the capelli identity tube domains and the generalized laplace transform the key result in this paper is a proof of a large class of identities generalizing the capelli identity cl the capelli identity is a centerpiece of 19th century invariant theory it asserts the equality of two differential operators on the n2 dimensional space m n r of n x n real matrices let e denote the ijth elementary matrix let xii denote the linear functions on m n r dual to e and write 3 for the partial differential operator a ax then this identity says

shimura invariant differential operators and their eigenvalues let bar d be the covariant cauchy riemann operator and mathcal d the covariant holomorphic differential operator on a line bundle over a hermitian symmetric space g k we study the shimura invariant differential operators defined via bar d and mathcal d we find the eigenvalues of a family of the shimura operators and of the generators

restriction to symmetric subgroups of unitary representations of rank one semisimple lie groups we consider the spherical complementary series of rank one lie groups h n so0 n 1 f for f r c h we prove that there exist finitely many discrete components in its restriction under the subgroup hn 1 so0 n 1 1 f this is proved by imbedding the complementary series into analytic continuation of holomorphic discrete series of g n su n 1 su n 1 x su n 1 and su 2n 2 and by the branching of holomorphic representations under the corresponding subgroup g n 1

berezin transform on real bounded symmetric domains let d be a bounded symmetric domain in a complex vector space vc with a real form v and d d v g k be the real bounded symmetric domain in the real vector space v we construct the berezin kernel and consider the berezin transform on the l2 space on d the corresponding representation of g is then unitarily equivalent to the restriction to g of a scalar holomorphic discrete series of holomorphic functions on d and is also called the canonical representation we find the spectral symbol of the berezin transform under the irreducible decomposition of the l2 space introduction the main purpose of the present paper is to calculate the spectral symbol of the berezin transform on real bounded symmetric domains to explain our results and motivations we let d be the unit disk in the complex plane with the lebesgue measure dm z we consider the weighted bergman space h 1 of holomorphic functions on d square integrable with respect to the measure 1 z 2 2dm z it has up to some positive constant the reproducing kernel kw z k z w 1 zw moreover the group gc su 1 1 of fractional transformations of d acts on the space h via f z 7 f gz g z 2 and it forms an irreducible unitary projective representation consider the subgroup so 1 1 consisting of transformations of the form z 7 az b bz a with a b r and a b 1 thus it is of interest to study the irreducible decomposition of the weighted bergman space under the subgroup g so 1 1 for that purpose we consider the unit interval d d r 1 1 as a trivial symmetric space g k so 1 1 1 and the restriction of holomorphic functions in h to the interval d more precisely consider r h c d rf x f x 1 x 2 x d let l d d 0 be the l space on d with the so 1 1 invariant measure d 0 x dx 1 x2 whose decomposition under so 1 1 can be done via the mellin transform see below the restriction r is a bounded operator from h into the space l d d 0 with dense image and intertwines the respective actions of so 1 1 received by the editors january 16 2000 and in revised form october 10 2000 2000 mathematics subject classification primary 22e46 43a85 32m15 53c35

berezin transform on the quantum unit ball we introduce and study in the framework of a theory of quantum cartan domains a q analog of the berezin transform on the unit ball we construct q analogs of weighted bergman spaces toeplitz operators and covariant symbol calculus in studying the analytical properties of the berezin transform we introduce also the q analog of the su n 1 invariant laplace operator the laplace beltrami operator and present related results on harmonic analysis on the quantum ball these are applied to obtain an analog of one result by a unterberger and h upmeier an explicit asymptotic formula expressing the q berezin transform via the q laplace beltrami operator is also derived at the end of the article we give an application of our results to basic hypergeometric q orthogonal polynomials

range characterization of the cosine transform on higher grassmannians abstract we characterize the range of the cosine transform on real grassmannians in terms of the decomposition under the action of the special orthogonal group so n we also give a geometric interpretation of this image in terms of valuations in addition we discuss the non archimedean analogues

certain hypergeometric series related to the root system bc we show that the generalized hypergeometric function 2 f 1 of matrix argument is the series expansion at the origin of a special case of the hypergeometric function associated with the root system of type bc in addition we prove that the jacobi polynomials of matrix argument correspond to the jacobi polynomials associated with the root system of type bc we also give a precise relation between jack polynomials and the jacobi polynomials associated with the root system of type a as a side result one obtains generalized hook length formulas which are related to harish chandra s c function and one can prove a conjecture due to macdonald relating two inner products on a space of symmetric functions

