonomy group for a stable vector bundle F on a variety in terms of the Narasimhan-Seshadri unitary representation of its restriction to curves.Next we relate the holonomy group to the minimal structure group and to the decomposition of tensor powers of F . Finally we illustrate the principle that either the holonomy is large or there is a clear geometric reason why it should be small. If X is a complex manifold and E a holomorphic vector bundle, then usually there are no holomorphic connections on E. One can, nonetheless, define a close analog of the holonomy representation in the complex setting if E is a stable vector bundle and X is projective algebraic.By Mehta-Ramanathan [MR82], if x ∈ C ⊂ X is a sufficiently general complex curve, then E| C is also stable and so, by a result of NarasimhanSeshadri [NS65], it corresponds to a unique unitary representation ρ :If c 1 (E| C ) = 0, one gets a special type of unitary representation ρ : π 1 (C \ x) → U (E x ), see (12). We call these the NarasimhanSeshadri representation of E| C .

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