representations p adiques et equations differentielles in this paper we associate to every p adic representation v a p adic differential equation d rig v that is to say a module with a connection over the robba ring we do this via the theory of fontaine s k modules this construction enables us to relate the theory of k modules to p adic hodge theory we explain how to construct dcris v and dst v from d rig v which allows us to recognize semi stable or crystalline representations the connection is then unipotent or trivial on d rig v 1 t in general the connection has an infinite number of regular singularities but v is de rham if and only if those are apparent singularities a structure theorem for modules over the robba ring allows us to get rid of all singularities at once and to obtain a classical differential equation with a frobenius structure using this we construct a functor from the category of de rham representations to that of classical p adic differential equations with frobenius structure a recent theorem of y andre gives a complete description of the structure of the latter object this allows us to prove fontaine s p adic monodromy conjecture every de rham representation is potentially semi stable as an application we can extend to the case of arbitrary perfect residue fields some results of hyodo h1g h1st of perrin riou the semi stability of ordinary representations of colmez absolutely crystalline representations are of finite height and of bloch and kato if the weights of v are 2 then bloch kato s exponential expv is an isomorphism

discrete integrable systems and deformations of associative algebras interrelations between discrete deformations of the structure constants for associative algebras and discrete integrable systems are reviewed theory of deformations for associative algebras is presented closed left ideal generated by the elements representing the multiplication table plays a central role in this theory deformations of the structure constants are generated by the deformation driving algebra and governed by the central system of equations it is demonstrated that many discrete equations such as discrete boussinesq equation discrete wdvv equation discrete schwarzian kp and bkp equations discrete hirota miwa equations for kp and bkp hierarchies are particular realizations of the central system an interaction between the theories of discrete integrable systems and discrete deformations of associative algebras is reciprocal and fruitful an interpretation of the menelaus relation discrete schwarzian kp equation discrete hirota miwa equation for kp hierarchy consistency around the cube as the associativity conditions and the concept of gauge equivalence for instance between the menelaus and kp configurations are particular examples

on a theorem of benson and curtis iwahori conjectured 5 that the hecke algebra over q of a chevalley group g over a finite field f with respect to a bore1 subgroup is isomorphic to the group algebra over q of the weyl group the analogous statement with q replaced by its algebraic closure was proved by tits 3 ex 27 p 561 using a deformation argument in 11 benson and curtis proved using a case by case analysis that iwahori s conjecture is true over q whenever g is simple of type ze e with the help of springer they proved that when g is of type e or e there exists irreducible representations of the hecke algebra whose character involve 4 and hence iwahori s conjecture over q is false in general the main result of this paper is an explicit construction of an isomorphism of the hecke algebra over q q with the group algebra over q q of the weyl group the construction makes essential use of the polynomials p w of 8 at some point in the proof i have to appeal to the interpretation of p l in terms of multiplicities in verma modules 8 i 51 which has recently been established by brylinski and kashiwara 4 and independently by belinson and bernstein

the non local delta problem and 2 1 dimensional soliton equations a method of constructing 2 1 dimensional non linear integrable equations and their solutions by means of the non local delta problem is developed a basic set of equations is obtained by using different normalisations of the non local delta problem and the lagrangian of the set is found other integrable equations which are degenerate cases of the basic set are also lagrangian

hecke algebras of finite type are cellular let mathcal h be the one parameter hecke algebra associated to a finite weyl group w defined over a ground ring in which bad primes for w are invertible using deep properties of the kazhdan lusztig basis of mathcal h and lusztig s a function we show that mathcal h has a natural cellular structure in the sense of graham and lehrer thus we obtain a general theory of specht modules for hecke algebras of finite type previously a general cellular structure was only known to exist in types an and bn

gauge transformation and symmetries of the commutative multicomponent bkp hierarchy in this paper we defined a new multi component b type kadomtsev petviashvili bkp hierarchy that takes values in a commutative subalgebra of after this we give the gauge transformation of this commutative multicomponent bkp cmbkp hierarchy meanwhile we construct a new constrained cmbkp hierarchy that contains some new integrable systems including coupled kdv equations under a certain reduction after this the quantum torus symmetry and quantum torus constraint on the tau function of the commutative multi component bkp hierarchy will be constructed

