canonical property of the miura maps between the mkp and kp hierarchies continuous and discrete apart from the case of the kp hierarchy all known miura maps between integrable hamiltonian systems had been proven to be canonical the remaining kp case is settled below as a corollary it is shown that the kp hierarchy is a factor hierarchy of the mkp one with the kernel consisting of a single scalar field a discrete mkp hierarchy and the associated miura map are constructed and the latter is shown to be canonical as well as in the continuous case this implies that one can extend the discrete kp hierarchy by a single new field into an extended discrete kp hierarchy in such a way that the extended discrete miura map mkp ekp is a canonical isomorphism

the application of generalized diagonal dominance to linear system stability theory abstract this paper shows that the well known diagonal dominance condition associated with rosenbrock s stability theorems is unnecessary and can be replaced by a less restrictive generalized diagonal dominance condition similar results are derived for generalized block diagonally dominant systems whenever possible graphical interpretations are given to facilitate the implementation of these ideas in computer aided design packages

modular forms on non linear double covers of algebraic groups we construct automorphic representations of non linear two fold covers of simply connected chevalley groups via residues of eisenstein series in the process we establish some basic results in representation theory of local groups

analysis and design of linear multivariable feedback systems in the presence of additive perturbations the paper introduces the concept of normal approximations to the analysis of uncertain linear multivariable feedback systems in order to derive a new criterion for the stability of plants subject to additive perturbations this criterion permits a complete assessment of stability margins in that it provides a measure of tolerance to uncertainty as well as providing a measure of gain phase margins the proposed approach leads to a systematic design procedure which minimizes the effects of uncertainty by attaining the condition of normality or which improves the stability margins by robust gain phase sharing the results presented in this paper are insensitive to small perturbations on the nominal plant model and provide the means for a consideration of information concerning the structure of uncertainty

unitary shimura correspondences for split real groups we find a relationship between certain complementary series representations for nonlinear coverings of split simple groups and spherical complementary series for different linear groups the main technique is barbasch s method of calculating some intertwining operators purely in terms of the weyl group

some small unipotent representations of indefinite orthogonal groups abstract we study a family of small unitary representations of indefinite orthogonal groups these representations arise as analytic continuations of the discrete series and were studied extensively by knapp in k3 we complete knapp s analysis by proving that they are irreducible in order to do so we prove that the representations are unipotent and have irreducible associated cycles in which all multiplicities are exactly one moreover we prove that the k type structure of each representation matches up to a shift the k type structure of the ring of functions on the closure a nilpotent k c orbit on p

characteristic cycles and wave front cycles of representations of reductive lie groups in the papers v1 and bv vogan and barbasch vogan attach two similar invariants to representations of a reductive lie group one by an algebraic process the other analytic they conjectured that the two invariants determine each other in a deflnite manner here we prove the conjecture our arguments involve two flner invariants the characteristic cycles of representations which are interesting in their own right to describe the invariants we consider a linear reductive lie group g and flx a maximal compact subgroup k g we denote their lie algebras by and and the complexifled lie algebras by an element of the dual space is said to be nilpotent if it corresponds to a nilpotent element of via the isomorphism induced by the killing form via the adjoint action the complexiflcation g of g acts with flnitely many orbits onn the cone of all nilpotents in like all coadjoint orbits each g orbit o n carries a distinguished complex algebraic symplectic structure the intersection ofo with i f 2 jh i i g consists of flnitely many g orbits each of which is lagrangian ino the choice of a maximal compact subgroup determines a cartan decomposition and dually the complexiflcationk of the groupk operates onn with flnitely many orbits and each of these orbits is lagrangian in the g orbit which contains it kr now let be an irreducible admissible representation of g for example an irreducible unitary representation to such a representation one can associate its harish chandra module v which is simultaneously and compatibly a module for the lie algebra and a locally flnite module for the algebraic group k then v admits k invariant good flltrations as module over the universal enveloping algebra u relative to its canonical flltered structure vogan v1 shows that the annihilator of the graded module deflnes an equidi