generalized goal programming polynomial methods and applications in this paper we address a general goal programming problem with linear objectives convex constraints and an arbitrary componentwise nondecreasing norm to aggregate deviations with respect to targets in particular classical linear goal programming problems as well as several models in location and regression analysis are modeled within this framework

asymptotic behaviour of reproducing kernels of weighted bergman spaces let q be a domain in c f a nonnegative and g a positive function on q such that 1 g is locally bounded a2 the space of all holomorphic functions on q square integrable with respect to the measure f0gda where da is the 2n dimensional lebesgue measure and k x y the reproducing kernel for a2 it has been known for a long time that in some special situations such as on bounded symmetric domains q with g 1 and f the bergman kernel function the formula lim k x x 1 0 1 f x ae 00 holds true this fact even plays a crucial role in berezin s theory of quantization on curved phase spaces in this paper we discuss the validity of this formula in the general case the answer turns out to depend on loosely speaking how well the function log f can be approximated by certain pluriharmonic functions lying below it for instance holds if log f is convex and hence can be approximated from below by linear functions for any function g counterexamples are also given to show that in general may fail drastically or even be true for some x and fail for the remaining ones finally we also consider the question of convergence of k0 x y 1 0 for x y which leads to an unexpected result showing that the zeroes of the reproducing kernels are affected by the smoothness of f for instance if f is not real analytic at some point then kc x y must have zeroes for all at sufficiently large

intertwining operators for the generalized principal series on a symmetric r space three questions about the intertwining operators for the generalized principal series on a symmetric r space are solved description of the functional kernel both in the noncompact and in the compact picture domain of convergence meromorphic continuation a large use is made of the theory of positive jordan triple systems the meromorphic continuation of the intertwining integral is achieved via a bernstein sato identity and a precise description of the poles is obtained 0 2000 mathematics subject classification 22e45 43a80

intersection bodies and generalized cosine transforms the paper is focused on intimate connection between geometric properties of intersection bodies in convex geometry and generalized cosine transforms in harmonic analysis a new concept of intersection body that unifies some known classes of geometric objects is introduced a parallel between trace theorems in function theory restriction onto lower dimensional subspaces of the spherical radon transforms and the generalized cosine transforms and sections of intersection bodies is established new integral formulas for different classes of cosine transforms are obtained and examples of intersection bodies are given we also revisit some known facts in this area and give them new simple proofs

geometric norm equality related to the harmonicity of the poisson kernel for homogeneous siegel domains in this paper we present a geometric norm equality involving an admissible linear form for the shilov boundary of a homogeneous siegel domain d we prove that the validity of this norm equality is equivalent to the symmetry of d and the reduction of essentially to the koszul form this in particular reveals a geometric reason that the poisson kernel is annihilated by the laplace beltrami operator if and only if d is symmetric a theorem due to hua look koranyi and xu

the busemann petty problem via spherical harmonics abstract the busemann petty problem asks whether symmetric convex bodies in r n with smaller central hyperplane sections necessarily have smaller n dimensional volume the solution has recently been completed and the answer is affirmative if n 4 and negative if n 5 in this article we present a short proof of the affirmative result and its generalization using the funk hecke formula for spherical harmonics

bmo in the bergman metric on bounded symmetric domains for bounded symmetric domains in cn a notion of bounded mean oscillation in terms of the bergman metric is introduced it is shown that for in l2 dv is in bmo if and only if the densely defined operator m p m p pm on l2 dv is bounded here m is multiplication by and p is the bergman projection with range the bergman subspace h2 dv la2 dv of holomorphic functions in l2 dv an analogous characterization of compactness for m p is provided by functions of vanishing mean oscillation at the boundary of

berezin kernels of tube domains abstract we study the berezin kernel decomposition into positive definite spherical functions on real forms of tube domains in other words we obtain for all tube domains associated to causal pairs g h the decomposition of holomorphic discrete series representations of g when restricted to the fully restrictive subgroup h

the p cones in dimension n 3 are not homogeneous when p 2 abstract using the t algebra machinery we show that up to linear isomorphism the only strictly convex homogeneous cones in r n n 3 are the 2 cones also known as lorentz cones or second order cones in particular this shows that the p cones are not homogeneous when p 2 1 p and n 3 thus answering a problem proposed by gowda and trott

