A natural generalization of vector bundles over a Riemann surface \(X\) arises when one endows the vector bundle with a parabolic structure. This entails the choice of a weighted flag on the fiber over each point from a finite collection \(D\) of different points on the Riemann surface. These objects, called parabolic vector bundles and introduced by C. Seshadri [Bull. Am. Math. Soc. 83, 124–126 (1977; Zbl 0354.14005)], now relate to representations of the fundamental group \({{\pi }_{1}}\left( X\backslash D \right)\) with fixed holonomy around the points in \(D\). The classical Narasimhan-Seshadri correspondence was established in this parabolic situation by V. Mehta and C. Seshadri [Math. Ann. 248, 205–239 (1980; Zbl 0454.14006)] for a natural generalization of the stability condition for the parabolic vector bundles on the one side of the correspondence and for fundamental group representations into the group \(G = \text{U} (n)\) on its other side.
In order to extend the Mehta-Seshadri correspondence in the case of compact groups \(G\), the main step was to understand the nonlinear analysis needed to construct the analytic objects involved in the correspondence. For the non-compact groups \(G\), the study of this correspondence for the group \(G = \text{GL} \left(n, \mathbb{C} \right)\) in particular was carried out by C. Simpson [J. Am. Math. Soc. 3, No. 3, 713–770 (1990; Zbl 0713.58012] with the introduction of filtered objects for the correct version of the correspondence; that included stable filtered regular Higgs bundles and stable filtered local systems.
From this point on, a central problem involved generalizing the notion of a stable parabolic vector bundle to the setting of principal \(G\)-bundles, for semi-simple or reductive structure groups \(G\). A central technical issue here was to clarify what the correct notion of parabolic weight should be in this principal \(G\)-bundle setting, in order to generalize the work of C. Simpson, with a variety of approaches and conventions having appeared in the literature.
In this article, the authors introduce a notion of weight for holomorphic principal bundles over a compact Riemann surface \(X\) with a collection \(D\) of finitely-many and different points of \(X\), and extend the correspondence of C. Simpson to the case of an arbitrary real reductive Lie group \(G\) (including the case in which \(G\) is complex). For \(H^{\mathbb{C}}\) a non-compact reductive Lie group, \(H\subset {{H}^{\mathbb{C}}}\) a fixed maximal compact subgroup and \(T \subset H\) a fixed maximal torus with Lie algebra \(\mathfrak{t}\), this notion of weight for a parabolic principal \(H^{\mathbb{C}}\)-bundle over the pair \(\left(X, D \right)\) involves a choice for each point in \(D\) of an element in a Weyl alcove \(\mathcal{A} \subset \mathfrak{t}\) of \(H\) whose closure contains 0. In their approach, the authors allow these elements to lie in a wall of the Weyl alcove, in order to establish the correspondence with representations having totally arbitrary fixed holonomy around the points in \(D\). However, under this definition of a parabolic principal bundle, to a representation corresponds not a single holomorphic bundle, but rather a class of holomorphic bundles equivalent under gauge transformations with meromorphic singularities.
The authors also relate their objects to a notion of parahoric bundles in the context of Hecke transformations. Such a parahoric bundle over the pair \(\left( X, D \right)\) of weight \(\alpha\) is defined as a sheaf of torsors over an appropriately defined sheaf of groups \(\mathcal{G}_{\alpha}\). The notion of weight \(\alpha\) here seems different than the notion of weight for parahoric bundles introduced by P. Boalch [Transform. Groups 16, No. 1, 27–50 (2011; Zbl 1232.34117)] or V. Balaji and C. Seshadri [J. Algebr. Geom. 24, No. 1, 1–49 (2015; Zbl 1330.14059)] involving a point in the affine apartment of the group \(G\).
For \(\left( G, H, \theta, B \right)\) a real reductive Lie group, a parabolic \(G\)-Higgs bundle over \(\left(X,D \right)\) is defined as a pair \(\left( E, \varphi \right)\), where \(E\) is a parabolic \(H^{\mathbb{C}}\)-principal bundle over \(\left(X,D \right)\) and \(\varphi\) is a holomorphic section of \(PE\left(\mathfrak{m}^{\mathbb{C}}\right) \otimes K \left(D \right)\); here is considered the Cartan decomposition \(\mathfrak{g} = \mathfrak{h} \oplus \mathfrak{m}\) of the Lie algebra of the group \(G\), \(PE\left(\mathfrak{m}^{\mathbb{C}}\right)\) denotes the sheaf of parabolic sections of \(E\left(\mathfrak{m}^{\mathbb{C}}\right)\) and \(K \left(D \right) := K \otimes \mathcal{O}_{X} \left(D\right)\), where \(K \simeq T^{*}X\) the canonical line bundle over \(X\). The authors define a notion of stability, semistability and polystability for a parabolic \(G\)-Higgs bundle depending on an element of \(\sqrt{-1} \mathfrak{\zeta} \), for \(\mathfrak{\zeta} \) the center of \(\mathfrak{h}\).
In Section 5 of the article the Hitchin-Kobayashi correspondence is proven, relating the polystability of a parabolic \(G\)-Higgs bundle to the existence of a specific Hermite-Einstein metric. For the proof it is important to first construct a model singular metric which gives an approximate solution to the Hermite-Einstein equations and next to minimize the Donaldson functional in this context under certain constraints.
Section 6 contains the other half of the non-abelian Hodge correspondence in this tame parabolic \(G\)-Higgs bundle case, that is, proving an existence theorem for harmonic reductions. The authors introduce a notion of parabolic \(G\)-local systems of weight \(\beta\) and a useful table describing the relation of weights and monodromies is provided for the parabolic \(G\)-Higgs bundles and parabolic \(G\)-local systems involved; this table is similar to the one of C. Simpson in the \(G=\text{GL}\left(n, \mathbb{C}\right)\)-case.
Despite the amount of technical expertise required in Lie theory, the article is very clearly written. Undoubtedly this work will inspire future research in investigating topological and geometric properties of the moduli spaces of these principal parabolic objects, some of which are already pointed out in the last two sections of this paper or have by now appeared in the literature.