optical orthogonal codes with unequal auto and cross correlation constraints an optical orthogonal code ooc is a collection of binary sequences with good auto and cross correlation properties they were defined by salehi and others as a means of obtaining code division multiple access on optical networks up to now all work on oocs have assumed that the constraint placed on the autocorrelation and that placed on the cross correlation are the same we consider codes for which the two constraints are not equal specifically we develop bounds on the size of such oocs and demonstrate constriction techniques for building them the results demonstrate that a significant increase in the code size is possible by letting the autocorrelation constraint exceed the cross correlation constraint these results suggest that for a given performance requirement the optimal ooc may be one with unequal constraints this paper also views oocs with unequal auto and cross correlation constraints as constant weight unequal error protection uep codes with two levels of protection the bounds derived are interpreted from this viewpoint

aggregation of space time processes abstract in this paper we compare the relative efficiency of different methods of forecasting the aggregate of spatially correlated variables small sample simulations confirm the asymptotic result that improved forecasting performance can be obtained by imposing a priori constraints on the amount of spatial correlation in the system one way to do so is to aggregate forecasts from a space time autoregressive model elements of spatial structure cambridge university press cambridge 1975 which offers a solution to the curse of dimensionality that arises when forecasting with vars we also show that ignoring spatial correlation even when it is weak leads to highly inaccurate forecasts finally if the system satisfies a poolability condition there is a benefit in forecasting the aggregate variable directly

discrete series representations for the orbit spaces arising from two involutions of real reductive lie groups leth gbe real reductive lie groups a discrete series representation for a homogeneous spaceg his an irreducible representation ofgrealized as a closedg invariant subspace ofl2 g h the condition for the existence of discrete series representations forg hwas not known in general except for reductive symmetric spaces this paper offers a sufficient condition for the existence of discrete series representations forg hin the setting thatg his a homogeneous submanifold of a symmetric space g hwhereg g h we prove that discrete series representations are non empty for a number of non symmetric homogeneous spaces such assp 2n r sp n0 c gl n1 c gl nk c nj n ando 4m n u 2m j 0 2j n

optical orthogonal codes new bounds and an optimal construction a technique for constructing optimal oocs optical orthogonal codes is presented it provides the only known family of optimal with respect to family size oocs having lambda 2 the parameters n omega lambda are respectively p sup 2m 1 p sup m 1 2 where p is any prime and the family size is p sup m 2 three distinct upper bounds on the size of an ooc are presented that for many values of the parameter set n omega lambda improve upon the tightest previously known bound

classes of nonseparable spatio temporal stationary covariance functions abstract suppose that a random process z s t indexed in space and time has spatio temporal stationary covariance c h u where h d d 1 is a spatial lag and u is a temporal lag separable spatio temporal covariances have the property that they can be written as a product of a purely spatial covariance and a purely temporal covariance their ease of definition is counterbalanced by the rather limited class of random processes to which they correspond in this article we derive a new approach that allows one to obtain many classes of nonseparable spatio temporal stationary covariance functions and fit several such classes to spatio temporal data on wind speed over a region in the tropical western pacific ocean

localization and standard modules for real semisimple lie groups i the duality theorem in this paper we relate two constructions of representations of semisimple lie groups constructions that appear quite different at first glance homogeneous vector bundles are one source of representations if a real semisimple lie group go acts on a vector bundle e m over a quotient space m go ho then go acts also on the space of sections c m e and on any subspace vcc m e defined by a go invariant system of differential equations ordinary induction so called cohomological induction and the construction of representations by quantizat ion all fit into the framework of homogeneous vector bundles for any complex semisimple lie algebra g there is an equivalence of categories due to beilinson bernstein 1 between modules on the one hand and sheaves of modules over the flag variety x of g on the other in the context of real semisimple lie groups this equivalence of categories associates infinitesimal representations to orbits in the flag variety of the complexified lie algebra orbits not of the group go itself but of the complexification of maximal compact subgroup ko c go we shall show that these beilinson bernstein modules are naturally dual to modules attached to certain homogeneous vector bundles in the special case of a compact group both the beilinson bernstein construction and the construction via homogeneous vector bundles reduce to the borel weil bott theorem our duality theorem is then a particular instance of serre duality

