shape description and matching using integral invariants on eccentricity transformed images matching occluded and noisy shapes is a problem frequently encountered in medical image analysis and more generally in computer vision to keep track of changes inside the breast for example it is important for a computer aided detection system to establish correspondences between regions of interest shape transformations computed both with integral invariants ii and with geodesic distance yield signatures that are invariant to isometric deformations such as bending and articulations integral invariants describe the boundaries of planar shapes however they provide no information about where a particular feature lies on the boundary with regard to the overall shape structure conversely eccentricity transforms ecc can match shapes by signatures of geodesic distance histograms based on information from inside the shape but they ignore the boundary information we describe a method that combines the boundary signature of a shape obtained from ii and structural information from the ecc to yield results that improve on them separately

integral geometry on trees in the hyperbolic disc or more generally in real hyperbolic spaces we consider the horocyclic radon transform r and the geodesic radon transform x composition with their respective adjoint operators yields two convolution operators on the disc with respect to the hyperbolic measure we describe their convolution kernels in comparison with those of the corresponding operators on a homogeneous tree

the averaging method and ergodic theory for pseudo differential operators on compact hyperbolic surfaces let x g be a riemannian manifold with laplacian and let a be a bounded do on l2 x the average a of a is defined as m e it aeit where mf limt 12t ttf t dt weak topology formally the principal symbol of a is m a gt where gt is the geodesic flow when gt is periodic or with some modifications completely integrable a differs from m op a gt op m a gt by a compact operator we take up the opposite extreme where gt is ergodic when x g is a compact hyperbolic surface we may use representation theory of psl2 r to directly compare these averages we find that m op a gt op m a gt op a ais the space average a constant however auk uk m op a gt uk uk k where uk is an onb of laplace eigenfunctions the latter is known to be a at least along a generic sequence of uk s

homogeneous asymptotic limits of haar measures of semisimple linear groups and their lattices we show that haar measures of connected semisimple groups embedded via a representation into a matrix space have a homogeneous asymptotic limit when viewed from far away and appropriately rescaled this is still true if the haar measure of the semisimple group is replaced by the haar measure of a irreducible lattice of the group and the asymptotic measure is the same in the case of an almost simple group of rank greater than 2 a remainder term is also obtained this extends and makes precise anterior results of duke rudnick and sarnak and eskin mcmullen in the case of a group variety

harmonic analysis as spectral theory of laplacians by writing the fourier inversion formula on euclidean space in polar coordinates we obtain f x 0 fgl x dx where fgl x 2fgl x which is the spectral theory of the laplacian how do properties of f relate to properties of the family of eigenfunctions fgl answers are provided for the following spaces l2 s s d in odd dimensions analogous results are obtained for harmonic analysis on hyperbolic space constant curvature semi riemannian spaces the heisenberg group and for differential forms on hyperbolic space for f l2 there is a plancherel formula f 22 limt o 1t bt z f x 2dx d where bt z denotes the ball of radius t about an arbitrary point z which is independent of the dimension and the identical formula holds in hyperbolic space if the parametrization of the eigenvalues is shifted for certain semi riemannian symmetric spaces we obtain a paley wiener theorem that explains the role of discrete series in producing functions of compact support and also involve certain non unitary representations that contain the discrete series representations for the heisenberg group the plancherel formula is of a different nature requiring that the eigenfunctions be almost periodic in one variable

the principal series for a reductive symmetric space ii eisenstein integrals in this paper we develop a theory of eisenstein integrals related to the principal series for a reductive symmetric space g h here g is a real reductive group of harish chandra s class an involution of g and h an open subgroup of the group g of xed points for the group g itself is a symmetric space for the left right action of g g we refer to this setting as the group case up to a normalization our eisenstein integrals generalize those of harish chandra 18 associated with a minimal parabolic subgroup in the group case

asymptotic expansions and boundary values of eigenfunctions on riemannian symmetric spaces soit x g k un espace de riemann symetrique de type non compacte a bord b k m et soit d x l algebre de tous les operateurs differentiels g invariants sur x on obtient des developpements asymptotiques pour une grande classe de fonctions jointives f sur x