from integrable nets to integrable lattices inspired by the results of jonas einsenhart demoulin and bianchi on the permutability property of classical geometrical transformations of conjugate nets and its reductions of pseudo orthogonal pseudo symmetric and pseudo egorov types dressing transformations of the n component kp hierarchy described within the grassmannian are used to generate quadrilateral lattices and its corresponding reductions as a byproduct we get the corresponding discrete dressing transformations in particular we characterize the vectorial fundamental discrete transformations preserving the symmetric lattice

characterization of jacobian varieties in terms of soliton equations pour dans le demi espace superieur de siegel h g soit x c g z g z g la variete abelienne principalement polarisee correspondante et soit z z m e z gexp 2 i t mz i t m m la fonction theta de riemann pour x les deux conditions a et b pour x sont equivalentes a 1 il existe des vecteurs a 1 a 2 a 3 c g a 1 0 et une forme quadratique q t i j 1 3 q ij t i t j q ij c tel que pour c g t t exp q t t 1 a 1 t 2 a 2 t 3 a 3 donne une solution a l equation de kadomtsev petviashvili sous la forme d hirota d 1 4 3d 2 2 4d 1 d 3 0 2 le diviseur de x est irreductible b x est isomorphe a la variete jacobienne d une courbe lisse complete de genre g sur c

integral crystalline cohomology over very ramified valuation rings in this paper we intend to show how fontaine s comparison theory fol fo2 between crystalline and p adic etale cohomology can be extented to very ramified base rings so far it had been developed either for qp coefficients or zp coefficients if the base is unramified over the witt vectors see fa2 for example it states that for any p adic discrete valuation ring v there exists a ring b v with the following property if x denotes a smooth and proper v scheme the p adic etale cohomology of the generic fiber x xv k is related to the crystalline cohomology of x v0 vo defined below by an isomorphism

analytic bilinear approach to integrable hierarchies i generalized kp hierarchy an analytic bilinear approach for construction and study of integrable hierarchies in particular the kp hierarchy is proposed it starts with the generalized hirota identity for the cauchy baker akhiezer cba function and leads to a generalized kp hierarchy in the form of compact functional equations containing a special shift operator a generalized kp hierarchy incorporates the basic kp hierarchy modified kp hierarchy kp singularity manifold equation hierarchy and corresponding hierarchies of linear problems different vertical levels of the generalized kp hierarchy are connected via invariants of the combescure symmetry group the resolution of functional equations also gives rise to the function and the addition formulas for it

generalized kp hierarchy mobius symmetry symmetry constraints and calogero moser system abstract analytic bilinear approach is used to study continuous and discrete non isospectral symmetries of the generalized kp hierarchy it is shown that mobius symmetry transformation for the singular manifold equation leads to continuous or discrete non isospectral symmetry of the basic scalar or multicomponent kp hierarchy connected with binary backlund transformation a more general class of multicomponent mobius type symmetries is studied it is demonstrated that symmetry constraints of kp hierarchy defined using multicomponent mobius type symmetries give rise to calogero moser system

on the one and two dimensional toda lattices and the painleve property the toda lattice and the two dimensional toda lattice 2 dtl are shown to possess a type of painleve property that is based on the use of separate singular manifolds for each dependent variable the isospectral problem for the 2 dtl found by both mikhailov and by fordy and gibbons can be simply and logically derived from this analysis some remarks are made about the connection between our work and independent work of kametaka and airhault on the relationship between the toda lattice and the second painleve transcendent

an algebraic description of generalized k constraints the generalized k constrained kp hierarchy is shown to correspond to a so called pseudo reduction of the two dimensional toda lattice hierarchy described in a free fermion approach which is adapted to the case of two singularities in the spectral parameter range wronskian solutions are discussed and in particular soliton solutions are recovered through a reduction of the toda solitons