restrictions of quaternionic representations in a paper by gross and wallach 1996 j reine angew math 481 73 123 the k types of the continuations of the quaternionic discrete series of a quaternionic lie group g are associated with projective orbits o of certain subgroups in g c in this paper we will show that the restrictions of the representations to quaternionic subgroups are closely related with the intersection of the zariski closure of o with hyperplanes we apply this to the minimal representations of the exceptional groups of real rank 4 and investigate the correspondences of certain compact dual pairs

geometric quantization and multiplicities of group representations the heisenberg uncertainty principle says that it is impossible to determine simultaneously the position and momentum of a quantum mechanical particle this can be rephrased as follows the smallest subsets of classical phase space in which the presence of a quantum mechanical particle can be detected are its lagrangian submanifolds for this reason it makes sense to regard the lagrangian submanifolds of phase space as being its true points see weinstein 17 now let g be a compact lie group and g x x x a hamiltonian action of g on x see w for definitions it is well known that the fixed points of this action form a symplectic submanifold of x see for instance guillemin and sternberg 5 however what can one say about the fixed points of g we will show that they are also the points of a symplectic manifold xc this manifold is the marsden weinstein reduction of x with respect to the zero orbit in g and will be described in sect 2 it was introduced in a completely different context fl om ours by marsden and weinstein 12 problems in classical mechanics can often be reduced to the study of hamiltonian systems on symplectic manifolds and problems in quantum mechanics to the study of linear operators on hilbert space this fact has inspired a number of efforts to quantize symplectic geometry by devising schemes for associating hilbert space to symplectic manifolds the no go theorems of groenwald and van hove impose some embarrassing limitations on all such schemes however it seems to be a useful idea heuristically to think of every symplectic manifold x a ic 1 as being symbiotically associated with a hilbert space xq a t in such a way that the classical observables on the first space correspond to quantum observables on the second space the heuristics further suggests that if g is a group of symmetries of x la i j it should also be a group of symmetries of xq a tum in this heuristic spirit we will state the main conjecture of this paper

discrete decomposability of the restriction of a_q λ with respect to reductive subgroups ii micro local analysis and asymptotic k support let g c g be real reductive lie groups this paper offers a criterion on the triplet g g ir that the irreducible unitary representation ir of g splits into a discrete sum of irreducible unitary representations of a subgroup g when restricted to g each of finite multiplicity furthermore we shall give an upper estimate of the multiplicity of an irreducible unitary representation of g occurring in wrig

differential operators on homogeneous spaces iii characteristic varieties of harish chandra modules and of primitive ideals letg be a semi simple complex algebraic group with lie algebra and flag varietyx g b for each primitive idealj with trivial central character in the enveloping algebrau we define a characteristic variety in the cotangent bundle ofx which projects under the springer resolution mapt x onto the closure of a nilpotent orbit we prove that this characteristic variety is theg saturation of the characteristic variety of a highest weight module with annihilatorj we conjecture that it is irreducible forg sl n our conjecture would provide a geometrical explanation for the classification of primitive ideals in terms of weyl group representations as achieved by a joseph the presentation of these ideas here is simultaneously used to some extent as an opportunity to continue our more general systematic discussion of differential operators on a complete homogeneous space and to study more generally characteristic varieties of harish chandra modules

sur la notion faiblement propre dans la conjecture de duflo in this article we verify duflo s conjecture for su 2 1 with respect to a borel subgroup it turns out that the non standard notion weakly proper faiblement propre in the formulation of duflo s conjecture is more reasonable than the standard notions proper and proper over image

nonseparable stationary covariance functions for space time data geostatistical approaches to spatiotemporal prediction in environmental science climatology meteorology and related fields rely on appropriate covariance models this article proposes general classes of nonseparable stationary covariance functions for spatiotemporal random processes the constructions are directly in the space time domain and do not depend on closed form fourier inversions the model parameters can be associated with the data s spatial and temporal structures respectively and a covariance model with a readily interpretable space time interaction parameter is fitted to wind data from ireland