filtration de certains espaces de fonctions sur un espace symetrique reductif we introduce a filtration of a g k module of some space of functions on a reductive symmetric space g h and compute the associated grading as a direct sum of induced representations as an application of this result to the reductive groups viewed as symmetric spaces we are able to realize any harish chandra module as a subquotient of a direct sum of induced representations from parabolic subgroups the inducing representations being trivial on the unipotent radical

band asymptotics in two dimensions let x be a rank one symmetric space e g s rp cp and let a be its laplace beltrami operator it is known that the spectrum of a is discrete and consists of a sequence of eigenvalues a k k a k 0 1 2 a 1 1 each occurring with multiplicity n b n l k o k 1 2 n being the dimension of x and a and b being positive integers which depend only on x let q be a zeroth order self adjoint pseudodifferential operator if we perturb a by adding q to it the eigenvalues 1 1 break up into bands

on kramer s sampling theorem associated with general strum liouville problems and lagrange interpolation kramer s sampling theorem which is a generalization of shannon s sampling theorem and its relationship with lagrange interpolation is studied kramer s sampling theorem is extended to the case where the interval a b is infinite and it is shown that any function that has a sampling expansion in the scope of kramer s theorem also has a lagrange type interpolation expansion provided that the kernel associated with kramer s theorem arises from a second order sturm liouville boundary value problem whether it is regular or singular this new technique not only reproduces many of the known sampling expansions such as those for the cosine the finite hankel and the continuous legendre transforms but also generates new ones such as those for the continuous jacobi laguerre and hermite transforms

a szego limit formula for conjugates of ψ do s by hecke operators let tg be a hecke operator and let op a be a do on l2 b further let be the projection onto span uj uj juj tguj j g uj j we will determine the szego limit measure g a for the operators tg op a tg xkd g a x lim 1 rank tr t g op a tg k comparison of g a with the szego limit measure of op tga shows that tg op a tg is quite remote spectrally from op tga

metric geometry in homogeneous spaces of the unitary group of a c algebra part i minimal curves abstract we study the metric geometry of homogeneous spaces p of the unitary group of a c algebra a modulo the unitary group of a c subalgebra b where the invariant finsler metric is induced by the quotient norm of a b the main results include the following in the von neumann algebra context for each tangent vector x at a point p there is a geodesic t 0 x which is obtained by the action on of a 1 parameter group in the unitary group of a this geodesic is minimizing up to length 2

intrinsic pseudo differential calculi on any compact lie group abstract in this paper we define in an intrinsic way operators on a compact lie group by means of symbols using the representations of the group the main purpose is to show that these operators form a symbolic pseudo differential calculus which coincides or generalises the local hormander pseudo differential calculus on the group viewed as a compact manifold

extended thermodynamics of fluids versus the revised enskog equation abstract the joint labours of specialists in mechanics and thermodynamics have suggested certain purely phenomenological hypotheses regarding the structure of the extended thermodynamics et of gases and fluids the microscopic counterparts to these propositions are however known for very few kinds of kinetic models of continuum systems boltzmann s equation while boltzmann s h theorem lies at the very basis of the kinetic origin of the et of dilute gases in particular in justifying the local constitutive assumption lca underlying the liu muller theory no such interrelation between macroscopic and microscopic approaches has been proved as yet for the revised enskog equation ree with the appropriate h theorem accordingly the purpose of this paper is to construct the et of dense gases and fluids within the framework of the ree and to investigate the nature of the resulting discrepancies with the existing phenomenological theories

integrals over grassmannians and random permutations testing the independence of two gaussian populations involves the distribution of the sample canonical correlation coefficients given that the actual correlation is zero the laplace transform as a function of x of this distribution is not only an integral over the grassmannian gr p f n of p dimensional planes in real complex or quaternion n space f n but is also related to a generalized hypergeometric function such integrals are solutions of painleve like equations in the complex case they are solutions to genuine painleve equations these integrals over gr p c n have remarkable expansions in x related to random words of length l formed with an alphabet of p letters 1 p the coefficients of these expansions are given by the probability that a word i contains a subsequence of letters p p 1 1 in that order and ii that the maximal length of the disjoint union of p 1 increasing subsequences of the word is less than or equal tok where k refers to the power of x note that if each letter appears in the word then the maximal length of the disjoint union of p increasing subsequences of the word is automatically l and is thus trivial c 2003 elsevier inc all rights reserved

