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

galerkin projected residual method applied to diffusion reaction problems a stabilized finite element method is presented for scalar and linear second order boundary value problems the method is obtained by adding to the galerkin formulation multiple projections of the residual of the differential equation at element level these multiple projections allow the generation of appropriate number of free stabilization parameters in the element matrix depending on the local space of approximation and on the differential operator the free parameters can be determined imposing some convergence and or stability criteria or by postulating the element matrix with the desired stability properties the element matrix of most stabilized methods such as gls and ggls methods can be obtained using this new method with appropriate choices of the stabilization parameters we applied this formulation to diffusion reaction problems optimal rates of convergency are numerically observed for regular solutions

multiscale phenomena green s functions the dirichlet to neumann formulation subgrid scale models bubbles and the origins of stabilized methods abstract an approach is developed for deriving variational methods capable of representing multiscale phenomena the ideas are first illustrated on the exterior problem for the helmholtz equation this leads to the well known dirichlet to neumann formulation next a class of subgrid scale models is developed and the relationships to bubble function methods and stabilized methods are established it is shown that both the latter methods are approximate subgrid scale models the identification for stabilized methods leads to an analytical formula for the intrinsic time scale whose origins have been a mystery heretofore

finite element solution of the helmholtz equation with high wave number part i the h version of the fem abstract the paper addresses the properties of finite element solutions for the helmholtz equation the h version of the finite element method with piecewise linear approximation is applied to a one dimensional model problem new results are shown on stability and error estimation of the discrete model in all propositions assumptions are made on the magnitude of hk only where k is the wavelength and h is the stepwidth of the fe mesh previous analytical results had been shown with the assumption that k 2 h is small for medium and high wavenumber these results do not cover the meshsizes that are applied in practical applications the main estimate reveals that the error in h 1 norm of discrete solutions for the helmholtz equation is polluted when k 2 h is not small the error is then not quasioptimal i e the relation of the fe error to the error of best approximation generally depends on the wavenumber k it is noted that the pollution term in the relative error is of the same order as the phase lead of the numerical solution in the result of this analysis thorough and rigorous understanding of error behavior throughout the range of convergence is gained numerical results are presented that show sharpness of the error estimates and highlight some phenomena of the discrete solution behavior the h p version of the fem is studied in part ii

the generalized finite element method for helmholtz equation theory computation and open problems in this paper we address the generalized finite element method for the helmholtz equation we obtain our method by employing the finite element method on cartesian meshes which may overlap the boundaries of the problem domain and by enriching the approximation by plane waves pasted into the finite element basis at each mesh vertex by the partition of unity method here we address the q convergence of the method where q is the number of plane waves added at each vertex for the class of smooth analytic solutions for which we get better than exponential convergence for sufficiently small h depending on p an important observation is that we can monitor the accuracy in any computed solution quantity of interest at negligible cost by using q extrapolation our results assume exact integration of all the employed integrals further studies are needed to analyze the effects of the numerical integrations and also the effect of the roundoff errors

reproducing kernels and composition series for spaces of vector valued holomorphic functions on tube domains abstract we study hilbert spaces of vector valued holomorphic sections on tube domains of type ii and iii we consider some small representations of the maximal compact subgroups and calculate the norm of each k type in the hilbert spaces as a consequence we get composition series of the analytic continuations of certain holomorphic discrete series and expansions of matrix valued reproducing kernels

nearly h1 optimal finite element methods abstract we examine the problem of finding the h 1 projection onto a finite element space of an unknown field satisfying a specified boundary value problem solving the projection problem typically requires knowing the exact solution we circumvent this issue and obtain a petrov galerkin formulation which achieves h 1 optimality requiring weighting functions to be defined locally on the element level permits only approximate h 1 optimality in multi dimensional configurations we investigate the relation between our formulation and other stabilized fem formulations we show in particular that our formulation leads to a derivation of the supg method in special cases the present formulation reduces to that of residual free bubbles finally we present guidelines for obtaining the petrov weight functions and include a numerical example for the helmholtz equation

