integration of quantum many body problems by affine knizhnik zamolodchikov equations abstract an equivalence of the affine analogues of knizhnik zamolodchikov and dunkl equations is established for arbitrary root systems as a corollary an isomorphism is obtained from the solutions of the above kz equations in the induced representations of hecke algebras to the eigenfunctions for the corresponding matrix quantum calogero sutherland operators which generalizes the matsuo map

a unification of knizhnik zamolodchikov and dunkl operators via affine hecke algebras some generalizations of the lusztig lascoux schutzenberger operators for affine hecke algebras are considered as corollaries we obtain lusztig s isomorphisms from affine hecke algebras to their degenerate versions a natural interpretation of the dunkl operators and a new class of differential difference operators generalizing dunkl s ones and the knizhnik zamolodchikov operators from the two dimensional conformal field theory

auto parametric resonance in cyclindrical shells using geometric averaging abstract a study is presented of internal auto parametric instabilities in the free non linear vibrations of a cylindrical shell focussed on two modes a concertina mode and a chequerboard mode whose non linear interaction breaks the in out symmetry of the linear vibration theory the two mode interaction leads to preferred vibration patterns with larger deflection inwards than outwards and at internal resonance significant energy transfer occurs between the modes a rayleigh ritz discretization of the von karman donnell equations leads to the hamiltonian and transformation into action angle co ordinates followed by averaging provides readily a geometric description of the modal interaction it was established that the interaction should be most pronounced when there are slightly less than 2 n square chequerboard panels circumferentially where n is the ratio of shell radius to thickness

some calculations of the lusztig vogan bijection for classical nilpotent orbits abstract in this manuscript we compute explicitly the lusztig vogan bijection for local systems of some classical special nilpotent orbits using these results we prove a conjecture of achar and sommers on regular functions of some covers of classical nilpotent orbits

dynamic instability and chaos of empty and fluid filled circular cylindrical shells under periodic axial loads abstract in the present paper the dynamic stability of circular cylindrical shells subjected to static and dynamic axial loads is investigated both donnell s nonlinear shallow shell and sanders koiter shell theories have been applied to model finite amplitude static and dynamic deformations results are compared in order to evaluate the accuracy of these theories in predicting instability onset and post critical nonlinear response the effect of a contained fluid on the stability and the post critical behaviour is analyzed in detail geometric imperfections are considered and their influence on the dynamic instability and post critical behaviour is investigated chaotic dynamics of pre compressed shells is investigated by means of nonlinear time series techniques extracting correlation dimension and lyapunov exponents

riesz distributions and laplace transform in the dunkl setting of type a abstract we study riesz distributions in the framework of rational dunkl theory associated with root systems of type a as an important tool we employ a laplace transform involving the associated dunkl kernel which essentially goes back to macdonald 18 but was so far only established at a formal level we give a rigorous treatment of this transform based on suitable estimates of the type a dunkl kernel our main result is a precise analogue in the dunkl setting of a well known result by gindikin stating that a riesz distribution on a symmetric cone is a positive measure if and only if its exponent is contained in the wallach set for riesz distributions in the dunkl setting we obtain an analogous characterization in terms of a generalized wallach set which depends on the multiplicity parameter on the root system

primitive ideals and orbital integrals in complex exceptional groups in 2 two related problems were studied for complex classical groups determination of the primitive spectrum of the enveloping algebra as a set and fourier inversion of unipotent orbital integrals the first of these was solved completely extending work of joseph and others in sl n c and small groups although our techniques presumably solve the second as well up to determination of normalizing constants we carried out the calculations only for special unipotent classes see 181 we had of course tried to apply the same methods to the exceptional groups but they seem to be inadequate since that work was done however brylinski and kashiwara and beilinson and bernstein have established the kazhdan lusztig conjecture giving character formulas for irreducible highest weight modules 3 71 because of a conjecture of joseph proved in 24 this determines in principle the primitive ideals with a fixed regular integral infinitesimal character they are in one to one correspondence with what kazhdan and lusztig call left cells in the weyl group however the algorithm given by kazhdan and lusztig to compute these cells is enormously complicated requiring one to compute roughly 1 i polynomials of degrees on the order of half the number of positive roots this is very unsatisfactory however it