risk optimization with p order conic constraints a linear programming approach the paper considers solving of linear programming problems with p order conic constraints that are related to a certain class of stochastic optimization models with risk objective or constraints the proposed approach is based on construction of polyhedral approximations for p order cones and then invoking a benders decomposition scheme that allows for efficient solving of the approximating problems the conducted case study of portfolio optimization with p order conic constraints demonstrates that the developed computational techniques compare favorably against a number of benchmark methods including second order conic programming methods

non perturbative deformation quantization of cartan domains abstract we construct families of non commuting c algebras of quantized functions for bounded irreducible hermitian symmetric spaces for this procedure we use algebras of toeplitz operators defined with respect to a perturbation of the ordinary bergman metric we prove the deformation quantization conditions for these algebras

on p norm linear discrimination abstract we consider a p norm linear discrimination model that generalizes the model of bennett and mangasarian 1992 and reduces to a linear programming problem with p order cone constraints the proposed approach for handling linear programming problems with p order cone constraints is based on reformulation of p order cone optimization problems as second order cone programming socp problems when p is rational since such reformulations typically lead to socp problems with large numbers of second order cones an economical representation that minimizes the number of second order cones is proposed a case study illustrating the developed model on several popular data sets is conducted

cone free primal dual path following and potential reduction polynomial time interior point methods we present a framework for designing and analyzing primal dual interior point methods for convex optimization we assume that a self concordant barrier for the convex domain of interest and the legendre transformation of the barrier are both available to us we directly apply the theory and techniques of interior point methods to the given good formulation of the problem as is without a conic reformulation using the very usual primal central path concept and a less usual version of a dual path concept we show that many of the advantages of the primal dual interior point techniques are available to us in this framework and therefore they are not intrinsically tied to the conic reformulation and the logarithmic homogeneity of the underlying barrier function

inversion of fractional integrals related to the spherical radon transform explicit inversion formulas are obtained for the analytic family of fractional integrals t f x n sn xy 1 f y dyon the unit sphere in rn 1 arbitrary complex andn 2 are considered in the ease 0 the integralt fcoincides with the spherical radon transform for 1 1 3 5 such integrals are known as the blaschke levy representations and arise in convex geometry probability and the banach space theory for 1 3 5 the integralt fis defined by continuity as the spherical convolution with the power logarithmic kernel different inversion methods are discussed

on the degenerate principal series of complex symplectic groups we apply techniques introduced by clerc kobayashi orsted and pevzner to study the degenerate principal series of sp n c an explicit description of the k types is provided and knapp stein normalised operators are realised as symplectic fourier transforms and their k spectrum explicitly computed reducibility phenomena are analysed in terms of k types and eigenvalues of intertwining operators we also construct a new model for these representations in which knapp stein intertwiners take an algebraic form

on the meromorphic extension of the spherical functions on noncompactly causal symmetric spaces abstract we determine integral formulas for the meromorphic extension in the parameter of the spherical functions on a noncompactly causal symmetric space the main tool is bernstein s theorem on the meromorphic extension of complex powers of polynomials the regularity properties of are deduced in particular the possible poles of are located among the translates of the zeros of the bernstein polynomial the translation parameter depends only on the structure of the symmetric space the expression of the bernstein polynomial is conjectured the relation between the bernstein polynomial and the product formula of the c function is analyzed the conjecture is verified in the rank one case the explicit formulas obtained in this case yield a detailed description of singularities of in the general higher rank case the integral formulas are applied to find asymptotic estimates for the spherical functions in the appendix the spherical functions on noncompactly causal symmetric spaces are regarded as a special instance of harish chandra type expansions associated with roots systems with arbitrary multiplicities we study expansions obtained by taking averages over arbitrary parabolic subgroups of the weyl group of the root system the possible singularities are located in this general context

hua operators and poisson transform for bounded symmetric domains let be a bounded symmetric domain of non tube type in c n with rank r and s its shilov boundary we consider the poisson transform p s f z for a hyperfunction f on s defined by the poisson kernel p s z u h z z n r h z u n r 2 s z u s s c for all s satisfying certain non integral condition we find a necessary and sufficient condition for the functions in the image of the poisson transform in terms of hua operators when is the type i matrix domain in m n m c n less than or slanted equal to m we prove that an eigenvalue equation for the second order m n n valued hua operator characterizes the image 2006 elsevier inc all rights reserved