dressing methods for geometric nets i conjugate nets the formalism of multicomponent kp hierarchies is applied to deriving efficient dressing methods for conjugate nets the notion of the cauchy propagator is used for characterizing these nets in terms of spectral data explicit examples in dimensions n 2 and 3 are given in particular periodic nets and cartesian nets with a gaussian localized deformation are exhibited

charged free fermions vertex operators and the classical theory of conjugate nets we show that the quantum field theoretical formulation of the function theory has a geometrical interpretation within the classical transformation theory of conjugate nets in particular we prove that i the partial charge transformations preserving the neutral sector are laplace transformations ii the basic vertex operators are levy and adjoint levy transformations and iii the diagonal soliton vertex operators generate fundamental transformations we also show that the bilinear identity for the multicomponent kadomtsev petviashvili hierarchy becomes through a generalized miwa map a bilinear identity for the multidimensional quadrilateral lattice equations

a sum rule for scale factors in the goldie rank polynomials this paper is a corollary to s and we adopt the same notation which is also that of s there we showed how two deep questions concerning the enveloping algebra of a semisimple lie algebra could be reduced to a purely combinatorial analysis of the kazhdan lusztig polynomials the first concerns the computation of sot l l wl l yl apart from its own intrinsic interest this question has deep interconnections with geometry via the component group a of the associated nilpotent orbit the second concerns the coefficients z which are ratios of goldie ranks these are central to the determination of the scale factors in the goldie rank polynomials which in turn relate to complete primeness and hence to unitarity see 7 1 4 16 7 121 it first seemed that the results in s did not give any easily extractable information however in 13 lusztig shows that for some canonical quotient a of a there exists a left cell of the weyl group w such that the character table of a can be read off in a well defined manner from data involving the hecke algebra module structure of the cell which we shall call a lusztig cell combined with s this gives several remarkable results first when rl z2 are involutions in a lusztig cell sot l l r a l 12 is determined by multiplying appropriate irreducible characters of a and identifying coefficients this implies that when a e s which occurs in type e it can happen that l l zl l r l is not multiplicity free providing thereby a counterexample to conjecture of 5 5 111 even in the diagonal case second if r is an involution in a lusztig cell then z is the degree of an irreducible character of a third we prove that z divides ial for every element w of a lusztig cell in partial answer to a conjecture in 9 3 231

generalized hirota bilinear identity and integrable q difference and lattice hierarchies abstract the hirota bilinear identity for the cauchy baker akhieser cba kernel is introduced as a basic tool to construct integrable hierarchies containing lattice and q difference times a determinant formula for the action of meromorphic function on the cba kernel is derived this formula gives the opportunity to construct generic solutions for integrable lattice equations

wach modules and critical slope p adic l functions we study kato and perrin riou s critical slope p adic l function attached to an ordinary modular form using the methods of a lei d loeffler and s l zerbes wach modules and iwasawa theory for modular forms asian j math 14 2010 475 528 we show that it may be decomposed as a sum of two bounded measures multiplied by explicit distributions depending only on the local properties of the modular form at p we use this decomposition to prove results on the zeros of the p adic l function and we show that our results match the behaviour observed in examples calculated by pollack and stevens in overconvergent modular symbols and p adic l functions ann sci ec norm super 4 44 2011 no 1 1 42

analytic bilinear approach to integrable hierarchies ii multicomponent kp and 2d toda lattice hierarchies an analytic bilinear approach for the construction and study of integrable hierarchies is discussed generalized multicomponent kp and 2d toda lattice hierarchies are considered this approach allows us to represent generalized hierarchies of integrable equations in a condensed form of finite functional equations a generalized hierarchy incorporates basic hierarchy modified hierarchy singularity manifold equation hierarchy and corresponding linear problems different levels of generalized hierarchy are connected via invariants of combescure symmetry transformation the resolution of functional equations also leads to the function and addition formulas to it