spatiotemporal prediction for log gaussian cox processes space time point pattern data have become more widely available as a result of technological developments in areas such as geographic information systems we describe a flexible class of space time point processes our models are cox processes whose stochastic intensity is a space time ornstein uhlenbeck process we develop moment based methods of parameter estimation show how to predict the underlying intensity by using a markov chain monte carlo approach and illustrate the performance of our methods on a synthetic data set

kirillov s orbit method the case of discrete series representations let be a discrete series representation of a real reductive lie group g and let g be a reductive subgroup of g in this paper we give a geometric expression of the g multiplicities in g when the representation is g admissible

optimum code structures for positive optical cdma using normalized divergence maximization criterion in this letter we consider optimum code structure for positive optical code division multiple access optical cdma systems positive systems are a class of systems that operate with positive real numbers only we consider the effect of multipleaccess interference in our model and show that code design for both on off keying ook and binary ppm optical cdma systems results in the same solutions furthermore we show that a class of codes known as optical orthogonal codes oocs are the best possible positive codes in obtaining the results we define normalized divergence based on signal to multipleaccess interference ratio sir for a multiple access system in a useful manner and use it as our criterion to maximize the multiple access capability of the codes finally we demonstrate that bppm ooc can be considered as the closest counterpart of plusmn1 pseudorandom sequence in radio cdma communication systems

harmonically induced representations of solvable lie groups soit un groupe de lie g d algebre de lie g et h une polarisation pour l orbite on etudie la representation obtenue quand h est totalement complexe non necessairement positive et g resoluble

branching laws for verma modules and applications in parabolic geometry i abstract we initiate a new study of differential operators with symmetries and combine this with the study of branching laws for verma modules of reductive lie algebras by the criterion for discretely decomposable and multiplicity free restrictions of generalized verma modules t kobayashi 2012 22 we are brought to natural settings of parabolic geometries for which there exist unique equivariant differential operators to submanifolds then we apply a new method f method relying on the fourier transform to find singular vectors in generalized verma modules which significantly simplifies and generalizes many preceding works in certain cases it also determines the jordan holder series of the restriction for singular parameters the f method yields an explicit formula of such unique operators for example giving an intrinsic and new proof of juhl s conformally invariant differential operators juhl 2009 16 and its generalizations to spinor bundles this article is the first in the series and the next ones include their extension to curved cases together with more applications of the f method to various settings in parabolic geometries

branching problems in reproducing kernel spaces for a semisimple lie group g satisfying the equal rank condition the most basic family of unitary irreducible representations is the discrete series found by harish chandra in our work here we study some of the branching laws for discrete series when restricted to a subgroup h of the same type by combining classical results with recent work of kobayashi in particular we prove discrete decomposability under harish chandra s condition of cusp form on the reproducing kernel we show a relation between discrete decomposability and representing certain intertwining operators in terms of differential operators

statistical methods for regular monitoring data summary meteorological and environmental data that are collected at regular time intervals on a fixed monitoring network can be usefully studied combining ideas from multiple time series and spatial statistics particularly when there are little or no missing data this work investigates methods for modelling such data and ways of approximating the associated likelihood functions models for processes on the sphere crossed with time are emphasized especially models that are not fully symmetric in space time two approaches to obtaining such models are described the first is to consider a rotated version of fully symmetric models for which we have explicit expressions for the covariance function the second is based on a representation of space time covariance functions that is spectral in just the time domain and is shown to lead to natural partially nonparametric asymmetric models on the sphere crossed with time various models are applied to a data set of daily winds at 11 sites in ireland over 18 years spectral and space time domain diagnostic procedures are used to assess the quality of the fits the spectral in time modelling approach is shown to yield a good fit to many properties of the data and can be applied in a routine fashion relative to finding elaborate parametric models that describe the space time dependences of the data about as well

