flat g bundles with canonical metrics on considere la relation entre la theorie invariante des actions de groupes algebriques semi simples sur des varietes algebriques complexes et le comportement de l application de moment et des donnees quantiques fournies par une structure symplectique associee on etudie un exemple de dimension infinie comprenant des fibres plats sur des varietes de riemann compactes

uniformization of mathcal g bundles we show some of the conjectures of pappas and rapoport concerning the moduli stack bung of g torsors on a curve c where g is a semisimple bruhat tits groupschemeon c inparticularweprovetheanalogoftheuniformizationtheoremof drinfeld simpson in this setting furthermore we apply this to compute the connected components of these moduli stacks and to calculate the picard group of bung in case g is simply connected

on brlek reutenauer conjecture brlek and reutenauer conjectured that any infinite word u with language closed under reversal satisfies the equality 2d u email protected n 0 t u n in which d u denotes the defect of u and t u n denotes c u n 1 c u n 2 p u n 1 p u n where c u and p u are the factor and palindromic complexity of u respectively brlek and reutenauer verified their conjecture for periodic infinite words using their result we prove the conjecture for uniformly recurrent words moreover we summarize results and some open problems related to defects which may be useful for the proof of the brlek reutenauer conjecture in full generality

convex functions on symmetric spaces side lengths of polygons and the stability inequalities for weighted configurations at infinity in a symmetric space of noncompact type x g k oriented geodesic segments correspond modulo isometries to vectors in the euclidean weyl chamber we can hence assign vector valued lengths to segments our main result is a system of homogeneous linear inequalities which we call the generalized triangle inequalities or stability inequalities describing the restrictions on the vector valued side lengths of oriented polygons it is based on the mod 2 schubert calculus in the real grassmannians g p for maximal parabolic subgroups p the side lengths of polygons in euclidean buildings are studied in the related paper klm2 applications of the geometric results in both papers to algebraic group theory are given in klm3

stable bundles and integrable systems on considere la geometrie symplectique des fibres cotangents aux espaces de modules de fibres vectoriels stables sur une surface de riemann on montre que ce sont des systemes dynamiques hamiltoniens algebriquement completement integrables

infinite grassmannians and moduli spaces of g bundles let c be a smooth projective irreducible algebraic curve over c of any genus and g a connected simply connected simple affine algebraic group over c in this paper we elucidate the relationship between 1 the space of vacua conformal blocks defined in conformal field theory using an integrable highest weight representation of the affine kac moody algebra associated to g and 2 the space of regular sections generalised theta functions of a line bundle on the moduli space ta of semistable principal g bundles on c fix a point p in c and let p resp kp be the completion of the local ring p of c at p resp the quotient field of c p let g kp the kp rational points of the algebraic group g be the loop group of g and let g p be the standard maximal parahoric subgroup of then the generalised flag variety x n is an inductive limit of projective varieties in fact of generalised schubert varieties one has a natural r line bundle s on x cf sect 2 2 and the picard group pic x is isomorphic to z which is generated by s proposition 2 3 where f is the universal central extension of by the multiplicative group c cf sect 2 2 by an analogue of the borel weil theorem proved in the kac moody setting by kumar and also by mathieu the space h 162 s of the regular sections of the line bundle y dxo y xo for any d 0 is canonically isomorphic with the full vector space dual v dxo of the integrable highest weight irreducible module v dxo of the affine kacmoody group or of the associated affine kac moody algebra which is a certain one dimensional central extension of the loop algebra lie g with highest weight dxo cf sect 6 1

the quantization conjecture revisited a strong version of the quantization conjecture of guillemin and sternberg is proved for a reductive group action on a smooth compact polarized variety x l the cohomologies of l over the git quotient x g equal the invariant part of the cohomologies over x this generalizes the theorem of gs on global sections and strengthens its subsequent extensions jk li to riemannroch numbers remarkable by products are the invariance of cohomology of vector bundles over x g under a small change in the defining polarization or under shift desingularization as well as a new proof of boutot s theorem also studied are equivariant holomorphic forms and the equivariant hodgeto de rham spectral sequences for x and its strata whose collapse is shown one application is a new proof of the borel weil bott theorem of ti for the moduli stack of c bundles over a curve and of analogous statements for the moduli stacks and spaces of bundles with parabolic structures collapse of the hodge to de rham sequences for these stacks is also shown