fourier transforms on a semisimple symmetric space let g h be a semisimple symmetric space that is g is a connected semisimple real lie group with an involution and h is an open subgroup of the group of xed points for in g the main purpose of this paper is to study an explicit fourier transform on g h in terms of general representation theory the abstract fourier transform of a compactly supported smooth function f 2 c 1 c g h is given by see 5 1 f f zg h f x x dx for h a unitary irreducible representation of g and 2 h an h invariant distribution vector for here dx is the invariant measure on g h thus f isa smooth vector for h depending linearly on our goal is to obtain an explicit version of the restriction of this fourier transform to representations h in the minimal unitary principal series h for g h under the assumption that the center of g is nite in a sequel 9 to this paper it will be proved that a function f 2 c 1 c g h is uniquely determined by the restriction of f to this series a priori it is known that f is determined by f

irreducible tensor description i a classical gas a classical moderately rarefied simple monatomic gas is considered and it is supposed that its behavior is governed by boltzmann s equation then after a brief review of the fundamental properties of the symmetric traceless tensors the irreducible equations of transfer for the relative symmetric traceless moments of the one point distribution function f are systematically derived and related to those of johnston subsequently grad s expansion of the distribution function in terms of reducible three dimensional hermite polynomials which does not fit in together and with the equations of transfer just mentioned is rigorously transformed into its irreducible counterpart fashioned by mathematical apparatus such as one dimensional laguerre polynomials and ikenberry s tensorial harmonics finally some useful conversion formulas between the relative symmetric traceless moments and the tensorial laguerre ikenberry expansion coefficients of f are deduced and the irreducible variant of grad s moment t

fonctions d g h finies sur un espace symetrique reductif abstract it is well known that on r n every smooth function annihilated by a finite codimensional ideal in the algebra of constant coefficient differential operators is a linear combination of polynomial exponential functions p x e x hom r n c furthermore the polynomial functions are obtained by applying to the exponential functions e x some constant coefficient differential operator in the parameter we generalize this fact to the reductive symmetric spaces case the role of the exponential functions being taken by the eisenstein integrals

on the inversion of integral transforms associated with sturm liouville problems consider the sturm liouville boundary value problem 1 1 y q x y t2y a x b 2 2 y a cos y a sin 0 3 3 y b cos y b sin 0 where q x is continuous on a b let x t be a solution of either the initial value problem 1 and 2 or 1 and 3 in this paper we develop two techniques to invert the integral f t abf x x t dx where f x l2 a b one technique is based on the construction of some biorthogonal sequence of functions and the other is based on poisson s summation formula

wigner transform and pseudodifferential operators on symmetric spaces of non compact type we obtain a general expression for a wigner transform wigner function on symmetric spaces of non compact type and study the weyl calculus of pseudodifferential operators on them

poisson transforms on vector bundles let g be a connected real semisimple lie group with finite center and k a maximal compact subgroup of g let r v be an irreducible unitary representation of k and g x k tv the associated vector bundle in the algebra of invariant differential operators on g xk v the center of the universal enveloping algebra of lie g induces a certain commutative subalgebra z we are able to determine the characters of z given such a character we define a poisson transform from certain principal series representations to the corresponding space of joint eigensections we prove that for most of the characters this map is a bijection generalizing a famous conjecture by helgason which corresponds to r the trivial representation introduction let g be a connected real semisimple lie group with finite center and k a maximal compact subgroup of g then g k is a riemannian symmetric space of noncompact type we fix an iwasawa decomposition g kan let m be the centralizer of a in k let g and a be the lie algebras of g and a respectiarely and s 0 a the root system for 0 a let s 0 a be the positiare roots in s a for the ordering giaren by n let d g k be the algebra of inarariant differential operators on g k it is well known that the characters of d g k are parametrized by a e a the complex dual space of a let a g k denote the space of joint eigenfunctions corresponding to a we write g k g exp h g n g for each g e g according to g kan for each x e c k m we define pxx e c g k by p x g t k e p h g lk dk k here p is the half sum of s 0 a including multiplicities it turns out that pxq e v g k one can easily extend the definition of px to the space d k m resp a k m of distributions resp analytic functionals on k m in this paragraph we fix a e a such that 2 a ct ct ct is not in n 0 for each ct c s 0 a it is proared by helgason helg2 that pa defines a bijection from c o k m k finite onto e g k k finite he also proares in the rank one case p is a bijection from a k m onto 6 g k he then conjectured this should be true for the higher rank case the conjecture was earentually proared by six japanese mathematicians in kkmoot it should be mentioned that a representation theoretic proof by schmid starting from the kfinite result is indicated in sch lewis then a student of helgason made the following obseraration let 6 g k be the subspace of e g k whereeach element increases at most exporlentially see 02 for definition received by the editors september 28 1994 and in revised form january 30 1995 1991 mathematics subject classification primary 22e46 secondary 43a85 1998 american mathematical society 857 this content downloaded from 157 55 39 198 on fri 17 jun 2016 05 05 10 utc all use subject to http about jstor org terms