minimal representations geometric quantization and unitarity in the framework of geometric quantization we explicitly construct in a uniform fashion a unitary minimal representation pio of every simply connected real lie group go such that the maximal compact subgroup of go has finite center and go admits some minimal representation we obtain algebraic and analytic results about pio we give several results on the algebraic and symplectic geometry of the minimal nilpotent orbits and then quantize these results to obtain the corresponding representations we assume lie go c is simple

unitarization of a singular representation of so p q a geometric construction of a certain singular unitary representation ofsoe p q withp q even is given the representation is realized geometrically as the kernel of asoe p q invariant operator on a space of sections over a homogeneous space forsoe p q thek structure of these representations is elucidated and we demonstrate their unitarity by explicitly writing down anso p q positive definite hermitian form finally we demonstrate that the annihilator inu g of this representation is the joseph ideal which is the maximal primitive ideal associated with the minimal coadjoint orbit

lie symmetry analysis and some new exact solutions of the wu zhang equation the lie symmetry analysis and the basic similarity reductions are performed for the wu zhang equation a 2 1 dimensional nonlinear dispersive wave equation some new exact solutions generated from the similarity transformation are provided they demonstrate some new three dimensional features of a single solitary wave and two interacting solitary waves

conformally invariant differential equations and projective geometry abstract we combine harmonic analysis on certain pseudo riemannian symmetric spaces with results on conformally invariant linear and non linear differential equations this gives in many cases part of the decomposition of certain representations of the conformal group of a manifold when restricted to the isometry group

poincare series for non riemannian locally symmetric spaces abstract we initiate the spectral analysis of pseudo riemannian locally symmetric spaces g h beyond the classical cases where h is compact automorphic forms or is trivial analysis on symmetric spaces for any non riemannian reductive symmetric space x g h on which the discrete spectrum of the laplacian is nonempty and for any discrete group of isometries whose action on x is sufficiently proper we construct l 2 eigenfunctions of the laplacian on x x for an infinite set of eigenvalues these eigenfunctions are obtained as generalized poincare series i e as projections to x of sums over the orbits of eigenfunctions of the laplacian on x we prove that the poincare series we construct still converge and define nonzero l 2 functions after any small deformation of inside g for a large class of groups thus the infinite set of eigenvalues we construct is stable under small deformations this contrasts with the classical setting where the nonzero discrete spectrum varies on the teichmuller space of a compact riemann surface we actually construct joint l 2 eigenfunctions for the whole commutative algebra of invariant differential operators on x

on the restriction of zuckerman s derived functor modules a q λ to reductive subgroups in this article we study the restriction of zuckerman s derived functor g k modules aq to gfor symmetric pairs of reductive lie algebras g g when the restriction decomposes into irreducible g k modules we give an upper bound for the branching law in particular we prove that each g k module occurring in the restriction is isomorphic to a submodule of aq for a parabolic subalgebra qof g and determine their associated varieties for the proof we realize aq on complex partial flag varieties by using d modules 1 introduction our object of study is branching laws of zuckerman s derived functor modules aq with respect to symmetric pairs of real reductive lie groups let g0 be a real reductive lie group with lie algebra g0 fix a cartan invo lution of g0 so that the fixed set k0 g0 is a maximal compact subgroup of g0 write k for the complexification of k0 g0 k0 p0 for the cartan de composition with respect to and g g0 rc for the complexification similar notation will be used for other lie algebras the cohomologically induced module aq is a g k module defined for a stable parabolic subalgebra q of g and a character t he g k module aq is unitarizable under certain conditions on the parameter and therefore plays a large part in the study of the unitary dual of real reductive lie groups one of the fundamental problems in representation theory is to decompose a given representation into irreducible constituents to begin with we consider the restriction of g k modules to k or equivalently to the compact group k0 i n this case it is known that any irreducible g k module decomposes as a direct sum of irreducible representations of k and each k type occurs with finite multi plicity for aq the following formula gives an upper bound for the multiplici ties