finite multiplicity theorems for induction and restriction abstract we find upper and lower bounds of the multiplicities of irreducible admissible representations of a semisimple lie group g occurring in the induced representations ind h g from irreducible representations of a closed subgroup h as corollaries we establish geometric criteria for finiteness of the dimension of hom g ind h g induction and of hom h h restriction by means of the real flag variety g p and discover that uniform boundedness property of these multiplicities is independent of real forms and characterized by means of the complex flag variety

analysis on the minimal representation of o p q abstract for the group o p q we give a new construction of its minimal unitary representation via euclidean fourier analysis this is an extension of the q 2 case where the representation is the mass zero spin zero representation realized in a hilbert space of solutions to the wave equation the group o p q acts as the mobius group of conformal transformations on r p 1 q 1 and preserves a space of solutions of the ultrahyperbolic laplace equation on r p 1 q 1 we construct in an intrinsic and natural way a hilbert space of solutions so that o p q becomes a continuous irreducible unitary representation in this hilbert space we also prove that this representation is unitarily equivalent to the representation on l2 c where c is the conical subvariety of the nilradical of a maximal parabolic subalgebra obtained by intersecting with the minimal nilpotent orbit in the lie algebra of o p q

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

wave front sets of reductive lie group representations iii abstract let g be a real reductive algebraic group and let x be a homogeneous space for g with a non zero invariant density we give an explicit description of a zariski open dense subset of the asymptotics of the tempered support of l 2 x under additional hypotheses this result remains true for vector bundle valued harmonic analysis on x these results follow from an upper bound on the wave front set of an induced lie group representation under a uniformity condition

optical orthogonal codes their bounds and new optimal constructions a v k spl lambda sub a spl lambda sub c optical orthogonal code ooc c is a family of 0 1 sequences of length v and weight k satisfying the following two correlation properties 1 spl sigma sub 0 spl les t spl les v 1 x sub t x sub t i spl les spl lambda sub a for any x x sub 0 x sub 1 spl middot spl middot spl middot x sub v 1 spl isin c and any integer i not equivalent 0 mod v and 2 spl sigma sub 0 spl les t spl les v 1 x sub t y sub t i spl les spl lambda sub b for any x x sub 0 x sub 1 spl middot spl middot spl middot x sub v 1 spl isin c y y sub 0 y sub 1 spl middot spl middot spl middot y sub v 1 spl isin c with x spl ne y and any integer i where the subscripts are taken modulo v the study of optical orthogonal codes is motivated by an application in optical code division multiple access communication systems in this paper upper bounds on the size of an optical orthogonal code are discussed several new constructions for optimal optical orthogonal codes with weight k spl ges 4 and correlation constraints spl lambda sub a spl lambda sub c 1 are described by means of optimal cyclic packings many new infinite series of such optimal optical orthogonal codes are thus produced

blur generated non separable space time models statistical space time modelling has traditionally been concerned with separable covariance functions meaning that the covariance function is a product of a purely temporal function and a purely spatial function we draw attention to a physical dispersion model which could model phenomena such as the spread of an air pollutant we show that this model has a non separable covariance function the model is well suited to a wide range of realistic problems which will be poorly fitted by separable models the model operates successively in time the spatial field at time t 1 is obtained by blurring the field at time t and adding a spatial random field the model is first introduced at discrete time steps and the limit is taken as the length of the time steps goes to 0 this gives a consistent continuous model with parameters that are interpretable in continuous space and independent of sampling intervals under certain conditions the blurring must be a gaussian smoothing kernel we also show that the model is generated by a stochastic differential equation which has been studied by several researchers previously

space time covariance functions this work considers a number of properties of space time covariance functions and how these relate to the spatial temporal interactions of the process first it examines how the smoothness away from the origin of a space time covariance function affects for example temporal correlations of spatial differences models that are not smoother away from the origin than they are at the origin such as separable models have a kind of discontinuity to certain correlations that one might wish to avoid in some circumstances smoothness away from the origin of a covariance function is shown to follow from the corresponding spectral density having derivatives with finite moments these results are used to obtain a parametric class of spectral densities whose corresponding space time covariance functions are infinitely differentiable away from the origin and that allows for essentially arbitrary and possibly different degrees of smoothness for the process in space and time second this work considers models that