some combinatorial properties of jack symmetric functions if 1 l n then write 2 n or 1 n if p is another partition then write p c j if pi 6 for all i i e if the diagram of 1 contains the diagram of p if ipl ii 1 then write p 2 1 reverse lexicographic order if either j i or the first nonvanishing difference e pi is positive for instance writing 2 1 i for ai i it 5 5 41 32 5 311 221 2111 11111 finally write p i if ipl lal and p1 pl2 pli a1 a2 li for all i macdonald m p 63 call the partial ordering the natural ordering but we will call it the dominance ordering let x x x2 be an infinite set of indeterminates as in mi we

global well posedness and scattering for a class of nonlinear dunkl schrodinger equations in this paper we introduce a class of nonlinear schrodinger equations associated with the dunkl operators in this regard we study local and global well posedness and the scattering theory associated with these equations

confluent hypergeometric orthogonal polynomials related to the rational quantum calogero system with harmonic confinement two families type a and type b of confluent hypergeometric polynomials in several variables are studied we describe the orthogonality properties differential equations and pieri type recurrence formulas for these families in the one variable case the polynomials in question reduce to the hermite polynomials type a and the laguerre polynomials type b respectively the multivariable confluent hypergeometric families considered here may be used to diagonalize the rational quantum calogero models with harmonic confinement for the classical root systems and are closely connected to the symmetric generalized spherical harmonics investigated by dunkl

nonlinear generalized dunkl wave equations and applications abstract in this paper we introduce a class of nonlinear wave equations associated with the generalized dunkl laplace operator we study local and global well posedness next we establish the linearization of bounded energy solutions in the spirit of gerard 1996 9 the proof uses strichartz type inequalities and the energy estimate

quantization of special symplectic nilpotent orbits and normality of their closures abstract we study the regular function ring r o for all symplectic nilpotent orbits o with even column sizes we begin by recalling the quantization model for all such orbits by barbasch using unipotent representations with this model we express the multiplicities of fundamental representations appearing in r o by a parabolically induced module finally we will use this formula to give a criterion on the normality of the zariski closure o of o

radon transform on symmetric matrix domains let k r c h be the field of real complex or quaternionic numbers and mp q k the vector space of all p q matrices let x be the matrix unit ball in mn r r k consisting of contractive matrices as a symmetric space x g k o n r r o n r o r u n r r u n r u r and respectively sp n r r sp n r sp r the matrix unit ball y0 in mr r r with r n 1 is a totally geodesic submanifold of x and let y be the set of all g translations of the submanifold y0 the set y is then a manifold and an affine symmetric space we consider the radon transform rf y for functions f c 0 x defined by integration of f over the subset y and the dual transform rt f x x x for functions f y on y for 2r n 2r r with a certain evenness condition in the case k r we find a g invariant differential operator m and prove it is the right inverse of rtr rtr mf cf for f c 0 x c 0 the operator f rtr f is an integration of f against a singular function determined by the root systems of x and y0 we study the analytic continuation of the powers of the function and we find a bernstein sato type formula generalizing earlier work of the author in the set up of the berezin transform when x is a rank one domain of hyperbolic balls in kn 1 and y0 is the hyperbolic ball in kr 1 1 r n we obtain an inversion formula for the radon transform namely mrtr f cf this generalizes earlier results of helgason for non compact rank one symmetric spaces for the case r n 1

low dimensional models of shell vibrations parametrically excited vibrations of cylinder shells vibrations of cylindrical shells parametrically excited by axial forcing are considered the governing system of two coupled non linear partial differential equations is discretized by using lagrange equations the computation is simplified significantly by the application of computer algebra and as a result low dimensional models of shell vibrations are readily obtained after applying numerical continuation techniques and ideas from dynamical systems theory complete bifurcation diagrams are constructed the principal aim is to investigate the interaction between different modes of shell vibration results for system models with two of the lowest modes are discussed

