complex geometry and the asymptotics of harish chandra modules for real reductive lie groups i let g be a connected semisimple real matrix group it is now apparent that the representation theory of g is intimately connected with the complex geometry of the flag variety 9 by studying appropriate orbit structures on 9 we are naturally led to representation theory in the category of harish chandra modules x or the representation theory of category 9 the jacquet functor j f 9 has proved a useful tool in converting yfw problems into 9 problems which are often more tractable in this paper we advance the philosophy that the complex geometry of 9 associated to yf and 9 interacts in a natural way with the functor j leading to deep new information on the structure of jacquet modules this in turn gives new insight into the structure of certain nilpotent cohomology groups associated to harish chandra modules our techniques are based upon many of the ideas present in the proof of the kazhdan lusztig conjectures and bernstein s proof of the jantzen conjecture

necessary conditions for existence of non degenerate hamiltonian structures the necessary criteria are pointed out for the exisence of hamiltonian and bi hamiltonian non degenerate structures for a nonlinear system of partial differential equations of first order the results are formulated in terms of the new invariants of the intrinsic geometry introduced in this paper connected with the nijenhuis and haantjes tensors of a 1 1 tensor field

on the structure of 2 1 dimensional commutative and noncommutative integrable equations we develop the symbolic representation method to derive the hierarchies of 2 1 dimensional integrable equations from the scalar lax operators and to study their properties globally the method applies to both commutative and noncommutative cases in the sense that the dependent variable takes its values in c or a noncommutative associative algebra we prove that these hierarchies are indeed quasi local in the commutative case as conjectured by mikhailov and yamilov j phys a 31 6707 1998 we propose a ring extension in the noncommutative case based on the symbolic representation as examples we give noncommutative versions of kadomtsev petviashvili kp modified kadomtsev petviashvili mkp and boussinesq equations

lie algebraic approach to the construction of 2 1 dimensional lattice field and field integrable hamiltonian equations two different methods for the construction of 2 1 dimensional integrable lattice field and field hamiltonian dynamical systems are presented the first method is based on the so called central extension procedure applied to the lie algebra of shift operators and the lie algebra of pseudodifferential operators the second method is the so called operand formalism both methods allow a construction of some new integrable nonlinear hamiltonian lattice field and field equations in 2 1 dimensional space

r matrices and higher poisson brackets for integrable systems the tri hamiltonian nature of lax equations is revealed starting with an r matrix on an associative algebra g equipped with a trace form there are g compatible poisson brackets with linear quadratic and cubic dependence on the coordinates the invariant functions casimir functions on g are in involution relative to these brackets they yield a hierarchy of integrable tri hamiltonian lax equations the results can be applied to solvable pde s such as the korteweg de vries equation as well as to finite integrable systems such as the toda lattice in these cases the poisson structures considered here turn out to be abstract versions of the first 3 hamiltonian operators of these equations obtained by their well known recursion operators

asymptotic behavior of matrix coefficients of admissible representations introduction 869 1 generalities on reductive groups 873 2 the infinitesimal cartan decomposition 876 3 the t radial components 880 4 differential equations satisfied by sphereical functions 885 5 asymptotic behavior of spherical functions on a 888 6 asymptotic behavior of spherical functions along the walls of a 892 7 leading characters and growth estimates on the group 898 8 admissible representations and their matrix coefficients 906 appendix 917

hamiltonian theory over noncommutative rings and integrability in multidimensions a generalization of the adler gel fand dikii scheme is used to generate bi hamiltonian structures in two spatial dimensions in order to implement this scheme a hamiltonian theory is built over a noncommutative ring namely the ring of formal pseudodifferential operators bi hamiltonian structures generated in this way can be used for the kadomtsev petviashvili equation as well as other integrable equations in 2 1

schouten tensor and bi hamiltonian systems of hydrodynamic type the necessary conditions are derived for existence of a bi hamiltonian structure for a given hydrodynamic type system one of the conditions is the vanishing of its haantjes tensor a theorem is proved on the canonical forms of the generic bi hamiltonian systems the schouten 2 1 tensor sijk is connected with any hamiltonian system of hydrodynamic type the complete symmetry of the 3 0 tensor sijk is demonstrated the necessary conditions for existence of a single nondegenerate hamiltonian structure are obtained in terms of the special differential k forms p1 pk u1 uk

an alternative approach to solve the mixed akns equations abstract the algebraic geometric solutions of the mixed akns equations are investigated through a finite dimensional lie poisson hamiltonian system which is generated by the nonlinearization of the adjoint equation related to the akns spectral problem first each mixed akns equation can be decomposed into two compatible lie poisson hamiltonian flows then the separated variables on the coadjoint orbit are introduced to study these lie poisson hamiltonian systems further based on the hamilton jacobi theory the relationship between the action angle coordinates and the jacobi inversion problem is established in the end using riemann jacobi inversion the algebraic geometric solutions of the first three mixed akns equations are obtained