processes of flats induced by higher dimensional processes stationary processes of k flats in mathbb e d can be thought of as point processes on the grassmannian mathcal l kd of k dimensional subspaces of mathbb e d if such a process is sampled by a d k j dimensional space f it induces a process of j flats in f in this work we will investigate the possibility of determining the original k process from knowledge of the intensity measures of the induced j processes we will see that this is impossible precisely when 1 k d 1 and j 0 2 r 2 1 where r is the rank of the manifold mathcal l kd we will show how the problem is equivalent to the study of the kernel of various integral transforms these will then be investigated using harmonic analysis on grassmannian manifolds

branching laws for discrete wallach points we consider the projective representations of the group of holomorphic automorphisms of a symmetric tube domain v i that are obtained by analytic continuation of the holomorphic discrete series for a representation corresponding to a discrete point in the wallach set we find the decomposition under restriction to the identity component of gl using riesz distributions an explicit intertwining operator is constructed as an analytic continuation of an integral operator the density of the plancherel measure involves quotients of functions and the c function for a symmetric cone of smaller rank

computing the shortest network under a fixed topology we show that in any given uniform orientation metric plane the shortest network interconnecting a given set of points under a fixed topology can be computed by solving a linear programming problem whose size is bounded by a polynomial in the number of terminals and the number of legal orientations when the given topology is restricted to a steiner topology our result implies that the steiner minimum tree under a given steiner topology can be computed in polynomial time in any given uniform orientation metric with spl lambda legal orientations for any fixed integer spl lambda spl ges 2 this settles an open problem posed by brazil thomas and weng 2000

a semigroup associated to a linear control system on a lie group abstract let us consider a linear control system on a connected lie group g it is known that the accessibility set a from the identity e is in general not a semigroup in this article we associate a new algebraic object s to which turns out to be a semigroup allowing the use of the semigroup machinery to approach in particular we obtain some controllability results

degenerate principal series representations and their holomorphic extensions let x h l be an irreducible real bounded symmetric domain realized as a real form in an hermitian symmetric domain d g k the intersection s of the shilov boundary of d with x defines a distinguished subset of the topological boundary of x and is invariant under h it can be realized as s h p for certain parabolic subgroup p of h we study the spherical representations of h induced from p we find formulas for the spherical functions in terms of the macdonald hypergeometric function this generalizes the earlier result of faraut koranyi for hermitian symmetric spaces d we consider a class of h invariant integral intertwining operators from the representations on l2 s to the holomorphic representations of g restricted to h we construct a new class of complementary series for the groups h so n m su n m with n m 2 and sp n m with n m 1 we realize them as discrete components in the branching rule of the analytic continuation of the holomorphic discrete series of g su n m su n m su n m and su 2n 2m respectively

kinematic formulas on the quaternionic plane we introduce different bases for the vector space of sp 2 sp 1 invariant translation invariant continuous valuations on the quaternionic plane and determine a complete set of kinematic formulas

toeplitz c algebras on bounded symmetric domains in this paper we develop a structure theory for the toeplitz c algebra y for an arbitrary bounded symmetric domain d in c classical or exceptional as the main result it is shown that 5y is a c algebra of type i which is solvable i e admits a finite composition sequence of closed ideals sk in 7 such that the subquotients fk i k are continuous fields of elementary c algebras of compact operators the length of this composition sequence coincides with the rank r of d as a hermitian symmetric space for 0 k r the spectral components sk of ay parametrizing the irreducible representations of y are realized in terms of the boundary components complex analytic faces of d the structure theory for y is based on the fact that the tangent space z c c of d carries a jordan algebraic structure z is a jordan triple system with triple product uv w describing the geometry of d in algebraic terms for example d is the open unit ball with respect to the so called spectral norm on z and the boundary components of d are in 1 1 correspondence with the tripotents e e z partial isometries satisfying ee e 4 e in particular the shilov boundary s of d consists of all tripotents of maximal rank r the jordan theoretic description of d was used in 17 to determine an explicit peter weyl decomposition of the hardy space h2 s cf 15 in terms of the norm functions generalized determinants associated with the jordan triple

the complex busemann petty problem on sections of convex bodies abstract the complex busemann petty problem asks whether origin symmetric convex bodies in c n with smaller central hyperplane sections necessarily have smaller volume we prove that the answer is affirmative if n 3 and negative if n 4