a finite continuous gegenbauer transform and its inverse ii the continuous gegenbauer transform is extended to noninteger values of the parameter 2 lambda the transformed function is no longer an entire function but is always holomorphic in a half plane the inverse transform is found for these parameter values its kernel is given both by a series and in closed form by an integral

comments on the grad procedure for the fokker planck equation if the kinetic equation of a macroscopic system is expanded with respect to the velocity in terms of orthogonal functions e g in terms of hermite functions one obtains an infinite hierarchy of equations for the expansion coefficients grad s method consists in truncating this hierarchy and investigating the remaining finite system in this paper we set up conditions under which this procedure is rigorously justified in case of the fokker planck equation

quantization of constrained systems and path integrals in curvilinear supercoordinates for systems with constraints the issue of non commutability of quantization and elimination of non physical variables is studied in the framework of path integrals pis it is shown that one should take into consideration the curvilinearity of physical variables and their phase space reduction in order to provide a one to one correspondence between the dirac scheme and pi description the latter leads to a modification of the standard pi pi with a gauge condition a general pi derivation is suggested for any method of picking out physical variables which corresponds to the dirac scheme

explicit hilbert spaces for certain unipotent representations ii letg be the universal cover of the group of automorphisms of a symmetric tube domain and letp ln be its shilov boundary parabolic subgroup this paper attaches an irreducible unitary representation ofg to each of the finitely many l orbits onn the hilbert space of the representation consists of functions on the orbit which are square integrable with respect to a certainl equivariant measure the representation remains irreducible when restricted top and descends to a quotient ofg which is at worst thedouble cover of a linear group if thel orbit isnot open inn the construction gives a unipotent representation ofg

subconvexity for half integral weight l functions we prove a subconvexity bound in the conductor aspect for the l function l s f chi where f is a half integer weight modular form this l function has analytic continuation and functional equation but no euler product due to the lack of an euler product one does not expect a riemann hypothesis for half integer weight modular forms nevertheless one may speculate a lindelof type hypothesis and this current subconvexity result is an indication towards its truth

asymptotic expansions and hua harmonic functions on bounded homogeneous domains let d be a homogeneous siegel domain of type ii we prove that every bounded hua harmonic function f on d is pluriharmonic the proof is based on asymptotic expansion of f

pseudo differential analysis on hyperbolic surfaces soit d le disque unite psu 1 1 un sous groupe discret et x d une surface hyperbolique on considere le comportement asymptotique des valeurs propres et des fonctions propres de sur x on presente un calcul d operateur pseudo differentiel pour obtenir ce resultat

random matrices beyond the cartan classification it is known that hermitean random matrix ensembles can be identified with symmetric coset spaces of lie groups or else with tangent spaces of the same this results in a classification of random matrix ensembles as well as applications in practical calculations of physical observables in this paper we show that a large number of non hermitean random matrix ensembles defined by physically motivated symmetries chiral symmetry time reversal invariance space rotation invariance particle hole symmetry or different reality conditions can likewise be identified with symmetric spaces we give explicit representations of the random matrix ensembles identified with lateral algebra subspaces and of the corresponding symmetric subalgebras spanning the group of invariance among the ensembles listed we identify as special cases all the hermitean ensembles identified with the cartan classes of symmetric spaces and the three ginibre ensembles with complex eigenvalues

patterson sullivan distributions in higher rank for a compact locally symmetric space x of non positive curvature we consider sequences of normalized joint eigenfunctions which belong to the principal spectrum of the algebra of invariant differential operators using an h pseudo differential calculus on x we define and study lifted quantum limits as weak limit points of wigner distributions the helgason boundary values of the eigenfunctions allow us to construct patterson sullivan distributions on the space of weyl chambers these distributions are asymptotic to lifted quantum limits and satisfy additional invariance properties which makes them useful in the context of quantum ergodicity our results generalize results for compact hyperbolic surfaces obtained by anantharaman and zelditch