the capelli identity the double commutant theorem and multiplicity free actions 0 the capelli identity cal 3 w p 39 is one of the most celebrated and useful formulas of classical invariant theory w d cl z the double commutant theorem w p 91 is likewise a basic result in the general theory of associative algebras both play key roles in weyl s book the classical groups the main purpose of this paper is to demonstrate a close connection between the two in the context of multiplicity free actions ekj of groups on vector spaces the focus of our discussion will be the structure of the differential operators which commute with a multiplicity free action applications include a new derivation of some formulas of shimura shl and rubenthaler and schiffmann rs for b functions associated to hermitian symmetric spaces and a construction of interesting sets of generators for the center of the universal enveloping algebra of gin we also give a detailed discussion of certain aspects of multiplicity free representations here is an overview of the contents of the paper in sect 1 we review the classical capelli identities we observe how their existence is predicted by a double commutant result 1 6 and 1 8 and we show how they can be used to compute b functions we remark that capelli understood that his operators generate the center of the algebra generated by the polarization operators see b1 p 77 also capelli s motivation in introducing his operators was to explain a formula of cayley essentially the computation of a b function capelli s point of view was remarkably modern and structuralist in certain ways more modern even than that of weyl sections 2 through 9 provide a conceptual context for understanding the capelli identities as a feature of multiplicity free actions they contain a general discussion of the structure of g the algebra of polynomial coefficient differential operators which commute with a given group g of linear transformations the results of the discussion are summarized in theorem 9 1 which says in particular that the polynomial coefficient differential operators commuting with a

fliess series a generalization of the ree s theorem and an algebraic approach to a homogeneous approximation problem in this paper an algebraic technique originated by m fliess is developed and applied to a homogeneous approximation problem for nonlinear affine control systems the central role is played by a generalization of the r ree s theorem on a connection of lie elements and the shuffle product operation in a free associative algebra

bessel operators on jordan pairs and small representations of semisimple lie groups abstract we provide a uniform construction of l 2 models for all small unitary representations in degenerate principal series of semisimple lie groups which are induced from maximal parabolic subgroups with abelian nilradical this generalizes previous constructions to the case of a maximal parabolic subgroup which is not necessarily conjugate to its opposite and hence the previously used jordan algebra methods have to be generalized to jordan pairs the crucial ingredients for the construction of the l 2 models are the lie algebra action and the spherical vector working in the so called fourier transformed picture of the degenerate principal series the lie algebra action is given in terms of bessel operators on jordan pairs we prove that precisely for those parameters of the principal series where small quotients occur the bessel operators are tangential to certain submanifolds further we show that the small quotients are unitarizable if and only if these submanifolds carry equivariant measures in this case we can express the spherical vectors in terms of multivariable k bessel functions and some delicate estimates for these bessel functions imply the existence of the l 2 models

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

on the support of graph lie algebras we deal with the support of graph lie algebras and we characterize the set of independence alphabets for which the noncommutative result can naturally be extended we also provide an algorithm which decides if an independence alphabet fulfills the preceding condition

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

remarks on classical invariant theory a uniform formulation applying to all classical groups simultaneously of the first fundamental theory of classical invariant theory is given in terms of the weyl algebra the formulation also allows skew symmetric as well as symmetric variables examples illustrate the scope of this formulation

some groups of transformations defined by jordan algebras i it is well known that the lie algebra of derivations of the exceptional jordan algebra m 3 8 over an algebraically closed field of characteristic 0 or over the field of reals is the exceptional lie algebra f 4 of killing cartan and that the lie algebra of linear transformations in m 3 8 leaving a certain cubic norm form defined on m 3 8 invariant is the exceptional lie algebra e 6 chevalley schafer 7 freudenthal 11 the program envisaged in the present series of papers is the extension of these results and their group analogues to arbitrary central simple jordan algebras the present paper gives the definition of the groups and lie algebras in the general case and gives their determination for special jordan algebras with some restrictions on the base field in a second paper we plan to consider the groups of automorphisms of arbitrary reduced exceptional simple jordan algebras and in a third paper we hope to study the generalized groups of types e 6 which arise from these jordan algebras

homogeneous vector bundles and intertwining operators for symmetric domains the main features of homogeneous cowen douglas operators well known for the unit disk are generalised to the setting of hermitian bounded symmetric domains of arbitrary rank c 2016 elsevier inc all rights reserved