controllability of linear systems on solvable lie groups linear systems on lie groups are a natural generalization of linear systems on euclidian spaces for such systems this paper studies controllability by taking into consideration the eigenvalues of an associated derivation mathcal d when the state space is a solvable connected lie group controllability of the system is guaranteed if the reachable set of the neutral element is open and the derivation mathcal d has only pure imaginary eigenvalues for bounded systems on nilpotent lie groups such conditions are also necessary

plancherel formula for berezin deformation of l 2 on riemannian symmetric space consider natural representations of the pseudounitary group u p q in the space of holomorphic functions on the cartan domain hermitian symmetric space u p q u p u q berezin representations of o p q are the restrictions of such representations to the subgroup o p q we obtain the explicit plancherel formula for the berezin representations the support of the plancherel measure is a union of many series of representations the density of the plancherel measure on each piece of the support is an explicit product of functions we also show that the berezin representations give an interpolation between l2 on noncompact symmetric space o p q o p o q and l2 on compact symmetric space o p q o p o q

transcending classical invariant theory let sp2 r sp2 sp be the real symplectic group of rank n let sp denote the two fold cover of sp there is a unitary representation constructed by shale sh and weil wa of sp that is of considerable interest in the theory of automorphic forms we denote this representation by co and call it the oscillator representation the purpose of this paper is to establish about c a fact that should be useful for clarifying the structure of spaces of automorphic forms constructed using co we begin by formulating the result let e be a reductive subgroup of sp and let e denote the inverse image of e in sp denote by r e the set of infinitesimal equivalence classes of continuous irreducible admissible representations of e on locally convex spaces let co be the smooth representation of sp associated to co let co be realized on a hilbert space and let the subspace of smooth vectors on which space co is defined be written denote by r9 e co the set of elements of e which are realized as quotients by co e invariant closed subspaces of t g consider a reductive dual pair g g c sp h2 it is not hard to show that g and g commute with one another hence we may regard co i g g as a representation of gx g it is well known f that r gx g s9 g xr9 g the identification associates to p e r g and p e i g the tensor product p x p we note that p x p is not defined as a topological vector space nevertheless the infinitesimal equivalence class of p op is well defined select p x p e g x g suppose that in fact p p e 9 g g c then clearly p e r g c and p e 9 g c hence g co defines the graph of a correspondence between certain subsets of r9 g cv and j g co in fact the situation is quite precise

laplace and segal bargmann transforms on hermitian symmetric spaces and orthogonal polynomials let d g k be a complex bounded symmetric domain of tube type in a complex jordan algebra v and let dr j d d be its real form in a formally real euclidean jordan algebra j v dr h l is a bounded realization of the symmetric cone in j we consider representations of h that are gotten by the generalized segal bargmann transform from a unitary g space of holomorphic functions on d to an l2 space on dr we prove that in the unbounded realization the inverse of the unitary part of the restriction map is actually the laplace transform we find the extension to d of the spherical functions on dr and find their expansion in terms of the l spherical polynomials on d which are jack symmetric polynomials we prove that the coefficients are orthogonal polynomials in an l2 space the measure being the harish chandra plancherel measure multiplied by the symbol of the berezin transform we prove the difference equation and recurrence relation for those polynomials by considering the action of the lie algebra and the cayley transform on the polynomials on d finally we use the laplace transform to study generalized laguerre functions on symmetric cones

quantization of kahler manifolds ii we use berezin s dequantization procedure to define a formal product on a dense subalgebra of the algebra ofsmooth functions on a compact homogeneous kahler manifold m we prove that this formal product isconvergent when m isa hermitian symmetric space 1993 american mathematical society

branching coefficients of holomorphic representations and segal bargmann transform let d g k be a complex bounded symmetric domain of tube type in a jordan algebra vc and let d h l d v be its real form in a jordan algebra v vc the analytic continuation of the holomorphic discrete series on dopf forms a family of interesting representations of g we consider the restriction on d and the branching rule under h of the scalar holomorphic representations the unitary part of the restriction map gives then a generalization of the segal bargmann transform the group l is a spherical subgroup of k and we find a canonical basis of l invariant polynomials in the components of the schmid decomposition and we express them in terms of the jack symmetric polynomials we prove that the segal bargmann transform of those l invariant polynomials are under the spherical transform on d multi variable wilson type polynomials and we give a simple alternative proof of their orthogonality relation we find the expansion of the spherical functions on d when extended to a holomorphic function in a neighborhood of 0 d in terms of the l spherical holomorphic polynomials on d the coefficients being the wilson polynomials