van den ban schlichtkrull wallach asymptotic expansions on nonsymmetric domains if it is annihilated by every element of dg x the algebra of all g invariant differential operators without constant term one of the most beautiful results in the harmonic analysis of symmetric spaces is the helgason conjecture which states that on a riemannian symmetric space of noncompact type a function is harmonic if and only if it is the poisson integral of a hyperfunction over the furstenberg boundary g po where po is a minimal parabolic subgroup see 14 17 one of the more remarkable aspects of this theorem is its generality one obtains a complete description of all solutions to the system of invariant differential operators on x without imposing any boundary or growth conditions if x is a hermitian symmetric space then one is typically interested in complex function theory in which case one is interested in functions whose boundary values are supported on the shilov boundary rather than the furstenberg boundary the shilov boundary is g p where p is a certain maximal parabolic containing po in this case it turns out that the algebra of g invariant differential operators is not necessarily the most appropriate one for

semiclassical limits for the hyperbolic plane we study the concentration properties of high energy eigenfunctions for the laplace beltrami operator of the hyperbolic plane with special consideration of automorphic eigenfunctions at the center of our investigation is the microlocal lift of an eigenfunction to sl 2 introduced by s zelditch the microlocal lift is based on s helgason s fourier transform and has a straightforward description in terms of the lie algebra sl 2 we begin with an elementary demonstration of zelditch s exact differential equation for the microlocal lift further for a sequence of suitably bounded eigenfunctions with eigenvalues tending to infinity we show that the limit of microlocal lifts is a geodesic flow invariant positive measure our main consideration is the microlocal lifts of the elementary eigenfunctions constructed from the macdonald bessel functions we find that with a scaling of auxiliary parameters the corresponding high energy limit converges to the positive dirac measure for the lift to sl 2 of a single geodesic on the upper half plane in particular at high energy the microlocal lift of a macdonald bessel function is concentrated along the lift of a single geodesic we further find that the convergence is uniform in parameters and thus our considerations can be applied to study automorphic eigenfunctions in particular the maass cusp forms for certain cofinite subgroups of sl 2 a formula is presented for the microlocal lift of a cusp form as an integral of a family of dirac measures and a measure given directly by sums of products of the classical fourier coefficients of the cusp form the formula establishes that questions on quantum chaos for automorphic eigenfunctions are equivalent to classically stated questions on twisted sums of coefficients in particular we find that the uniform distribution for the microlocal limit is equivalent to a uniform distribution for the limit of twisted coefficient sums we extend the considerations to the case of the nonholomorphic eisenstein series for the modular group we find that a result of w luo and p sarnak is equivalent to a limit sum formula involving the riemann zeta function and the elementary divisor function the formula is suggestive of a formula of s ramanujan

intertwining the geodesic flow and the schrodinger group on hyperbolic surfaces we construct an explicit intertwining operator mathcal l between the schrodinger group e it frac triangle2 and the geodesic flow on certain hilbert spaces of symbols on the cotangent bundle t x of a compact hyperbolic surface x d we also define invariant eigendistributions of the geodesic flow ps j k nu j nu k patterson sullivan distributions out of pairs of triangle eigenfunctions generalizing the diagonal case j k treated in anantharaman and zelditch ann henri poincare 8 2 361 426 2007 the operator mathcal l maps ps j k nu j nu k to the wigner distribution w gamma j k studied in quantum chaos we define hilbert spaces mathcal h ps whose dual is spanned by ps j k nu j nu k resp mathcal h w whose dual is spanned by w gamma j k and show that mathcal l is a unitary isomorphism from mathcal h w to mathcal h ps

meromorphic extension of the spherical functions on a class of ordered symmetric spaces we discuss a conjecture of olafsson and pasquale published in j funct anal 181 2001 346 this conjecture gives the bernstein sato polynomial associated with the poisson kernel of the ordered or non compactly causal symmetric spaces the bernstein sato polynomials allow to locate the singularities of the spherical functions on the considered spaces we prove that this conjecture does not hold in general and propose a slight improvement of it finally we prove that the new conjecture holds for a class of ordered symmetric spaces called both the makarevic spaces of type i and the satellite cones