the local converse theorem for so 2n 1 and applications in this paper we characterize irreducible generic representations of so 2n 1 k where k is a p adic field by means of twisted local gamma factors the local converse theorem as applications we prove that two irreducible generic cuspidal automorphic representations of so 2n 1 a where a is the ring of adeles of a number field are equivalent if their local components are equivalent at almost all local places the rigidity theorem and prove the local langlands reciprocity conjecture for generic supercuspidal representations of so 2n 1 k

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

une representation unipotente associee a l orbite minimale le cas deso 4 3 resume d vogan dans vo 4 a etabli l existence d un element du dual unitaire du revetement universel deso0 4 3 dont l annulateur infinitesimal est l ideal de joseph deso 4 3 ce travail propose tout d abord une autre methode permettant de retrouver ce resultat et s inspirant essentiellement d une parametrisation du dual unitaire d un groupe presque algebrique reel due a m duflo du 1 on donne ensuite une realisation explicite de dansl2 r 3 r on determine ses parametres de langlands puis on s interesse enfin a sa restriction ag2 d vogan in vo 4 has proved the existence of an element of the unitary dual of the universal covering ofso0 4 3 of which the annihilator is the joseph ideal in this work we present a new method for obtaining this result which essentially uses a parametrization of the unitary dual of a real almost semialgebraic group given by m duflo du 1 then we give an explicit realization of inl2 r 3 r we find the langlands s parameters and at last we study its restriction tog2

classification of minimal representations of real simple lie groups based on an idea by gan and savin represent theory 9 46 93 2005 we give a classification of minimal representations of connected simple real lie groups not of type a actually we prove that there exist no new minimal representations up to infinitesimal equivalence

a supersymmetric extension of the kadomtsev petviashvili hierarchy an extension of the kadomtsev petviashvili hierarchy by odd variables is given conservation laws and formal integrability are proved

a normalizing precompensator for the design of effective and reliable commutative controllers sensitivity to parameter perturbation represents the main caution regarding the use of the characteristic locus method on the design of multivariable control systems the method is not effective when the condition number of the plant eigenvector matrix is high or equivalently when the plant transfer matrix differs a great deal from normality with the view to coping with this problem it is proposed in this paper a precompensator structure and in the sequel two optimization problems are formulated and solved the first one for 2 x 2 systems aims at minimizing the eigenvector matrix condition number the second one for the general m x m case is intended to make the precompensated system as normal as possible by minimizing a defined measure of normality once the precompensated system matrix has been made close to a normal one the characteristic locus method can then be applied effectively leading to reliable control systems as far as stability in face of uncertainty is concerned

nilpotent orbits and some small unitary representations of indefinite orthogonal groups abstract for 2 m l 2 let g be a simply connected lie group with g 0 so 2m 2l 2m as lie algebra let g k p be the complexification of the usual cartan decomposition let k be the analytic subgroup with lie algebra k g 0 and let u g be the universal enveloping algebra of g this work examines the unitarity and k spectrum of representations in the analytic continuation of discrete series of g relating these properties to orbits in the nilpotent radical of a certain parabolic subalgebra of g the roots with respect to the usual compact cartan subalgebra are all ei ej with 1 i q l u be the parabolic subalgebra of g for which em em 1 contributes to u and the other simple roots contribute to l let l be the analytic subgroup of g with lie algebra l g 0 let l c int g l let 2 u be the sum of the roots contributing to u and let q l u be the parabolic subalgebra opposite to q the members of u p are nilpotent members of g the group l c acts on u p with finitely many orbits and the topological closure of each orbit is an irreducible algebraic variety if y is one of these varieties let r y be the dual coordinate ring of y this is a quotient of the algebra of symmetric tensors on u p that carries a fully reducible representation of l c for s z let s k 1 m l s 2 e k then s defines a one dimensional l l module c s extend this to a q l module by having u act by 0 and define n s 2 u u g q c s 2 u let n s 2 u be the unique irreducible quotient of n s 2 u the representations under study are s s n s 2 u and s s n s 2 u where s dim u k and s is the sth derived bernstein functor for s 2l 2 it is known that s s and that s is in the discrete series enright parthsarathy wallach and wolf showed for m s 2l 2 that s s and that s is still unitary the present paper shows that s is unitary for 0 s m 1 even though s s and it relates the k spectrum of the representations s to the representation of l c on a suitable r y with y depending on s use of a branching formula of d e littlewood allows one to obtain an explicit multiplicity formula for each k type in s the variety y is indispensable in the proof the chief tools involved are an idea of b gross and wallach a geometric interpretation of littlewood s theorem and some estimates of norms it is shown further that the natural invariant hermitian form on s does not make s unitary for s l c on any r y a final section of the paper treats in similar fashion the simply connected lie group with lie algebra g 0 so 2m 2l 2m 1 2 m l 2