a symmetric trefftz dg formulation based on a local boundary element method for the solution of the helmholtz equation a general symmetric trefftz discontinuous galerkin method is built for solving the helmholtz equation with piecewise constant coefficients the construction of the corresponding local solutions to the helmholtz equation is based on a boundary element method a series of numerical experiments displays an excellent stability of the method relatively to the penalty parameters and more importantly its outstanding ability to reduce the instabilities known as the pollution effect in the literature on numerical simulations of long range wave propagation

a nearly optimal galerkin projected residual finite element method for helmholtz problem a finite element formulation for scalar and linear second order boundary value problems is introduced the new method relies on a variational formulation obtained following the usual path of appending to the galerkin variational formulation a balanced residual form of the governing partial differential equation computed within each element the novelty consists of projecting the residual in a subspace defined for each element which gives rise to the name of the method galerkin projected residual gpr this subspace is built by systematically exploring some a priori criteria either based on the physics or on the underlying mathematics the method can be used to stabilize a variety of problems here it is applied to helmholtz equation where standard galerkin formulations are known to present poor approximations for high wave numbers the method is formally introduced along with some numerical examples that are used to assess the improvements achieved

a fourth order compact scheme for the helmholtz equation alpha interpolation of fem and fdm stencils we propose a fourth order compact scheme on structured meshes for the helmholtz equation given by r f x 2 0 the scheme consists of taking the alpha interpolation of the galerkin finite element method and the classical central finite difference method in 1d this scheme is identical to the alpha interpolation method j comput appl math 1982 8 1 15 19 and in 2d making the choice 0 5 we recover the generalized fourth order compact pade approximation j comput phys 1995 119 252 270 comput meth appl mech engrg 1998 163 343 358 therein using the parameter 2 we follow siam rev 2000 42 3 451 484 comput meth appl mech engrg 1995 128 325 359 for the analysis of this scheme and its performance on square meshes is compared with that of the quasi stabilized fem comput meth appl mech engrg 1995 128 325 359 in particular we show that the relative phase error of the numerical solution and the local truncation error of this scheme for plane wave solutions diminish at the rate o l 4 where l represent the wavenumber and the mesh size respectively an expression for the parameter is given that minimizes the maximum relative phase error in a sense that will be explained in section 4 5 convergence studies of the error in the l2 norm the h1 semi norm and the l euclidean norm are done and the pollution effect is found to be small copyright 2010 john wiley sons ltd

a discontinuous finite element formulation for helmholtz equation abstract a discontinuous finite element formulation for helmholtz problems is presented with c 0 continuity across inter element boundaries enforced in a weak sense depending on two parameters and whose choice is crucial for the performance of the proposed method these parameters are chosen through one dimension numerical experiments aiming at minimizing the pollution error optimal values of these parameters are then adopted in more general situations the accuracy and stability of the proposed formulation for linear and bilinear shape functions are demonstrated in several numerical examples in one and two dimensions

a galerkin least squares finite element method for the two dimensional helmholtz equation in this paper a galerkin least squares gls finite element method in which residuals in least squares form are added to the standard galerkin variational equation is developed to solve the helmholtz equation in two dimensions an important feature of gls methods is the introduction of a local mesh parameter that may be designed to provide accurate solutions with relatively coarse meshes previous work has accomplished this for the one dimensional helmholtz equation using dispersion analysis in this paper the selection of the gls mesh parameter for two dimensions is considered and leads to elements that exhibit improved phase accuracy for any given direction of wave propagation an optimal gls mesh parameter is determined using two dimensional fourier analysis in general problems the direction of wave propagation will not be known a priori in this case an optimal gls parameter is found which reduces phase error for all possible wave vector orientations over elements the optimal gls parameters are derived for both consistent and lumped mass approximations several numerical examples are given and the results compared with those obtained from the galerkin method the extension of gls to higher order quadratic interpolations is also presented