geometric realization of discrete series for semisimple symmetric spaces let g be a connected semisimple lie group a an involution of g and h the connected component of the fixed point set of a containing the identity then g h is a semisimple symmetric space we assume g is a real form of a complex lie group and rank g h rank k k c h where k is a a stable maximal compact subgroup of g flensted jensen using his duality principle and poisson transform constructed countably many discrete series in u g h f j i then using deep results of hyperfunction theory and microlocal analysis oshima and matsuki gave a detailed characterization of these discrete series including a proof of flensted jensen s conjecture c 0 om based on these results we give the discrete series a geometric interpretation in locally symmetric spaces f c where g k and f cg is a cocompact discrete group such that f c h c h is cocompact our geometric interpretation is the following first of all to a flensted jensen function which is the generating function of a discrete series v we associate a harmonic form cb z on g with values in a finite dimensional g module v right g translates of coefficients of 5z realizes the contragredient t of r and tba is left invariant under h and under right k action it transforms according to the minimal k type with highest weight of 3 is the pull back of a v valued from coz on and v determines a locally constant bundle e on f so that if convergence holds

the plancherel decomposition for a reductive symmetric space i spherical functions we prove the plancherel formula for spherical schwartz functions on a reductive symmetric space our starting point is an inversion formula for spherical smooth compactly supported functions the latter formula was earlier obtained from the most continuous part of the plancherel formula by means of a residue calculus in the course of the present paper we also obtain new proofs of the uniform tempered estimates for normalized eisenstein integrals and of the maass selberg relations satisfied by the associated c functions

local boundary data of eigenfunctions on a riemannian symmetric space let x g k be a riemannian symmetric space of non compact type and let d x be the algebra of invariant differential operators on x in a previous paper 1 we developed a theory of asymptotic expansions for joint eigenfunctions of d x of at most exponential growth in the present paper we show that local asymptotic data determine the eigenfunctions completely we also develop a theory of asymptotic expansions along walls both results are of importance for the theory of the discrete series of a semisimple symmetric space let a be a maximal abelian split subspace of the lie algebra 9 of g and denote by a its complexified dual let e be the restricted root system of a in g and w the associated weyl group then harish chandra s isomorphism d x s a w determines a bijection 2 z from a w to the set d x of algebra homomorphisms d x c given 2ca let ga x denote the joint eigenspace of functions f e c o x satisfying

rank of quantized universal enveloping algebras and modular functions we compute an intrinsic rank invariant for quasitriangular hopf algebras in the case of general quantum groupsu q g as a function ofq the rank has remarkable number theoretic properties connected with modular covariance and galois theory a number of examples are treated in detail including rank u q su 3 and rank u q e 8 we briefly indicate a physical interpretation as relating chern simons theory with the theory of a quantum particle confined to an alcove ofg

entropy and the localization of eigenfunctions we study the large eigenvalue limit for the eigenfunctions of the laplacian on a compact manifold of negative curvature in fact we only assume that the geodesic flow has the anosov property in the semi classical limit we prove that the wigner measures associated to eigenfunctions have positive metric entropy in particular they cannot concentrate entirely on closed geodesies 1 introduction statement of results we consider a compact riemannian manifold m of dimension d 2 and assume that the geodesic flow ter acting on the unit tangent bundle of m has a chaotic behaviour this refers to the asymptotic properties of the flow when time t tends to infinity ergodicity mixing hyperbolicity we assume here that the geodesic flow has the anosov property the main example being the case of negatively curved manifolds the words quantum chaos express the intuitive idea that the chaotic features of the geodesic flow should imply certain special features for the corresponding quantum dynamical system that is according to schrodinger the unitary flow exp iht t r acting on the hilbert space l2 m where a stands for the laplacian on m and h is proportional to the planck constant recall that the quantum flow converges in a sense to the classical flow g in the so called semi classical limit h 0 one can imagine that for small values ofh the quantum system will inherit certain qualitative properties of the classical flow one expects for instance a very different behaviour of eigenfunctions of the laplacian or the distribution of its eigenvalues if the geodesic flow is anosov or in the other extreme completely integrable see sa95 the convergence of the quantum flow to the classical flow is stated in the egorov theorem consider one of the usual quantization procedures op which associates an operator oph a acting on l2 m to every smooth compactly supported function a e c t m on the cotangent bundle t m according to the egorov theorem we have for any fixed t