the berezin transform and invariant differential operators the berezin calculus is important to quantum mechanics creation annihilation operators and operator theory toeplitz operators we study the basic berezin transform linking the contravariant and covariant symbol for all bounded symmetric domains and express it in terms of invariant differential operators

a weiszfeld algorithm for the solution of an asymmetric extension of the generalized fermat location problem the generalized fermat problem in the plane is given n 3 destination points find the point email protected which minimizes the sum of euclidean distances from email protected to each of the destination points the weiszfeld iterative algorithm for this problem is globally convergent independent of the initial guess also a test is available email protected priori to determine when email protected a destination point this paper generalizes earlier work by the first author by introducing an asymmetric euclidean distance in which at each destination the x component is weighted differently from the y component a weiszfeld algorithm is studied to compute email protected and is shown to be a descent method which is globally convergent except possibly for a denumerable number of starting points local convergence properties are characterized when email protected is not a destination point the iteration matrix at email protected is shown to be convergent and local convergence is always linear when email protected is a destination point local convergence can be linear sub linear or super linear depending upon a computable criterion a test which does not require iteration for email protected to be a destination is derived comparisons are made between the symmetric and asymmetric problems numerical examples are given

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

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

orthogonal polynomials of types a and b and related calogero models there are examples of calogero sutherland models associated to the weyl groups of type a and b when exchange terms are added to the hamiltonians the systems have non symmetric eigenfunctions which can be expressed as products of the ground state with members of a family of orthogonal polynomials these polynomials can be defined and studied by using the differential difference operators introduced by the author in trans am math soc 311 167 183 1989 after a description of known results particularly from the works of baker and forrester and sahi there is a study of polynomials which are invariant or alternating for parabolic subgroups of the symmetric group the detailed analysis depends on using two bases of polynomials one of which transforms monomially under group actions and the other one is orthogonal there are formulas for norms and point evaluations which are simplifications of those of sahi for any parabolic subgroup of the symmetric group there is a skew operator on polynomials which leads to evaluation at 1 1 1 of the quotient of the unique skew polynomial in a given irreducible subspace by the minimum alternating polynomial analogously to a weyl character formula the last section concerns orthogonal polynomials for the type b weyl group with an emphasis on the hermite type polynomials these can be expressed by using the generalized binomial coefficients a complete basis of eigenfunctions of yamamoto s b n spin calogero model is obtained by multiplying these polynomials by the ground state

fourier and poisson transformation associated to a semisimple symmetric space soit g un groupe de lie semisimple connexe contenu dans un groupe de lie semisimple simplement connexe dont l algebre de lie est la complexification de celle de g soit une involution non triviale de g et une involution de cartan commutant avec soit h un groupe point fixe de on etudie les representations h spheriques de dimension finie de g contenant un vecteur k fixe ou k est un sous groupe compact maximal stable de g sous representations de dimension finie de le noyau de poisson la distribution u i la transformation de fourier et poisson formules integrales

toeplitz quantization and asymptotic expansions for real bounded symmetric domains an analogue of the star product familiar from deformation quantization is studied in the setting of real bounded symmetric domains the analogue turns out to be a certain invariant operator which one might call star restriction from functions on the hermitification of the domain into functions on the domain itself in particular we establish the usual i e semiclassical asymptotic expansion of this star restriction and further obtain a real variable analogue of a theorem of arazy and orsted concerning the analogous expansion for the berezin transform

generalizations of the busemann petty problem for sections of convex bodies abstract we present generalizations of the busemann petty problem for dual volumes of intermediate central sections of symmetric convex bodies it is proved that the answer is negative when the dimension of the sections is greater than or equal to 4 for two three dimensional sections both negative and positive answers are given depending on the orders of dual volumes involved and certain cases remain open for bodies of revolution a complete solution is obtained in all dimensions