compositions of theta correspondences theta correspondence y over r is established by howe j amer math soc 2 1989 535 in he j funct anal 199 2003 92 we prove that y preserves unitarity under certain restrictions generalizing the result of li invent math 97 1989 237 the goal of this paper is to elucidate the idea of constructing unitary representation through the propagation of theta correspondences we show that under a natural condition on the sizes of the related dual pairs which can be predicted by the orbit method j algebra 190 1997 518 representation theory of lie groups park city 1998 pp 179 238 the orbit correspondence for real and complex reductive dual pairs preprint 2001 one can compose theta correspondences to obtain unitary representations we call this process quantum induction r 2004 published by elsevier inc msc primary 22e45 22e46

low dimensional models for the nonlinear vibration analysis of cylindrical shells based on a perturbation procedure and proper orthogonal decomposition in formulating mathematical models for dynamical systems obtaining a high degree of qualitative correctness i e predictive capability may not be the only objective the model must be useful for its intended application and models of reduced complexity are attractive in many cases where time consuming numerical procedures are required this paper discusses the derivation of discrete low dimensional models for the nonlinear vibration analysis of thin cylindrical shells in order to understand the peculiarities inherent to this class of structural problems the nonlinear vibrations and dynamic stability of a circular cylindrical shell subjected to static and dynamic loads are analyzed this choice is based on the fact that cylindrical shells exhibit a highly nonlinear behavior under both static and dynamic loads geometric nonlinearities due to finite amplitude shell motions are considered by using donnell s nonlinear shallow shell theory a perturbation procedure validated in previous studies is used to derive a general expression for the nonlinear vibration modes and the discretized equations of motion are obtained by the galerkin method using modal expansions for the displacements that

yangian gelfand zetlin bases gl n jack polynomials and computation of dynamical correlation functions in the spin calogero sutherland model we consider the invariant calogero sutherland models with n 1 2 3 in the framework of symmetric polynomials

stability and vibration of empty and fluid filled circular cylindrical shells under static and periodic axial loads abstract in the present study the dynamic stability of simply supported circular cylindrical shells subjected to dynamic axial loads is analysed geometric nonlinearities due to finite amplitude shell motion are considered by using the donnell s nonlinear shallow shell theory the effect of structural damping is taken into account a discretization method based on a series expansion involving a relatively large number of linear modes including axisymmetric and asymmetric modes and on the galerkin procedure is developed axisymmetric modes are included indeed they are essential in simulating the inward deflection of the mean oscillation with respect to the equilibrium position and in describing the axisymmetric deflection due to axial loads a finite length simply supported shell is considered the boundary conditions are satisfied including the contribution of external axial loads acting at the shell edges the effect of a contained liquid is investigated the linear dynamic stability and nonlinear response are analysed by using continuation techniques and direct simulations

the unitary dual for complex classical lie groups in this paper we classify the unitary spectrum of the complex classical groups let g be any complex simply connected semisimple lie group there are two well known ways of constructing unitary representations from lower groups unitary induction from a parabolic subgroup and complementary series a reasonable way to describe the unitary dual is to find a distinguished set of representations for each group such that every other unitary representation is obtained from this set by the above constructions from strictly smaller parabolic subgroups this is entirely similar to the classification of admissible modules where the distinguished representations are the discrete series for integral infinitesimal character such a set is provided by the special unipotent representations introduced in bv1 their existence and unitarity was conjectured earlier by a they were shown to have many interesting properties such as the fact that their annihilator is a maximal primitive ideal from this it was possible to obtain character formulas for them however it is clear that this set is not large enough to describe the entire unitary spectrum for example in type c the metaplectic representation is not special unipotent or even one of the larger class of representations considered in a the definition of special unipotent representations suggests that one should consider representations with maximal primitive ideal and infinitesimal character as small as possible it is not clear which of them should be unitary imposing the condition that the annihilator be a completely prime ideal which is necessary for unitarity does not seem to be very helpful either thus one is left with the task of explicitly computing the g invariant form and ruling out the cases when it is indefinite this is an almost impossible calculation a rough estimate says that one needs to compute signatures on the k types in the principal series which have length ii 2p ql