on the problem and integrable equations using the problem and dual problem we derive bilinear relations which allow us to construct integrable hierarchies in different parametrizations their darboux backlund transformations and to analyse constraints for them in a very simple way scalar kp bkp and ckp hierarchies are considered as examples

transformations of quadrilateral lattices motivated by the classical studies on transformations of conjugate nets we develop the general geometric theory of transformations of their discrete analogs the multidimensional quadrilateral lattices i e lattices x zn rm n m whose elementary quadrilaterals are planar our investigation is based on the discrete analog of the theory of the rectilinear congruences which we also present in detail we study in particular the discrete analogs of the laplace combescure levy radial and fundamental transformations and their interrelations the composition of these transformations and their permutability is also investigated from a geometric point of view the deep connections between transformations and discretizations is also investigated for quadrilateral lattices we finally interpret these results within the formalism

decomposition numbers for generic iwahori hecke algebras of noncrystallographic type we compute the e modular decomposition matrices for the generic iwahori hecke algebras of typei2 m form n m 2 h3 andh4 for alle n leading to nontrivial decomposition maps the results are obtained by a combined use of different ideas from computational representation theory and by application of the computer algebra systemsgap chevie vectorenumeratorandmeataxe

morphisms from hecke algebras to graphic temperley lieb algebras abstract in this paper we construct many surjective morphisms from the hecke algebras h of uniform type infinite type mostly to certain graphic temperley lieb algebras which provide a way to obtain representations of those hecke algebras when is the coxeter matrix i 2 2 n 1 of the dihedral group d 2 n 1 we show there are altogether n such morphisms exactly realize all n different two dimensional irreducible representations of h in the study the chebyshev polynomials appear and play a prominent role in the computation

combinatorial gelfand models abstract a combinatorial construction of a gelfand model for the symmetric group and its iwahori hecke algebra is presented

darboux transformations for linear operators on two dimensional regular lattices darboux transformations for linear operators on regular two dimensional lattices are reviewed the six point scheme is considered as the master linear problem whose various specifications reductions and sublattice combinations lead to other linear operators together with the corresponding darboux transformations the second part of the review deals with multidimensional aspects of basic reductions of the four point scheme as well as the three point scheme

dieudonne crystals and wach modules for p divisible groups let k be a perfect field of characteristic p 2 and k an extension of f fracw k contained in some f pr using crystalline dieudonne theory we provide a classification of p divisible groups over r ok t1 td in terms of finite height modules over s w k u t1 td when d 0 such a classification is a consequence of a special case of the theory of kisin ren in this setting our construction gives an independent proof of this result and moreover allows us to recover the dieudonne crystal of a p divisible group from the wach module associated to its tate module by berger breuil or by kisin ren

dispersionful analog of the whitham hierarchy the transition from the zero genus universal whitham hierarchy to its dispersionful counterpart making use only of the lax representations is presented this is an alternative approach to that of takasaki who has recently shown that the dispersionless limit of the charged multicomponent kadomtsev petviashvili kp hierarchy is the whitham hierarchy the advantage of the presented approach is the construction of finite field reductions which are the main concern in this paper the theory is illustrated by several significant examples

a new proof for classification of irreducible modules of a hecke algebra of type an 1 abstract in this paper we give a new proof for the classification of irreducible modules of a hecke algebra of type a n 1 which was obtained by dipper and james in 1986

a novel generalization of clifford s classical point circle configuration geometric interpretation of the quaternionic discrete schwarzian kadomtsev petviashvili equation the algebraic and geometric properties of a novel generalization of clifford s classical 4 pointcircle configuration are analysed a connection with the integrable quaternionic discrete schwarzian

functional representations of integrable hierarchies we consider a general framework for integrable hierarchies in lax form and derive certain universal equations from which functional representations of particular hierarchies such as kp discrete kp mkp akns i e formulations in terms of functional equations are systematically and quite easily obtained the formalism genuinely applies to hierarchies where the dependent variables live in a noncommutative typically matrix algebra the obtained functional representations can be understood as noncommutative analogues of fay identities for the kp hierarchy

