International Congress of Mathematicians

Spectral Problems 607i.e. £ be topologically trivial. Narasimhan-Seshadri theorem [27] claims that everystable bundle carries unique flat metric, and hence defines unitary monodromyrepresentationpe : m(X,x 0 ) -^SY(E), E = £(x 0 ).This gives rise to equivalence_ /stable bundles A _ /irreducible uitary represent9 ' \of degree zero J Stations p : ni ^-SY(E) JThis theorem is an ancestor of the Donaldson-Yau generalization [7] to higher dimensions,and may be seen as a geometric version of Langlands correspondence.In algebraic terms the theorem describes stable bundles in terms of solutionof equation[Ui,V 1 ][U 2 ,V 2 ]---[U g ,V g ] = lin unitary matrices Ui,Vj £ SU(£'). This is not the matrix problem we arecurrently interested in. To modify it let's consider punctured Riemann surfaceX = X\{pi,p 2 ,... ,pi). It has distinguished classes7 Q = (small circle around p Q )in fundamental group m(X), and we can readily define an analogue of RHS of (4.2):M g (\W,\W,--- ,X W ) = {P •• MX) -+ SUCE) | A(p( 7 a)) = \ (a) }, (4.3)where A^a^ is a given spectrum of monodromy around puncture p Q . C. S. Seshadri[31] manages to find an analogue of more subtle holomorphic LHS of (4.2) in termsof so called parabolic bundles.Parabolic bundle £ on X is actually a bundle on compactification X togetherwith R-filtration in every special fiber E a = £(p a ) with support in an interval oflength < 1. The filtration is a substitution for spectral decomposition of p(-y a ), cf.(4.1). Seshadri also defines (semi)stability of parabolic bundle £ by inequalitiesPar deg T Par deg £^ k ^ - ^ ^ k ^ 'V^ C £ '(44)where the parabolic degree is given by equation Par deg £ = deg£ + ^2 ai X\ a .Aletha-Seshadri theorem [24] claims that every stable parabolic bundle £ on Xcarries unique flat metric with given spectra of monodromies A(7 Q ) = A^a^. Thisgives a holomorphic interpretation of the space (4.3)M r\(!) \( 2 ) \W\ — fiable parabolic bundles of degree zeroA , .9' ' ' y with given types of the filtrations J 'In the simplest case of projective line with three punctures (4.3) amounts to spaceof solutions of equation UVW = 1 in unitary matrices U, V, W £ SU(n) with given