support properties and holmgren s uniqueness theorem for differential operators with hyperplane singularities let w be a finite coxeter group acting linearly on rn in this article we study the support properties of a w invariant partial differential operator d on rn with real analytic coefficients our assumption is that the principal symbol of d has a special form related to the root system corresponding to w in particular the zeros of the principal symbol are supposed to be located on hyperplanes fixed by reflections in w we show that conv suppdf conv suppf holds for all compactly supported smooth functions f so that conv suppf is w invariant the main tools in the proof are holmgren s uniqueness theorem and some elementary convex geometry several examples and applications linked to the theory of special functions associated with root systems are presented

the calogero sutherland model and generalized classical polynomials multivariable generalizations of the classical hermite laguerre and jacobi polynomials occur as the polynomial part of the eigenfunctions of certain schrodinger operators for calogero sutherland type quantum systems for the generalized hermite and laguerre polynomials the multidimensional analogues of many classical results regarding generating functions differentiation and integration formulas recurrence relations and summation theorems are obtained we use this and related theory to evaluate the global limit of the ground state density obtaining in the hermite case the wigner semi circle law and to give an explicit solution for an initial value problem in the hermite and laguerre case

reduction to real infinitesimal character in affine hecke algebras the main result of this paper is to show that the problem of the determination of the unramified unitary dual of a split p adic group is equivalent to the problem of determining the unitary dual of the corresponding graded hecke algebra in bm the authors established this equivalence in the case of iwahori spherical representations under a certain restriction namely it was essential for the infinitesimal character to be real in the terminology of bmj in terms of the langlands deligne lusztig parameters s u p kul the restriction is that s e lg be a purely hyperbolic element the technique used in bm was to combine the notion of the signature of a k character in v with some facts which follow from kl namely that the 2w characters of tempered representations are linearly independent this is essentially true precisely when the infinitesimal character is real for if not s has an elliptic part se such that

unipotent representations of complex semisimple groups algebraic group over r or c in this paper we restrict attention to c we generalize arthur s definition slightly or perhaps simply make it more precise all of the resulting representations except for a finite set are then unitarily induced from representations of the same kind on proper parabolic subgroups we call the finite set remaining special unipotent representations a precise definition will be given later definition 1 17 our main result theorem iii of this introduction is a character formula for any special unipotent representation of course such a formula can be deduced from the kazhdan lusztig conjecture cf v3 the advantages of theorem iii are that it is in closed form and that it lends itself to verification of some conjectures of arthur in a so let g be a connected complex semisimple lie group and q its lie algebra choose

kazhdan lusztig algorithms for nonlinear groups and applications to kazhdan patterson lifting we establish an algorithm to compute characters of irreducible harish chandra modules for a large class of nonalgebraic lie groups roughly speaking the class of groups consists of those obtained as nonlinear double covers of linear groups in harish chandra s class we then apply this theory to study a particular group the universal cover of the real general linear group and discover a symmetry of the character computations encoded in a character multiplicity duality using this duality theory we reinterpret a kind of representation theoretic shimura correspondence for the general linear group geometrically and find that it is dual to an analogous lifting for indefinite unitary groups it seems likely that this example is illustrative of a general framework for studying similar correspondences

differential difference operators associated to reflection groups there is a theory of spherical harmonics for measures invariant under a finite reflection group the measures are products of powers of linear functions whose zero sets are the mirrors of the reflections in the group times the rotation invariant measure on the unit sphere in rn a commutative set of differential difference operators each homogeneous of degree 1 is the analogue of the set of first order partial derivatives in the ordinary theory of spherical harmonics in the case of r2 and dihedral groups there are analogues of the cauchy riemann equations which apply to gegenbauer and jacobi polynomial expansions the analysis of orthogonality structures for polynomials in several variables is a problem of vast dimensions this paper is part of an ongoing program to establish a workable theory for one particular class the underlying structure is based on finite coxeter groups these are finite groups acting on euclidean space generated by reflections in the zero sets of a collection of linear functions the roots the weight functions for the orthogonality are products of powers of these linear functions restricted to the surface of the unit sphere in addition the weight function is required to be invariant under the action of the group the resulting theory has strong similarities to the theory of spherical harmonics this was established in previous papers of the author 3 4 5 most notably a homogeneous polynomial is orthogonal to all polynomials of lower degree if and only if it is annihilated by a certain second order differential difference operator ordinary partial differentiation acts as an endomorphism on ordinary harmonic functions the use of such operators and their adjoints leads to recurrence formulas and orthogonal decompositions for harmonic polynomials in this paper we construct a commutative set of first order differentialdifference operators associated to the second order operator previously mentioned received by the editors june 1 1987 1980 mathematics subject classification 1985 revision primary 33a45 33a65 20h15 secondary 20c30 42c10 51f15