principal gains and principal phases in the analysis of linear multivariable feedback systems the concepts of principal gain and principal phase are introduced for linear multivariable systems and their use in the analysis of feedback behavior is demonstrated a sufficient nyquist type stability criterion is presented in terms of these quantities and is used to characterize the robustness of the closed loop stability property when the system model is subjected to a linear perturbation either multiplicative or additive at any point in the feedback configuration the results presented are less conservative than those obtained via the small gain theorem

a new robust stability criterion for linear and non linear multivariable feedback systems earlier eigenvalue inclusion regions are replaced by new regions which are tight and thus lead to sufficient and necessary stability conditions for uncertain linear multivariate systems in the presence of unstructured perturbations the approach provides an exact and immediate assessment of gain and phase margins when applied to systems with sector bounded but unstructured non linearities the inclusion regions produce a necessary and sufficient test for the circle criterion conditions to hold in the multivariable case as such the resulting stability criterion gives the best possible nyquist type extension of the circle criterion to the multivariable case for unstructured non linearities

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

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

a supersymmetric benney hierarchy a semiintegrable system it is shown that supersymmetry can weaken integrability a supersymmetric extension of the quasiclassical limit of the kp hierarchy has flows that no longer commute between themselves but still have an infinite common set of conserved densities this extension has thus no hamiltonian structure

conservation laws and symmetries of generalized sine gordon equations we study some systems of non linear pde s eqs 1 1 below which can be regarded either as generalizations of the sine gordon equation or as two dimensional versions of the toda lattice equations we show that these systems have an infinite number of non trivial conservation laws and an infinite number of symmetries the second result is deduced from the first by a variant of the hamiltonian formalism for evolution equations we also consider some specializations of the systems

minimal unitary representation of su 2 2 and its deformations as massless conformal fields and their supersymmetric extensions we study the minimal unitary representation minrep of so 4 2 over a hilbert space of functions of three variables obtained by quantizing its quasiconformal action on a five dimensional space the minrep of so 4 2 which coincides with the minrep of su 2 2 similarly constructed corresponds to a massless conformal scalar in four space time dimensions there exists a one parameter family of deformations of the minrep of su 2 2 for positive negative integer values of the deformation parameter one obtains positive energy unitary irreducible representations corresponding to massless conformal fields transforming in 0 2 2 0 representation of the sl 2 c subgroup we construct the supersymmetric extensions of the minrep of su 2 2 and its deformations to those of su 2 2 n the minimal unitary supermultiplet of su 2 2 4 in the undeformed case simply corresponds to the massless n 4 yang mills supermultiplet in four dimensions for each given nonzero integer value of one obtains a uniq

theta distinguished representations inflation and the symmetric square l function a theta distinguished representation is a quotient of a tensor of exceptional representations where exceptional is in the sense of kazhdan and patterson we study relations between theta distinguished representations of mathrm gl n and mathrm gspin 2n 1 in the case of mathrm gspin 2n 1 or mathrm so 2n 1 exceptional or small representations were constructed by bump friedberg and ginzburg we prove a rodier type hereditary property a tempered representation tau is distinguished if and only if the representation mathrm i tau induced to mathrm gspin 2n 1 is distinguished and the multiplicity of both quotients is at most one if tau is supercuspidal and distinguished then so is the langlands quotient of mathrm i tau as a corollary we characterize supercuspidal distinguished representations in terms of the pole of the symmetric square l function at s 0