the plancherel decomposition for a reductive symmetric space ii representation theory we obtain the plancherel decomposition for a reductive symmetric space in the sense of representation theory our starting point is the plancherel formula for spherical schwartz functions obtained in part i the formula for schwartz functions involves eisenstein integrals obtained by a residual calculus in the present paper we identify these integrals as matrix coefficients of the generalized principal series

solution to the nonlinear boltzmann equation for maxwell models for nonisotropic initial conditions the known solution to the spatially homogeneous nonlinear boltzmann equation for maxwell models in a series of laguerre polynomials is extended to include nonisotropic initial conditions existence proofs for a class of solutions are supplied the equations for the generalized nonisotropic laguerre moments are derived in explicit form for two and three dimensional models further it is shown that the ordinary moments satisfy the same set of equations as the hermite polynomial moments

holomorphic l p type for sub laplacians on connected lie groups we study the problem of determining all connected lie groups g which have the following property hlp every sub laplacian l on g is of holomorphic lp type for 1 p p 2 first we show that semi simple non compact lie groups with finite center have this property by using holomorphic families of representations in the class one principal series of g and the kunze stein phenomenon we then apply an lp transference principle essentially due to anker to show that every connected lie group g whose semi simple quotient by its radical is non compact has property hlp for the convenience of the reader we give a self contained proof of this transference principle which generalizes the well known coifman weiss principle one is thus reduced to studying those groups for which the semi simple quotient is compact i e to compact extensions of solvable lie groups in this article we consider semi direct extensions of exponential solvable lie groups by connected compact lie groups it had been proved in joint work by w hebisch and the two first named authors that every exponential solvable lie group s which has a non regular co adjoint orbit whose restriction to the nilradical is closed has property hlp and we show here that hlp remains valid for compact extensions of these groups

classical and quantum resonances for hyperbolic surfaces for compact and for convex co compact oriented hyperbolic surfaces we prove an explicit correspondence between classical ruelle resonant states and quantum resonant states except at negative integers where the correspondence involves holomorphic sections of line bundles

on the fundamental solution for higher spin dirac operators abstract in this paper we will determine the fundamental solution for the higher spin dirac operator q which is a generalisation of the classical rarita schwinger operator to more complicated irreducible half integer representations for the spin group in m dimensions this will allow us to generalise the stokes theorem the cauchy pompeiu theorem and the cauchy integral formula which lie at the very heart of the function theory behind arbitrary elliptic higher spin operators

unitary representations of lie groups with reflection symmetry we consider the following class of unitary representations of some real lie groupgwhich has a matched pair of symmetries described as follows i supposeghas a period 2 automorphism and that the hilbert spaceh carries a unitary operatorjsuch thatj j i e selfsimilarity ii an added symmetry is implied ifh further contains a closed subspacek0having a certainorder covarianceproperty and satisfying thek0 restricted positivity v jv 0 v k0 where is the inner product inh from i ii we get an induced dual representation of an associated dual groupgc all three properties selfsimilarity order covariance and positivity are satisfied in a natural context whengis semisimple and hermitean but whengis the ax b group or the heisenberg group positivity is incompatible with the other two axioms for the infinite dimensional irreducible representations we describe a class ofg containing the latter two which admits a classification of the possible spacesk0 h satisfying the axioms of selfsimilarity and order covariance

the modulus of continuity for γ0 m semi classical limits we study the behavior of a large eigenvalue limit of eigenfunctions for the hyperbolic laplacian for the modular quotient sl 2 fejer summation and results of s zelditch are used to show that the microlocal lifts of eigenfunctions have large eigenvalue limit a geodesic flow invariant measure for the modular unit cotangent bundle the limit is studied for hecke maass forms joint eigenfunctions of the hecke operators and the hyperbolic laplacian the first modulus of continuity result is presented for the limit the singular concentration set of the limit cannot be a compact union of closed geodesics and measured geodesic laminations