from akns to derivative nls hierarchies via deformations of associative products using deformations of associative products derivative nonlinear schrodinger dnls hierarchies are recovered as akns type hierarchies since the latter can also be formulated as gelfand dickey type lax hierarchies a recently developed method to obtain functional representations can be applied we actually consider hierarchies with dependent variables in any possibly noncommutative associative algebra e g an algebra of matrices of functions this also covers the case of hierarchies of coupled derivative nls equations

menelaus theorem clifford configurations and inversive geometry of the schwarzian kp hierarchy it is shown that the integrable discrete schwarzian kp dskp equation which constitutes an algebraic superposition formula associated with for instance the schwarzian kp hierarchy the classical darboux transformation and quasi conformal mappings encapsulates nothing but a fundamental theorem of ancient greek geometry thus it is demonstrated that the connection with menelaus theorem and more generally clifford configurations renders the dskp equation a natural object of inversive geometry on the plane the geometric and algebraic integrability of dskp lattices and their reductions to lattices of menelaus darboux schwarzian kdv schwarzian boussinesq and schramm type are discussed the dskp and discrete schwarzian boussinesq equations are shown to represent discretizations of families of quasi conformal mappings

representations galoisiennes p adiques et phi tau modules let p be an odd prime number and k be a p adic field in this paper we develop an analogue of fontaine s theory of phi gamma modules replacing the p cyclotomic extension by the extension k infty obtained by adding to k a compatible system of p n th roots of a fixed uniformizer pi of k as a result we obtain a new classification of p adic representations of g k gal kbar k by some phi tau modules we then make a link between the theory of phi tau modules discussed above and the so called theory of phi n nabla modules developped by kisin as a corollary we answer a question of tong liu we prove that if k is a finite extension of q p every representation of g k of e u finite height is potentially semi stable

tetrahedra octahedra and cubo octahedra integrable geometry of multi ratios geometric and algebraic aspects of multi ratios m2n are investigated in detail connections with menelaus theorem clifford configurations and maxwell s reciprocal quadrangles are utilized to associate the multi ratios m4 m8 and m8 with tetrahedra octahedra and cubo octahedra respectively integrable maps defined on face centred fcc lattices and irregular lattices composed of the face centres of simple cubic lattices are constructed and related to the discrete kp and bkp equations and the integrable discrete darboux system governing conjugate lattices an interpretation in terms of integrable irregular lattices of slopes on the plane is also given

generalized integrable hierarchies and combescure symmetry transformations unifying hierarchies of integrable equations are discussed they are constructed via the generalized hirota identity it is shown that the combescure transformations known for a long time for the darboux system and having a simple geometrical meaning are in fact the symmetry transformations of generalized integrable hierarchies though the connection with geometry in the general case is not clear the generalized equation written in terms of invariants of combescure transformations are the usual integrable equations and their modified partners the kp mkp ds mds hierarchies and darboux system are considered

the minimal basis for the centre of an iwahori hecke algebra abstract this paper arose out of an attempt to generalize the q q q 1 basis for the centre of an iwahori hecke algebra h q found by jones to z q q 1 and to other types considering the iwahori hecke algebra h over a subring z of z q1 2 q 1 2 where q1 2 q 1 2 we use a new and natural definition of positivity on h to describe the minimal z basis for z h in terms of a partial order on the positive part of h the main result is to show that this minimal basis is the set of primitive minimal elements of the positive part of the centre for any weyl group in addition the primitive minimal positive central elements can be characterized as exactly those positive central elements which specialize on setting 0 the equivalent of setting q 1 in z q1 2 q 1 2 to the sum of elements in a conjugacy class and which apart from the shortest elements from that conjugacy class sum contain no other terms corresponding to shortest elements of any conjugacy class a constructive algorithm is provided for obtaining the minimal basis we use the results of jones to achieve the result in type an without the need for character theory and give the result for all weyl groups using the character theoretical results of geck and rouquier finally we discuss the non crystallographic cases and give some explicit examples