unitary highest weight modules the unitary highest weight modules of a semisimple lie group g have been classified s 93 unfortunately this classification does not fit well with a more general picture of the unitary dual of g we give such a classification for g a classical group using derived functors and the philosophy of unipotent representations in particular these representations are associated to coadjoint orbits of g let 8 lie g 0 a cartan involution of 6 8 c a derived functor module is defined by the data q i u a o stable parabolic subalgebra of 0 and 17 an irreducible unitary representation of l the stabilizer of l in g considering z7 as a representation of i write z7 a rc where a e c c center of i denotes dual and rt z7l r we define a rc i rc cf definition 6 1 now fix n and vary 1 under certain restrictions on i which amount to a positivity condition on the corresponding bundle over g l a i rr is known to be irreducible and unitary as ii varies within this range the langlands parameters of a a rr vary smoothly however one cannot obtain all unitary highest weight modules under these restrictions the orbit method suggests how to obtain the remaining unitary highest weight modules the modules a rr are associated to coadjoint orbits as follows if rc is trivial we say a rr is associated to the elliptic orbit 0 g ii not all unitary highest weight modules may be obtained with rc trivial an example of which is the oscillator representation of 2n r the metapletic group we define 2n irreducible unitary highest weight modules rc 1 2 n k r is 113 oool 8708 87 7 50

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

discrete decomposability of the restriction of aq λ with respect to reductive subgroupsiii restriction of harish chandra modules and associated varieties let h g be real reductive lie groups and an irreducible unitary representation of g we introduce an algebraic formulation discretely decomposable restriction to single out the nice class of the branching problem breaking symmetry in physics in the sense that there is no continuous spectrum in the irreducible decomposition of the restriction h this paper offers basic algebraic properties of discretely decomposable restrictions especially for a reductive symmetric pair g h and for the zuckerman vogan derived functor module and proves that the sufficient condition invent math 94 is in fact necessary a finite multiplicity theorem is established for discretely decomposable modules which is in sharp contrast to known examples of the continuous spectrum an application to the restriction h of discrete series for a symmetric space g h is also given

classification of discretely decomposable aq λ with respect to reductive symmetric pairs abstract we give a classification of the triples g g q such that zuckerman s derived functor g k module a q for a stable parabolic subalgebra q is discretely decomposable with respect to a reductive symmetric pair g g the proof is based on the criterion for discretely decomposable restrictions by the first author and on berger s classification of reductive symmetric pairs

discrete decomposability of the restriction of a q λ with respect to reductive subgroups and its applications letg g be real reductive lie groups and q a stable parabolic subalgebra of lie g this paper offers a sufficient condition on g g q that the irreducible unitary representation mathop a q limits ofg with non zero continuous cohomology splits into a discrete sum of irreducible unitary representations of a subgroupg each of finite multiplicity as an application to purely analytic problems new results on discrete series are also obtained for some pseudo riemannian non symmetric spherical homogeneous spaces which fit nicely into this framework some explicit examples of a decomposition formula are also found in the cases wherea q is not necessarily a highest weight module

analysis on the minimal representation of o p q ii branching laws this is a second paper in a series devoted to the minimal unitary representation of o p q by explicit methods from conformal geometry of pseudo riemannian manifolds we find the branching law corresponding to restricting the minimal unitary representation to natural symmetric subgroups in the case of purely discrete spectrum we obtain the full spectrum and give an explicit parseval plancherel formula and in the general case we construct an infinite discrete spectrum r 2003 published by elsevier science usa msc 22e45 22e46 53a30 43a85