the oshima sekiguchi and liouville theorems on heintze groups abstract let l be an elliptic operator on a riemannian manifold m a function f annihilated by l is said to be l harmonic f is said to have moderate growth if and only if f grows at most exponentially in the riemannian distance if m is a rank one symmetric space and l is the laplace beltrami operator for m the oshima sekiguchi theorem t oshima j sekiguchi eigenspaces of invariant differential operators on an affine symmetric space invent math 57 1980 1 81 states that a l harmonic function f has moderate growth if and only if f is the poisson integral of a distribution on the furstenberg boundary in this work we prove that this result generalizes to a very large class of homogeneous riemannian manifolds of negative curvature we also i prove a liouville type theorem that characterizes the polynomial like harmonic functions which vanish on the boundary in terms of their growth properties ii describe all polynomial like harmonic functions and iii give asymptotic expansions for the poisson kernel one consequence of this work is that every schwartz distribution on the boundary is the boundary value for a l harmonic function f which is uniquely determined modulo polynomial like harmonic functions

plongements de series discretes pour un espace symetrique reductif resume using the realization of a discrete series a of a semi simple symmetric space g h as a space of boundary values in the dual space gd kd we construct an explicit embedding of a in a principal series ind 1 associated to an open subgroup q1 of a minimal stable parabolic subgoup q of g the construction of q1 and uses the properties of what we call a fundamental system of roots

irreducible tensor description iii thermodynamics of a low temperature phonon gas let one assume that the interacting phonon gas whose behavior is governed by the boltzmann peierls equation inhabits an insulating crystal at sufficiently low temperature then within the framework of a single acoustic phonon branch and of an isotropic long wavelength approximation to the dispersion relation the simplest acceptable version of extended irreversible thermodynamics based upon the nine moment system of differential equations for the slow and fast gas state variables is carefully investigated it is clearly demonstrated that in virtue of the structure simplifications just mentioned the conceptually different macroscopic kinetic and variational procedures which are discussed in this paper appear complementary to each other finally with the help of a suitable contraction of the nine moment system of field equations for callaway s relaxation model a slightly generalized nonlinear variant of ordinary low temperature phonon hydrodynamics is explicitly derived

vecteurs distributions h invariants pour les series principales generalisees d espaces symetriques reductifs et prolongement meromorphe d integrales d eisenstein

uniform distribution of eigenfunctions on compact hyperbolic surfaces on demontre que les fonctions propres k du laplacien sur une surface hyperbolique compacte x devient uniformement distribuee sur x quand k

scattering theory for the laplacian on symmetric spaces of noncompact type and its application to a conjecture of strichartz abstract in this paper we develop the scattering theory for the laplacian on symmetric spaces of noncompact type we study the asymptotic properties of the resolvent in the framework of the agmon hormander space our approach is based on a detailed analysis of the helgason fourier transform and generalized spherical functions on symmetric spaces of noncompact type as an application of our scattering theory we prove a conjecture by strichartz concerning a characterization of a family of generalized eigenfunctions of the laplacian

support d asymptotiques et croissance des fonctions propres sur un espace symetrique semi simple abstract we associate several distribution boundary values to an eigenfunction with moderate growth on a riemannian symmetric space g k the associated character of the algebra d g k of invariant differential operators is allowed to be non regular we prove results on the support of these boundary values these allow us to recover the theorems of matsuki oshima and oshima on the equivalence between growth of an eigenfunction and limitations on the supports of its boundary values our approach is based on an asymptotic analysis that makes no use of hyperfunction theory

laplace beltrami operator on a riemann surface and equidistribution of measures we consider the laplace beltrami operator on a compact riemann surface of a constant negative curvature for any eigenvalue of the laplace beltrami operator there is an associated sequence of measures on the riemann surface these measures naturally appear in quantum chaos type questions in the theory of electro magnetic flow on a riemann surface the main result of the paper is the claim that this sequence of measures has the liouville measure as the weak limit we prove a quantitative version of this equidistribution claim

a characterization of the l2 range of the poisson transform related to strichartz conjecture on symmetric spaces of noncompact type abstract the main purpose of this paper is to solve strichartz conjecture concerning an image characterization of the poisson transform in the l 2 theory on symmetric spaces of noncompact type we prove that the poisson transform provides an isomorphism between the l 2 space on the boundary and a certain weighted l 2 space consisting of joint eigenfunctions on symmetric spaces of noncompact type our approach is based on the scattering theory for the schrodinger operator moreover we give a fourier restriction estimate and an asymptotic formula for the poisson transform