integrable connections related to zonal spherical functions we define a system of differential equations of first order for a function valued in the group algebra of the weyl group associated with an arbitrary root system this is equivalent to the system of differential equations given by heckman and opdam which is a deformation of the system satisfied by the zonal spherical function of the riemannian symmetric spaceg k of non compact type when the root system is a n type our equation is related to the knizhnik zamolodchikov equation in conformal field theory

generalized hermite polynomials and the heat equation for dunkl operators based on the theory of dunkl operators this paper presents a general concept of multivariable hermite polynomials and hermite functions which are associated with finite reflection groups on n the definition and properties of these generalized hermite systems extend naturally those of their classical counterparts partial derivatives and the usual exponential kernel are here replaced by dunkl operators and the generalized exponential kernel k of the dunkl transform in the case of the symmetric group s n our setting includes the polynomial eigenfunctions of certain calogero sutherland type operators the second part of this paper is devoted to the heat equation associated with dunkl s laplacian as in the classical case the corresponding cauchy problem is governed by a positive one parameter semigroup this is assured by a maximum principle for the generalized laplacian the explicit solution to the cauchy problem involves again the kernel k which is on the way proven to be nonnegative for real arguments

the importance of the selberg integral it has been remarked that a fair measure of the impact of atle selberg s work is the number of mathematical terms that bear his name one of these is the selberg integral an n dimensional generalization of the euler beta integral we trace its sudden rise to prominence initiated by a question to selberg from enrico bombieri more than thirty years after its initial publication in quick succession the selberg integral was used to prove an outstanding conjecture in random matrix theory and cases of the macdonald conjectures it further initiated the study of q analogues which in turn enriched the macdonald conjectures we review these developments and proceed to exhibit the sustained prominence of the selberg integral as evidenced by its central role in random matrix theory calogero sutherland quantum many body systems knizhnik zamolodchikov equations and multivariable orthogonal polynomial theory

an external approach to unitary representations the principal ideas of harmonic analysis on a locally compact group g which is not necessarily compact or commutative were developed in the 1940 s and early 1950 s in this theory the role of the classical fundamental harmonics is played by the irreducible unitary representations of g the set of all equivalence classes of such representations is denoted by and is called the dual object of g or the unitary dual of g since the 1940 s an intensive study of the foundations of harmonic analysis on complex and real reductive groups has been in progress for a definition of reductive groups the reader may consult the appendix at the end of the second section the motivation for this development came from mathematical physics differential equations differential geometry number theory etc through the 1960 s progress in the direction of the plancherel formula for real reductive groups was great due mainly to harish chandra s monumental work while at the same time the unitary duals of only a few groups had been parametrized with f mautner s work ma a study of harmonic analysis on reductive groups over other locally compact non discrete fields was started we shall first describe such fields in the sequel a locally compact non discrete field will be called a local field if we have a non discrete absolute value on the field q of rational numbers then it is equivalent either to the standard absolute value and the completion is the field r of real numbers or to a p adic absolute value for some prime number p for r 2 q write r pa b where a and b are integers and neither a nor b are divisible by p then the p adic absolute value of r is jrjp p a completion of q with respect to the p adic absolute value is denoted by qp it is called a field of p adic numbers each finite dimensional extension f of qp has a natural topology of a vector space over qp with this topology f becomes a local field the topology of f can be also introduced with an absolute value which is denoted by j jf in the fifth section we shall fix a natural absolute value the fields of real and complex numbers together with the finite extensions of p adic numbers exhaust all local fields of characteristic zero up to isomorphisms we let f be a finite field denote by f x the field of formal power series over f elements of this field are series of the form f p n k anx an 2 f for some integer k fix q 1