the moduli of flat pu p p structures with large toledo invariants for a compact riemann surface x of genus g 1 rm hom pi 1 x rm pu p q rm pu p q is the moduli space of flat rm pu p q connections on x there are two invariants the chern class c and the toledo invariant tau associated with each element in the moduli the toledo invariant is bounded in the range 2min p q g 1 le tau le 2min p q g 1 this paper shows that the component associated with a fixed tau 2 max p q 1 g 1 resp tau 2 max p q 1 g 1 and a fixed chern class c is connected the restriction on tau implies p q

on bi hamiltonian structures the definition of a bi hamiltonian structure is reviewed and it is shown that for systems of differential equations of the form x v x on even dimensional manifolds there always exists locally a bi hamiltonian structure if this structure is global then the system of equations is integrable furthermore the geometry and canonical forms for such structures are discussed

parabolic higgs bundles and representations of the fundamental group of a punctured surface into a real group abstract we study parabolic g higgs bundles over a compact riemann surface with fixed punctures when g is a real reductive lie group and establish a correspondence between these objects and representations of the fundamental group of the punctured surface in g with arbitrary holonomy around the punctures this generalizes simpson s results for gl n c to arbitrary complex and real reductive lie groups three interesting features are the relation between the parabolic degree and the tits geometry of the boundary at infinity of the symmetric space the treatment of the case when the logarithm of the monodromy is on the boundary of a weyl alcove and the correspondence of the orbits encoding the singularity via the kostant sekiguchi correspondence we also describe some special features of the moduli spaces when g is a split real form or a group of hermitian type

complexity and palindromic defect of infinite words in this note we state a conjecture and prove it in the periodic case which is an equality relating the number of factors and palindromic factors of infinite words this equality establishes a link between two inequalities one due to droubay justin and pirillo and the other to balazi masakova and pelantova by means of the palindromic defect

factor versus palindromic complexity of uniformly recurrent infinite words we study the relation between the palindromic and factor complexity of infinite words we show that for uniformly recurrent words one has p n p n 1 email protected n 2 for all email protected n for a large class of words this is a better estimate of the palindromic complexity in terms of the factor complexity than the one presented in j p allouche m baake j cassaigne d damanik palindrome complexity theoret comput sci 292 2003 9 31 we provide several examples of infinite words for which our estimate reaches its upper bound in particular we derive an explicit prescription for the palindromic complexity of infinite words coding r interval exchange transformations if the permutation p connected with the transformation is given by p k r 1 k for all k then there is exactly one palindrome of every even length and exactly r palindromes of every odd length

moduli of parabolic higgs bundles and atiyah algebroids in this paper we study the geometry of the moduli space of non strongly parabolic higgs bundles over a riemann surface with marked points we show that this space possesses a poisson structure extending the one over the dual of an atiyah al gebroid over the moduli space of parabolic vector bundles by con sidering the case of full flags we get a grothendieck springer reso lution for all other flag types in particular for the moduli spaces of twisted higgs bundles as studied by markman and bottacin and used in the recent work of laumon ngo we discuss the hitchin system and demonstrate that all these moduli spaces are inte grable systems in the poisson sense

a hilbert mumford criterion for polystability in kaehler geometry consider a hamiltonian action of a compact lie group k on a kaehler manifold x with moment map x assume that the action of k extends to a holomorphic action of the complexification g of k we characterize which g orbits in x intersect 1 0 in terms of the maximal weights lim t e its x s where s we do not impose any a priori restriction on the stabilizer of x under some mild restrictions on the action k x we view the maximal weights as defining a collection of maps for each x x x k g where k g is the boundary at infinity of the symmetric space k g we prove that g x 1 0 0 if 1 x is everywhere nonnegative 2 any boundary point y such that x y 0 can be connected with a geodesic in k g to another boundary point y satisfying x y 0 we also prove that the maximal weight functions are g equivariant for any g g and any y k g we have g x y x y g

surface group representations with maximal toledo invariant we develop the theory of maximal representations of the fundamental group 1 of a compact connected oriented surface possibly with boundary into lie groups g of hermitian type for any homomorphism p 1 g we define the toledo invariant t p a numerical invariant which has both topological and analytical interpretations we establish important properties of t among which continuity uniform boundedness on the representation variety additivity under connected sum of surfaces and congruence relations mod we thus obtain information about the representation variety as well as striking geometric properties of maximal representations that is representations whose toledo invariant achieves the maximum value moreover we establish properties of boundary maps associated to maximal representations which generalize naturally monotonicity properties of semiconjugations of the circle we define a rotation number function for general locally compact groups and study it in detail for groups of hermitian type properties of the rotation number together with the existence of boundary maps lead to additional invariants for maximal representations and show that the subset of maximal representations is always real semialgebraic