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

analysis on the minimal representation of o p q i realization via conformal geometry abstract this is the first in a series of papers devoted to an analogue of the metaplectic representation namely the minimal unitary representation of an indefinite orthogonal group this representation corresponds to the minimal nilpotent coadjoint orbit in the philosophy of kirillov kostant we begin by applying methods from conformal geometry of pseudo riemannian manifolds to a general construction of an infinite dimensional representation of the conformal group on the solution space of the yamabe equation by functoriality of the constructions we obtain different models of the unitary representation as well as giving new proofs of unitarity and irreducibility the results in this paper play a basic role in the subsequent papers where we give explicit branching formulae and prove unitarization in the various models

spectrum generating operators and intertwining operators for representations induced from a maximal parabolic subgroup abstract we present a new method of calculating intertwining operators between principal series representations of semisimple lie groups g working in the compact realization we find the eigenvalues of the operators on the k types and give several examples among the advantages of our method is its applicability to bundle valued cases

an efficient algorithm for minimizing a sum of p norms we study the problem of minimizing a sum of p norms where p is a fixed real number in the interval 1 infty several practical algorithms have been proposed to solve this problem however none of them has a known polynomial time complexity in this paper we transform the problem into standard conic form unlike those in most convex optimization problems the cone for the p norm problem is not self dual unless p 2 nevertheless we are able to construct two logarithmically homogeneous self concordant barrier functions for this problem the barrier parameter of the first barrier function does not depend on p the barrier parameter of the second barrier function increases with p using both barrier functions we present a primal dual potential reduction algorithm to compute an epsilon optimal solution in polynomial time that is independent of p computational experiences with a matlab implementation are also reported

the cos λ and sin λ transforms as intertwining operators between generalized principal series representations of sl n 1 k abstract in this article we connect topics from convex and integral geometry with well known topics in representation theory of semisimple lie groups by showing that the cos and sin transforms on the grassmann manifolds gr p k su n 1 k s u p k u n 1 p k are standard intertwining operators between certain generalized principal series representations induced from a maximal parabolic subgroup p p of sl n 1 k the index p indicates the dependence of the parabolic on p the general results of knapp and stein 1971 1980 20 21 and vogan and wallach 1990 44 then show that both transforms have meromorphic extension to c and are invertible for generic c furthermore known methods from representation theory combined with a selberg type integral allow us to determine the k spectrum of those operators

toeplitz operators on bounded symmetric domains in this paper jordan algebraic methods are applied to study toeplitz operators on the hardy space h2 s associated with the shilov boundary s of a bounded symmetric domain d in c of arbitrary rank the jordan triple system z c associated with d is used to determine the relationship between toeplitz operators and differential operators further it is shown that each jordan triple idempotent e e z induces an irreducible representation e symbol of the c algebra generated by all toeplitz operators tf with continuous symbol function f 0 introduction toeplitz operators on the boundary t aa of the open unit disc a c c play an important role in function theory of one complex variable cf 9 chapter 7 in several dimensions toeplitz operators have been mainly studied for strictly pseudo convex domains d c c 4 13 14 24 28 in particular for the hilbert ball 7 using the relationship with pseudo differential operators another class of domains in c generalizing the unit disc is the class of bounded symmetric domains cartan domains and exceptional domains which have a more complicated boundary structure compared to the strictly pseudo convex case the aim of this paper is the study of toeplitz operators t1 with continuous symbol functionf e c s on the shilov boundary s of a bounded symmetric domain d of arbitrary rank r in the special cases of the hilbert ball r 1 and the lie ball r 2 the structure of the operators ti and of the toeplitz c algebra

a homogeneous interior point algorithm for nonsymmetric convex conic optimization a homogeneous interior point algorithm for solving nonsymmetric convex conic optimization problems is presented starting each iteration from the vicinity of the central path the method steps in the approximate tangent direction and then applies a correction phase to locate the next well centered primal dual point features of the algorithm include that it makes use only of the primal barrier function that it is able to detect infeasibilities in the problem and that no phase i method is needed we prove convergence to epsilon accuracy in mathcal o sqrt nu log 1 epsilon o log 1 iterations to improve performance the algorithm employs a new runge kutta type second order search direction suitable for the general nonsymmetric conic problem moreover quasi newton updating is used to reduce the number of factorizations needed implemented so that data sparsity can still be exploited extensive and promising computational results are presented for the p p cone problem the facility location problem entropy maximization problems and geometric programs all formulated as nonsymmetric convex conic optimization problems

canonical representations related to hyperbolic spaces abstract we extend the notion of canonical representation introduced by a m vershik i m gel fand and m i graev 1982 in representation theory cambridge univ press cambridge u k to all classical rank one semisimple lie groups we determine the spectral decomposition of the canonical representations in detail