stochastic partial differential equation based modelling of large space time data sets summary increasingly larger data sets of processes in space and time ask for statistical models and methods that can cope with such data we show that the solution of a stochastic advection diffusion partial differential equation provides a flexible model class for spatiotemporal processes which is computationally feasible also for large data sets the gaussian process defined through the stochastic partial differential equation has in general a non separable covariance structure its parameters can be physically interpreted as explicitly modelling phenomena such as transport and diffusion that occur in many natural processes in diverse fields ranging from environmental sciences to ecology to obtain computationally efficient statistical algorithms we use spectral methods to solve the stochastic partial differential equation this has the advantage that approximation errors do not accumulate over time and that in the spectral space the computational cost grows linearly with the dimension the total computational cost of bayesian or frequentist inference being dominated by the fast fourier transform the model proposed is applied to post processing of precipitation forecasts from a numerical weather prediction model for northern switzerland in contrast with the raw forecasts from the numerical model the post processed forecasts are calibrated and quantify prediction uncertainty moreover they outperform the raw forecasts in the sense that they have a lower mean absolute error

duflo s conjecture for the branching problem to the iwasawa an subgroup of a semisimple lie group the purpose of this paper is to prove duflo s conjecture for g pi an where g is a real connected simple lie group of hermitian type and pi is a discrete series of g and an is the maximal exponential solvable subgroup for an iwasawa decomposition g kan this is essentially deduced from the following general theorem which we prove in this paper let g be a real connected semisimple lie group with lie algebra mathfrak g then a strongly elliptic g coadjoint orbit mathcal o is holomorphic if and only if text p mathcal o is an open an coadjoint orbit here text p mathfrak g longrightarrow mathfrak a oplus mathfrak n is the natural projection

distinguished nilpotent orbits kostant pairs and normalizers of lie algebras abstract a pair of lie algebras g g 1 will be called a kostant pair if g is semisimple g 1 is reductive in g and the restriction of the killing form b g to g 1 is nondegenerate we study the class of such nonsymmetric pairs and obtain some useful and new structural results we study the structure of the normalizers n g g 1 and as a consequence we obtain some corresponding worthy results about algebraic groups in particular we consider an interesting case when g 1 is a distinguished sl 2 subalgebra of g combined with the research due to v l popov we observe that the notions of self normalizing reductive subalgebras of a semisimple lie algebra and projective self dual algebraic subvarieties of the usual nilpotent cones are closely related

fock model and segal bargmann transform for minimal representations of hermitian lie groups abstract for any hermitian lie group g of tube type we construct a fock model of its minimal representation the fock space is defined on the minimal nilpotent k c orbit x in p c and the l 2 inner product involves a k bessel function as density here k g is a maximal compact subgroup and g c k c p c is a complexified cartan decomposition in this realization the space of k finite vectors consists of holomorphic polynomials on x the reproducing kernel of the fock space is calculated explicitly in terms of an i bessel function we further find an explicit formula of a generalized segal bargmann transform which intertwines the schrodinger and fock model its kernel involves the same i bessel function using the segal bargmann transform we also determine the integral kernel of the unitary inversion operator in the schrodinger model which is given by a j bessel function

dual pair correspondences for non linear covers of orthogonal groups abstract in this paper we study compact dual pair correspondences arising from smallest representations of non linear covers of odd orthogonal groups we identify representations appearing in these correspondences with subquotients of cohomologically induced representations

the simulation smoother for time series models summary recently suggested procedures for simulating from the posterior density of states given a gaussian state space time series are refined and extended we introduce and study the simulation smoother which draws from the multivariate posterior distribution of the disturbances of the model so avoiding the degeneracies inherent in state samplers the technique is important in gibbs sampling with non gaussian time series models and for performing bayesian analysis of gaussian time series

geometric analysis on small unitary representations of gl n r abstract the most degenerate unitary principal series representations i r z 2 z of g gl n r attain the minimum of the gelfand kirillov dimension among all irreducible unitary representations of g this article gives an explicit formula of the irreducible decomposition of the restriction i h branching law with respect to all symmetric pairs g h for n 2 n with n 2 the restriction i h remains irreducible for h sp n r if 0 and splits into two irreducible representations if 0 the branching law of the restriction i h is purely discrete for h gl n c consists only of continuous spectrum for h gl p r gl q r p q n and contains both discrete and continuous spectra for h o p q p q 1 our emphasis is laid on geometric analysis which arises from the restriction of small representations to various subgroups