the multiplicative eigenvalue problem and deformed quantum cohomology abstract we construct deformations of the small quantum cohomology rings of homogeneous spaces g p and obtain an irredundant set of inequalities determining the multiplicative eigen polytope for the compact form k of g

parabolic higgs bundles and teichmuller spaces for punctured surfaces in this paper we study the relation between parabolic higgs vector bundles and irreducible representations of the fundamental group of punctured riemann surfaces established by simpson we generalize a result of hitchin identifying those parabolic higgs bundles that correspond to fuchsian representations we also study the higgs bundles that give representations whose image is contained after conjugation in sl k r we compute the real dimension of one of the components of this space of representations which in the absence of punctures is the generalized teichmuller space introduced by hitchin and which in the case of k 2 is the usual teichmuller space of the punctured surface

episturmian words and some constructions of de luca and rauzy abstract in this paper we study infinite episturmian words which are a natural generalization of sturmian words to an arbitrary alphabet a characteristic property is they are closed under reversal and have at most one right special factor of each length they are first obtained by a construction due to de luca which utilizes the palindrome closure they can also be obtained by the way of extended rauzy rules

integrable dynamical systems with hierarchy ii solutions a semisystematic method of finding solutions that satisfy the zero nijenhuis tensor condition with dual symplectic structure is developed in this manner a few solutions including that of the toda lattice have been found it is also shown that the general solution for the linear problem is intimately connected with nontriviality of the second cohomology group of solvable lie algebras moreover the case of the quadratic problem requires studies of an algebra involving the triple linear product system

the self duality equations on a riemann surface in this paper we shall study a special class of solutions of the self dual yang mills equations the original self duality equations which arose in mathematical physics were defined on euclidean 4 space the physically relevant solutions were the ones with finite action the so called instantons the same equations may be dimensionally reduced to euclidean 3 space by imposing invariance under translation in one direction these equations also have physical relevance the solutions which have finite action in three dimensions are the magnetic monopoles if we take the reduction process one step further and consider solutions which are invariant under two translations we obtain a set of equations in the plane here however there is no clear physical meaning and indeed attempts to find finite action solutions have failed nevertheless these are the equations we shall consider despite the lack of interesting solutions in r2 the equations have the important property conformal invariance which allows them to be defined on manifolds modelled on r2 by conformal maps namely riemann surfaces we shall consider here solutions of the self duality equations defined on a compact riemann surface there are in fact solutions as we shall show and the moduli space of all solutions turns out to be a manifold with an extremely rich geometric structure which will be the focus of our study it brings together in a harmonious way the subjects of riemannian geometry topology algebraic geometry and symplectic geometry illuminating all these facets of the same object accounts for the length of this paper the self duality equations are equations from gauge theory geometrically they are defined in terms of connections on principal bundles while the group of the principal bundle may be chosen arbitrarily for the equations to make sense we restrict attention here to the simplest case of su 2 or so 3 there are two reasons for this the first and most obvious is that it simplifies calculations and avoids the use of inductive processes which are inherent in the consideration of a general lie group of higher rank the second reason is that solutions for su 2 have an intimate relationship with the internal structure of the riemann surface as a consequence of results we shall prove about solutions to the self duality equations we learn something about the moduli space of complex structures on the surface itself namely teichmuller space

integrable dynamical systems with hierarchy i formulation it is shown that any model with the zero generalized nijenhuis tensor under some additional assumptions will automatically satisfy the hierarchy equation with an infinite number of conserved quantities in involution especially if there is a dual symplectic structure with zero nijenhuis tensor then there exist infinite numbers of poisson brackets and of lagrangians giving the same equation of motion toda lattice is one such example

a note on the paper on brlek reutenauer conjecture in this short note we point to an error in the proof of a theorem stated in l balkova e pelantova s starosta on brlek reutenauer conjecture theoret comput sci 412 2011 5649 5655 by constructing a counterexample we show that the assertion of the theorem is actually incorrect although this theorem is of a technical character it was used in an argument leading to a corollary of a general interest to the brlek reutenauer conjecture and thus as a consequence of this note we have that the proof of the mentioned corollary is also flawed