nijenhuis g manifolds and lenard bicomplexes a new approach to kp systems we suggest a method to extend the theory of recursion operators to integrable hamiltonian systems in two space dimensions like kp systems the approach aims to stress the conceptual unity of the theories in one and two space dimensions a sound explanation of the appearance of bilocal operators is also given

reduction techniques for infinite dimensional hamiltonian systems some ideas and applications in the language of tensor analysis on differentiable manifolds we present a reduction method of integrability structures and apply it to recover some well known hierarchies of integrable nonlinear evolution equations

an inversion formula for recursion operator of kp hierarchy an inversion formula is established for the recursion operator of the kp hierarchy the formula explains in part how operators y i are cancelled each other in certain combination of i y j that is essential for generating the kp hierarchy

the hamiltonian structure of the 2 1 dimensional ablowitz kaup newell segur hierarchy by considering hamiltonian theory over a suitable noncommutative ring the nonlinear evolution equations of the ablowitz kaup newell segur 2 1 hierarchy are incorporated into a hamiltonian framework and a modified lenard scheme

on the theory of recursion operator the general structure and properties of recursion operators for hamiltonian systems with a finite number and with a continuum of degrees of freedom are considered weak and strong recursion operators are introduced the conditions which determine weak and strong recursion operators are found

punctual hilbert schemes and resolutions of multiple point singularities if f x yp is a morphism of smooth complex analytic varieties with n p then the multiple points of order k of f i n the target are those y e yp with k preimages each preimage counted with proper multiplicity it is clear that to understand the geometry o f f it is necessary to understand these multiple point sets and their closures much work has also been done recently to calculate characteristic class formulas for these sets and their relatives in the source off cf 1 10 17 one approach to the problem of calculating the characteristic class formula of a singularity involves finding resolutions of the singularities 4 15 18 traditionally in singularities of maps one uses multi jet bundles to study and control multiple point singularities jet bundles often enter also in the constructions made to resolve singularities unfortunately as will be discussed in detail later multi jet bundles are difficult to work with if not useless when the problem involves the closure of the set of multiple points in this paper we give a new construction using punctual hilbert schemes which we offer as an alternative to multi jets in the study of multiple point singularities as an illustration of its usefulness we use it to find a resolution of the closure of the triple point set of any good map f theorem 2 3 and of the multiple point set of a g o o d f o f any order provided the m a p f has kernel rank at most 2 theorem 1 6 recall that the kernel rank of f a t x is the dimension of the kernel of df x this construction also provides a useful starting point for finding resolutions of multiple point sets for general f by reducing the problem of resolving the singularity of f t o the problem of resolving the singularities of the corresponding hilbert schemes in a later paper we will show how a modification of our construction allows one to resolve triple points in the source as well

enumerating stationary multiple points given a map f x y an r fold point off is a point xi of x such that there exist points x2 x of x which all have the same image underf we say that an r fold point is strict if x x are all distinct points of x and stationary if some of the xi have coalesced to lie infinitely near the infinitely near points determine tangent directions along the iibref f x in the sense that they constitute ramification points of f this paper enlarges on the work of 9 to develop under appropriate hypotheses enumerative formulas for loci of arbitrary stationary r fold points in terms of invariants of the map multiple point theory has a long and rich history we may trace work back to clebsch 1864 who gave a formula for the number of nodes of a plane curve in terms of the degree and genus of the curve perhaps the most famous example of a stationary multiple point result is the riemann hurwitz formula let f x y be a finite separable surjective map between smooth projective curves the riemann hurwitz theorem gives a formula for deg r where r is the ramification divisor of j it may be easily seen that deg r is simply a weighted number of stationary doublepoints off modern multiple point theory is concerned with developing formulas for general classes of maps the subject has been of considerable interest in differential geometry and topology as well as in algebraic geometry surveys of the work in these fields can be found in lo 9 introduction or 8 chap v pp 365 3911 in algebraic geometry multiple point theory has presently culiminated in two basic approaches the hilbert scheme see lo for a survey and the method of iteration we shall only be concerned with the latter technique in the body of this work both approaches share a 149 oool 8708 87 7 50

hamiltonian structure for the modulation equations of a sine gordon wavetrain on etudie l equation sinus gordon e 2 t 2 x 2 u sin u 0 ou e est un petit parametre on construit une solution asymptotique sous la forme d un train d onde n phase variant lentement