canonical semi invariants and the plancherel formula for parabolic groups a parabolic subgroup of a reductive lie group is called good if the center of the universal enveloping algebra of its nilradical contains an element that is semi invariant of weight proportional to the modular function the good case is characterized here by invariance of the set of simple roots defining the parabolic under the negative of the opposition element of the weyl group in the good case the unbounded dixmier pukanszky operator of the parabolic subgroup is described the conditions under which it is a differential operator rather than just a pseudodifferential operator are specified and an explicit plancherel formula is derived for that parabolic 1 introduction this paper continues our work on the plancherel formula for parabolic subgroups of reductive lie groups earlier we considered maximal parabolic subgroups 7 and 11 minimal parabolic subgroups 4 and 6 and parabolic subgroups whose unipotent radical has square integrable representations modulo its center 7 and 12 in each case we described the representation theory the plancherel measure and the dixmier pukanszky operator that appears in the plancherel formula to compensate for nonunimodularity we now consider arbitrary parabolic subgroups we isolate the common features of all previously treated cases which enabled us to describe the fundamental semi invariant polynomial that gives rise to the dixmier pukanszky operator that is our first main result here the second main result is a simple characterization of which parabolics have the requisite property and the third is the construction in that case of the canonical semi invariant polynomial we explicitly compute the dixmier pukanszky operator and the plancherel formula for parabolics that have the requisite property here is a brief summary of the contents of this paper in 2 we describe the key property see definition 2 1 for parabolic groups that guarantees the existence of a unique appropriate semi invariant theorem 2 2 in the center of the universal enveloping algebra of the nilradical we call such parabolics good let p be a good parabolic and p nam its langlands decomposition we prove proposition 2 4 that the generic stability groups si for

the nearly optimal petrov galerkin method for convection diffusion problems abstract the nearly optimal petrov galerkin nopg method is employed to improve finite element computation of convection dominated transport phenomena the design of the nopg method for convection diffusion is based on consideration of the advective limit nonetheless the resulting method is applicable to the entire admissible range of problem parameters an investigation of the stability properties of this method leads to a coercivity inequality the convergence features of the nopg method for convection diffusion are studied in an error analysis that is based on the stability estimates the proposed method compares favorably to the performance of an established technique on several numerical tests

a generalized finite element method for solving the helmholtz equation in two dimensions with minimal pollution when using the galerkin fem for solving the helmholtz equation in two dimensions the error of the corresponding solution differs substantially from the error of the best approximation and this effect increases with higher wave number k in this paper we will design a generalized finite element method gfem for the helmholtz equation such that the pollution effect is minimal

orthogonal projection onto the free lie algebra abstract as an answer to a question of garsia we give a universal formula in a normal form for the orthogonal projection of free associative algebra onto free lie algebra

a residual based finite element method for the helmholtz equation a new residual based finite element method for the scalar helmholtz equation is developed this method is obtained from the galerkin approximation by appending terms that are proportional to residuals on element interiors and inter element boundaries the inclusion of residuals on inter element boundaries distinguishes this method from the well known galerkin least squares method and is crucial to the resulting accuracy of this method in two dimensions and for regular bilinear quadrilateral finite elements it is shown via a dispersion analysis that this method has minimal phase error numerical experiments are conducted to verify this claim as well as test and compare the performance of this method on unstructured meshes with other methods it is found that even for unstructured meshes this method retains a high level of accuracy copyright 2000 john wiley sons ltd

differential equations and recursion relations for laguerre functions on symmetric cones we obtain the differential equation and recurrence relations satisfied by the laguerre functions l m on an arbitrary symmetric cone

bubble based stabilization for the helmholtz equation a comparison of two bubble enriched methods derived by different considerations indicates that the methods are identical in some cases thus series representations of auxiliary functions derived independently for the two methods turn out to be equivalent prior to truncation three such series for time harmonic acoustics are considered dispersion analysis points to the more efficient series representation and provides guidelines for the number of terms to be retained numerical tests confirm the validity of these practical guidelines copyright 2006 john wiley sons ltd

the shuffle algebra and its derivations abstract the shuffle algebra on a partially commutative alphabet and its derivations are studied the derivation of the shuffle algebra is characterized in terms of admissible mappings the module of all derivations of the shuffle algebra is proved to be isomorphic to the free partially commutative power series lie algebra with the coefficient ring extended the orthogonality between the lie elements and the proper shuffles is established