a recursion and a combinatorial formula for jack polynomials the jack polynomials j x form a remarkable class of symmetric polynomials they are indexed by a partition and depend on a parameter one of their properties is that several classical families of symmetric functions can be obtained by specializing e g the monomial symmetric functions m the elementary functions e 0 the schur functions s 1 and nally the two classes of zonal polynomials 2 1 2 the jack polynomials can be de ned in various ways e g a as an orthogonal family of functions which is compatible with the canonical ltration of the ring of symmetric functions or b as simultaneous eigenfunctions of certain di erential operators the sekiguchi debiard operators recently opdam o constructed a similar family f x of nonsymmetric polynomials the index runs now through all compositions nn they are de ned in a completely similar fashion e g the sekiguchi debiard operators are being replaced by the cherednik di erential re ection operators see sect 3 it is becoming more and more clear that these polynomials are as important as their symmetric counterparts the purpose of this paper is to add to the existing characterizations of jack polynomials two further ones c a recursion formula among the f together with two formulas to obtain j from them d combinatorial formulas of both j and f in terms of certain generalized tableaux

orthogonal polynomials on the sphere with octahedral symmetry for any finite reflection group g acting on rn there is a family of g invariant measures h2 dw where h is a certain product of linear functions whose zero sets are the reflecting hyperplanes for g on the unit sphere and an associated partial differential operator lh f a fh fa h a is the laplacian analogously to spherical harmonics there is an orthogonal with respect to h2 do decomposition of homogeneous polynomials that is if p is of degree n then i n 21 p x j ix12jpn 2 x 0 where lh p 0 and deg p i for each i but with the restriction that p and p must all be g invariant the main topic is the hyperoctahedral group with h x x ix2 i n u x xj i j the special case n 2 leads to jacobi polynomials a detailed study of the case n 3 is made an important result is the construction of a third order differential operator that maps polynomials associated to h with indices a i to those associated with a 2 fi 1 introduction regular figures have long been objects of fascination we study here the interplay between them and the theory of orthogonal polynomials this will also touch on finite groups generated by reflections and their polynomial invariants as well as a generalization of spherical harmonics the underlying situation throughout will consist of spaces of homogeneous polynomials on rn that satisfy a second order differential equation and whose restrictions to the unit sphere have an orthogonality relation with respect to a weight function and of a finite reflection group for which all these objects are invariant the weight measure h2 x dw x on the unit sphere s in r3 where h x xlx2x3 a x 23 3 a 2 dw is surface area and the associated polynomials will be studied most intensely some of the highlights of the theory are as follows define the differential operator lhf f fah received by the editors april 19 1982 and in revised form december 15 1982 1980 mathematics subject classification primary 33a45 33a65 secondary 20h15 this research was partially supported by nsf grant mcs 81 02581 d 984 american mathematical society 0002 9947 84 1 00 25 per page

biorthogonal polynomials associated with reflection groups and a formula of macdonald abstract dunkl operators are differential difference operators on r n which generalize partial derivatives they lead to generalizations of laplace operators fourier transforms heat semigroups hermite polynomials and so on in this paper we introduce two systems of biorthogonal polynomials with respect to dunkl s gaussian distributions in a canonical way these systems called appell systems admit many properties known from classical hermite polynomials and turn out to be useful for the analysis of dunkl s gaussian distributions in particular these polynomials lead to a new proof of a generalized formula of macdonald due to dunkl the ideas for this paper are taken from recent works on non gaussian white noise analysis and from the umbral calculus

the multiplicities of certain k types in spherical representations abstract let g be a connected semisimple lie group with finite center g 0 its lie algebra g 0 k 0 p 0 a cartan decomposition of g 0 and let k be the corresponding maximal compact subgroup we denote the complexification of p 0 by p for an irreducible spherical representation x of g we compute dim hom k p x the result is given explicitly in terms of the langlands parameters of x