moduli spaces of holomorphic triples over compact riemann surfaces a holomorphic triple over a compact riemann surface consists of two holomorphic vector bundles and a holomorphic map between them after fixing the topological types of the bundles and a real parameter there exist moduli spaces of stable holomorphic triples in this paper we study non emptiness irreducibility smoothness and birational descriptions of these moduli spaces for a certain range of the parameter our results have important applications to the study of the moduli space of representations of the fundamental group of the surface into unitary lie groups of indefinite signature 5 7 another application that we study in this paper is to the existence of stable bundles on the product of the surface by the complex projective line

the gromov norm of the kaehler class of symmetric domains let d be bounded symmetric domain of rank p and let o9 be the krhler form of its bergmann metric where the metric is normalized so that the minimum holomorphic sectional curvature is 1 i fx is a compact manifold whose universal cover is d to defines a cohomology class to e h 2 x r the purpose of this paper is to compute its sup norm in the sense of gromov 5 we show ii to ii co pn strictly speaking we only prove this for three of the four classes of classical domains cf sect 2 this theorem has the following topological corollary let s be a riemann surface of genus 0 1 and f s x a continuous map then f t o 4 p 1 n

symplectic structures their backlund transformations and hereditary symmetries abstract it is shown that compatible symplectic structures lead in a natural way to hereditary symmetries we recall that a hereditary symmetry is an operator valued function which immediately yields a hierarchy of evolution equations each having infinitely many commuting symmetries all generated by this hereditary symmetry furthermore this hereditary symmetry usually describes completely the soliton structure and the conservation laws of these equations this result then provide us with a method for constructing hereditary symmetries and hence exactly solvable evolution equations in addition we show how symplectic structures transform under backlund transformations this leads to a method for generating a whole class of symplectic structures from a given one several examples and applications are given illustrating the above results also the connection of our results with those of gelfand and dikii and of magri is briefly pointed out

surface group representations and u p q higgs bundles using the l2 norm of the higgs field as a morse function we study the moduli spaces of u p q higgs bundles over a riemann surface we require that the genus of the surface be at least two but place no constraints on p q a key step is the identification of the function s local minima as moduli spaces of holomorphic triples in a companion paper 7 we prove that these moduli spaces of triples are nonempty and irreducible because of the relation between flat bundles and fundamental group representations we can interpret our conclusions as results about the number of connected components in the moduli space of semisimple pu p q representations the topological invariants of the flat bundles are used to label subspaces these invariants are bounded by a milnor wood type inequality for each allowed value of the invariants satisfying a certain coprimality condition we prove that the corresponding subspace is nonempty and connected if the coprimality condition does not hold our results apply to the closure of the moduli space of irreducible representations 1 members of vbac vector bundles on algebraic curves which is partially supported by eager ec fp5 contract no hprn ct 2000 00099 and by edge ec fp5 contract no hprn ct 2000 00101 2 partially supported by the national science foundation under grant dms0072073 3 partially supported by the ministerio de ciencia y tecnolo ia spain under grant bfm2000 0024 4 partially supported by the fundacao para a ciencia e a tecnologia portugal through the centro de matematica da universidade do porto and through grant no sfrh bpd 1606 2000 5 partially supported by the portugal spain bilateral programme acciones integradas grant nos hp2000 0015 and ai 01 24 6 partially supported by a british epsrc grant october december 2001 received 11 27 2002

corrigendum on brlek reutenauer conjecture basic 2012 in 1 pointed to a gap in the proof of corollary 5 10 in balkova et al 2011 2 related to the brlek reutenauer conjecture in this corrigendum we correct the proof and show that the corollary remains valid

proof of the brlek reutenauer conjecture brlek and reutenauer conjectured that any infinite word u with language closed under reversal satisfies the equality 2d u email protected n 0 t u n in which d u denotes the defect of u and t u n denotes c u n 1 c u n 2 p u n 1 p u n where c u and p u are the factor and palindromic complexity of u respectively this conjecture was verified for periodic words by brlek and reutenauer themselves using their results for periodic words we have recently proved the conjecture for uniformly recurrent words in the present article we prove the conjecture in its general version by a new method without exploiting the result for periodic words

twisted harmonic maps and the self duality equations si m est une surface de riemann compacte on utilise des resultats de la theorie des applications harmoniques pour trouver une solution aux equations d auto dualite associees a une representation irreductible p 1 m psl 2 c