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

residual free bubbles for the helmholtz equation the galerkin method enriched with residual free bubbles is considered for approximating the solution of the helmholtz equation two dimensional tests demonstrate the improvement over the standard galerkin method and the galerkin least squares method using piecewise bilinear interpolations

a discontinuous galerkin method with lagrange multipliers for the solution of helmholtz problems in the mid frequency regime we present a discontinuous galerkin method dgm for the solution of the helmholtz equation in the mid frequency regime our approach is based on the discontinuous enrichment method in which the standard polynomial field is enriched within each finite element by a non conforming field that contains free space solutions of the homogeneous partial differential equation to be solved hence for the helmholtz equation the enrichment field is chosen here as the superposition of plane waves we enforce a weak continuity of these plane waves across the element interfaces by suitable lagrange multipliers preliminary results obtained for two dimensional model problems discretized by uniform meshes reveal that the proposed dgm enables the development of elements that are far more competitive than both the standard linear and the standard quadratic galerkin elements for the discretization of helmholtz problems

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

on differential operators attached to certain representations of classical groups classification et construction de sous espaces irreductibles de sr integrales contenant les applications polynomiales l operateur differentiel d operateurs differentiels de type arithmetique applications aux series de eisenstein

a quasi optimal petrov galerkin method for helmholtz problem a petrov galerkin finite element formulation is introduced for helmholtz problem in two dimensions using polynomial weighting functions at each node of the mesh a global basis function for the weighting space is obtained adding to the bilinear c 0 lagrangian weighting function linear combinations of polynomial bubbles defined on a macroelement containing this node quasi optimal weighting functions with the same support of the corresponding global test functions are obtained after computing the coefficients of these linear combinations attending to optimality criteria this is done numerically through a simple preprocessing technique naturally applied to non uniform and unstructured meshes in particular for uniform mesh a quasi optimal interior stencil of the same order of the quasi stabilized finite element method stencil derived by babuska et al comput methods appl mech engrg 1995 128 325 359 is obtained numerical results are presented illustrating the great stability and accuracy of this formulation with uniform and non uniform meshes

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 discontinuous finite element method at element level for helmholtz equation abstract a discontinuous finite element formulation is presented for helmholtz equation continuity is relaxed locally in the interior of the element instead of across the element edges the interior shape functions can be viewed as discontinuous bubbles and the corresponding degrees of freedom can be eliminated at element level by static condensation yielding a global matrix topologically equivalent to those of classical c0 finite element approximations the discontinuous bubbles are introduced to consistently add stability to the formulation preserving the optimal rates of convergence and improving accuracy a crucial point of the discontinuous formulation relies on the choice of the stabilization parameters related to the weak enforcement of continuity inside each element explicit values of these stabilization parameters minimizing the pollution effect are presented for uniform meshes the accuracy and stability of the proposed formulation for bilinear shape functions are demonstrated in several numerical examples

quasi optimal finite difference method for helmholtz problem on unstructured grids a quasi optimal finite difference method qofd is proposed for the helmholtz problem the stencils coefficients are obtained numerically by minimizing a least squares functional of the local truncation error for plane wave solutions in any direction in one dimension this approach leads to a nodally exact scheme with no truncation error for uniform or non uniform meshes in two dimensions when applied to a uniform cartesian grid a 9 point sixth order scheme is derived with the same truncation error of the quasi stabilized finite element method qsfem introduced by babuska et al comp meth appl mech eng 1995 128 325 359 similarly a 27 point sixth order stencil is derived in three dimensions the qofd formulation proposed here is naturally applied on uniform non uniform and unstructured meshes in any dimension numerical results are presented showing optimal rates of convergence and reduced pollution effects for large values of the wave number copyright 2009 john wiley sons ltd

eigendistributions for orthogonal groups and representations of symplectic groups we consider the action of h o p q on the matrix space mp q n r we study a certain orbit o of h in the null cone n mp q n r which supports an eigendistribution d o for h using some identities of capelli type developed in the appendix we determine the structure of g sp 2n r cyclic module generated by d o under the oscillator representation of g the metaplectic cover of g sp 2n p q r applications to local theta correspondence and a generalized huygens principle are given

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

tensor product of degenerate principal series and local theta correspondence abstract we construct an intertwining map from the oscillator representation to the tensor product of two degenerate principal series of g and g which form a reductive dual pair in the sense of howe we derive results on local theta correspondence for the dual pairs o p q sp 2 q r u p q u q q and sp p q o 4 q where p q we also describe the correspondence completely for the pairs o p q sl 2 r u p q u 1 1 and sp p q o 4