function spaces and reproducing kernels on bounded symmetric domains there are several natural hilbert spaces of holomorphic functions on a bounded symmetric domain such are the bergman type spaces on which the holomorphic discrete series operates and the hardy type spaces which are related to the analytic continuation of the holomorphic discrete series also closely related is the bergmann space of entire functions on the ambient which arises as the closure of the polynomials with respect to a natural inner product the space of holomorphic polynomials decomposes into irreducible subspaces under the action of the isotropy group k of the domain the main facts about this decomposition were proved by schmid 21 for another proof see 22 each irreducible subspace contains a unique normalized l invariant spherical polynomial where l is the isotropy group of the shilov boundary in k our first main result is the explicit computation of the norms of the spherical polynomials with respect to each of the hilbert spaces considered for the domains of classical type they were considered by hua 9 and for some of the hilbert spaces this was done before by upmeier 24 using different methods for certain others there are partial results in 22 we are able to do this in a fairly simple unified way by making strong use of gindikin s generalized gamma function s next we obtain a description of the reproducing kernels of the k irreducible subspaces in each of our hilbert spaces and an expansion in terms of these for

structure of the degenerate principal series on symmetric r spaces and small representations abstract let g be a simple real lie group with maximal parabolic subgroup p whose nilradical is abelian then x g p is called a symmetric r space we study the degenerate principal series representations of g on c x in the case where p is not conjugate to its opposite parabolic we find the points of reducibility the composition series and all unitarizable constituents among the unitarizable constituents we identify some small representations having as associated variety the minimal nilpotent k c orbit in p c where k c is the complexification of a maximal compact subgroup k g and g k p the corresponding cartan decomposition

a review on some classes of algebraic systems in this paper we review some algebraic control system precisely linear and bilinear systems on euclidean spaces and invariant and linear systems on lie groups the fourth classes of systems have

two pairs of families of polyhedral norms versus ell _p lp norms proximity and applications in optimization this paper studies four families of polyhedral norms parametrized by a single parameter the first two families consist of the cvar norm which is equivalent to the d norm or the largest k k norm and its dual norm while the second two families consist of the convex combination of the ell 1 l1 and ell infty l norms referred to as the deltoidal norm and its dual norm these families contain the ell 1 l1 and ell infty l norms as special limiting cases these norms can be represented using linear programming lp and the size of lp formulations is independent of the norm parameters the purpose of this paper is to establish a relation of the considered lp representable norms to the ell p lp norm and to demonstrate their potential in optimization on the basis of the ratio of the tight lower and upper bounds of the ratio of two norms we show that in each dual pair the primal and dual norms can equivalently well approximate the ell p lp and ell q lq norms respectively for p q in 1 infty p q 1 satisfying 1 p 1 q 1 1 p 1 q 1 in addition the deltoidal norm and its dual norm are shown to have better proximity to the ell p lp norm than the cvar norm and its dual numerical examples demonstrate that lp solutions with optimized parameters attain better approximation of the ell 2 l2 norm than the ell 1 l1 and ell infty l norms do

classification of invariant valuations on the quaternionic plane abstract we describe the orbit space of the action of the group sp 2 sp 1 on the real grassmann manifolds gr k h 2 in terms of certain quaternionic matrices of moore rank not larger than 2 we then give a complete classification of valuations on the quaternionic plane h 2 which are invariant under the action of the group sp 2 sp 1

a smoothing newton method for minimizing a sum of euclidean norms we consider the problem of minimizing a sum of euclidean norms f x sum i 1 m b i a i tx this problem is a nonsmooth problem because f is not differentiable at a point x when one of the norms is zero in this paper we present a smoothing newton method for this problem by applying the smoothing newton method proposed by qi sun and zhou math programming 87 2000 pp 1 35 directly to a system of strongly semismooth equations derived from primal and dual feasibility and a complementarity condition this method is globally and quadratically convergent as applications to this problem smoothing newton methods are presented for the euclidean facilities location problem and the steiner minimal tree problem under a given topology preliminary numerical results indicate that this method is extremely promising

a primal dual algorithm for minimizing a sum of euclidean norms we study the problem of minimizing a sum of euclidean norms this nonsmooth optimization problem arises in many different kinds of modern scientific applications in this paper we first transform this problem and its dual problem into a system of strongly semismooth equations and give some uniqueness theorems for this problem we then present a primal dual algorithm for this problem by solving this system of strongly semismooth equations preliminary numerical results are reported which show that this primal dual algorithm is very promising