block diagonalizability problem for hydrodynamic type systems the necessary and sufficient conditions are derived for the reducibility of a hydrodynamic type system into a block diagonal form with k mutually interacting blocks applications to the perturbations of the benney system and to the hamiltonian systems of partial differential equations are presented the algebraic identities connecting the nijenhuis tensors nb a u v and na u v and haantjes tensors hb a u v and ha u v are discovered

bi hamiltonian structure in 2 d field theory we exhibit the bi hamiltonian structure of the equations of associativity witten dijkgraaf verlinde verlinde dubrovin equations in 2 d topological field theory which reduce to a single equation of monge ampere type f ttt f xxt 2 f xxx f xtt in the case of three primary fields the first hamiltonian structure of this equation is based on its representation as a 3 component system of hydrodynamic type and the second hamiltonian structure follows from its formulation in terms of a variational principle with a degenerate lagrangian

on the decoupling problem of general quasilinear first order systems in two independent variables abstract the paper deals with the decoupling problem of general quasilinear first order systems in two independent variables we consider either the case of homogeneous and autonomous systems or the one of nonhomogeneous and or nonautonomous systems necessary and sufficient conditions for the partial or full decoupling of the systems at hand are provided the conditions involve the properties of eigenvalues and eigenvectors of the coefficient matrix and provide the differential constraints whose integration leads to the decoupling transformation some applications of physical interest are also given

recursion operators and bi hamiltonian structures in multidimensions ii the algebraic properties of exactly solvable evolution equations in one spatial and one temporal dimensions have been well studied in particular the factorization of certain operators called recursion operators establishes the bi hamiltonian nature of all these equations recently we have presented the recursion operator and the bi hamiltonian formulation of the kadomtsev petviashvili equation a two spatial dimensional analogue of the korteweg devries equation here we present the general theory associated with recursion operators for bi hamiltonian equations in two spatial and one temporal dimensions as an application we show that general classes of equations which include the kadomtsev petviashvili and the davey stewartson equations possess infinitely many commuting symmetries and infinitely many constants of motion in involution under two distinct poisson brackets furthermore we show that the relevant recursion operators naturally follow from the underlying isospectral eigenvalue problems

hamiltonian structure of real monge ampere equations the variational principle for the real homogeneous monge ampere equation in two dimensions is shown to contain three arbitrary functions of four variables there exist two different specializations of this variational principle where the lagrangian is degenerate and furthermore contains an arbitrary function of two variables the hamiltonian formulation of these degenerate lagrangian systems requires the use of dirac s theory of constraints as in the case of most completely integrable systems the constraints are second class and dirac brackets directly yield the hamiltonian operators thus the real homogeneous monge ampere equation in two dimensions admits two classes of infinitely many hamiltonian operators namely a family of local as well as another family non local hamiltonian operators and symplectic 2 forms which depend on arbitrary functions of two variables the simplest non local hamiltonian operator corresponds to the kac moody algebra of vector fields and functions on the unit circle hamiltonian operators that belong to either class are compatible with each other but between classes there is only one compatible pair in the case of real monge ampere equations with constant right hand side this compatible pair is the only pair of hamiltonian operators that survives then the complete integrability of all these real monge ampere equations follows by magri s theorem some of the remarkable properties we have obtained for the hamiltonian structure of the real homogeneous monge ampere equation in two dimensions turn out to be generic to the real homogeneous monge ampere equation and the geodesic flow for the complex homogeneous monge ampere equation in arbitrary number of dimensions hence among all integrable nonlinear evolution equations in one space and one time dimension the real homogeneous monge ampere equation is distinguished as one that retains its character as an integrable system in multiple dimensions

decoupling problem for systems of quasi linear pde s the necessary and sufficient conditions for the decoupling of a quasi linear system of partial differential equations into k non interacting subsystems are derived several necessary conditions for the decoupling are found and applied to the benney system

embeddings of harish chandra modules homology and the composition series problem the case of real rank one let g be a connected semisimple matrix group of real rank one fix a minimal parabolic subgroup p man and form the normalized principal series representations 1i u in the case of regular infinitesimal character we explicitly determine in terms of langlands classification all irreducible submodules and quotients of 1i u as a corollary all embeddings of an irreducible harish chandra module into principal series are computed the number of such embeddings is always less than or equal to three these computations are equivalent to the determination of zero n homology

yang baxter equations and intermediate long wave hierarchies in the framework of the gn manifold approach the algebraic structures of two representations of the ilw hierarchy are investigated it is shown that this hierarchy can be obtained by different realizations of two abstract structures strictly related with the yang baxter equations a new reduction theorem for the second representation of ilw is also formulated