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

uniqueness and disjointness of klyachko models we show the uniqueness and disjointness of klyachko models for gln over a non archimedean local field this completes in particular the study of klyachko models on the unitary dual our local results imply a global rigidity property for the discrete automorphic spectrum of gln

mathematics of dispersive water waves a commuting hierarchy of dispersive water wave equations makes a three hamiltonian system which belongs to a general class of nonstandard integrable systems whose theory is developed the modified water wave hierarchy is a bi hamiltonian system its modification bifurcates the water wave hierarchy and the hierarchies of the korteweg de vries and the modified korteweg de vries equations as well as the classical miura map are given new representations through various specializations of nonstandard systems

the twisted symmetric square l function of gl r in this paper we consider the partial symmetric square l function ls s sym2 of an irreducible cuspidal automorphic representation of glr a twisted by a hecke character in particular we will show that the l function ls s sym2 is holomorphic for the region re s 1 1 r with the exception that if r 2 1 a pole might occur at s 1 where is the central character of our method of proof is essentially a nontrivial modification of the one by bump and ginzburg in which they considered the case 1

backlund transformations for fourth painleve hierarchies abstract backlund transformations bts for ordinary differential equations odes and in particular for hierarchies of odes are a topic of great current interest here we give an improved method of constructing bts for hierarchies of odes this approach is then applied to fourth painleve p iv hierarchies recently found by gordoa et al publ res inst math sci kyoto 37 2001 327 347 we show how the known pattern of bts for p iv can be extended to our p iv hierarchies remarkably the bts required to do this are precisely the miura maps of the dispersive water wave hierarchy we also obtain the important result that the fourth painleve equation has only one nontrivial fundamental bt and not two such as is frequently stated

application of exp function method to the whitham broer kaup shallow water model using symbolic computation in this letter the exp function method is applied to the whitham broer kaup shallow water model with the help of symbolic computation several kinds of new solitary wave solutions are formally derived

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

a lie algebraic setting for miura maps related to an energy dependent linear problem we study the affine semi direct product vect s 1 s 1 1 and show how the coadjoint action on its extended dual leads immediately to a generalisation of the miura map and also to the existence of several compatible hamiltonian structures

the choice and use of normal approximations to transfer function matrices of multivariable control systems approximating the transfer function matrix of a multivariable system by a normal matrix provides a reliable way for assessing the stability of feedback systems in the presence of uncertainty or non linearities the appropriate stability results however must take into account the error in the approximation and if this is large the stability assessment becomes conservative the present paper proposes techniques for obtaining improved normal approximations and presents results which for the 2 2 case minimize the error of approximation the approach is amenable to a numerically robust implementation on a digital computer and has obvious application in the analysis of feedback stability

the smallest representations of nonlinear covers of odd orthogonal groups we construct the smallest genuine representations of a nonlinear cover of the group rm so 0 p q where p q is odd we determine correspondences of infinitesimal characters arising from restricting the smallest representations to dual pairs frak s frak o p a oplus frak s frak o b where a b q

hermitian symmetric spaces and their unitary highest weight modules abstract the purpose of this article is to determine the set of unitarizable highest weight modules corresponding to hermitian symmetric spaces of the noncompact type the major step is that of proving unitarity at the last possible place with this established the description of the full set of unitarizable highest weight modules follows by a straightforward tensor product argument combined with the main ingredients of the proof of the key theorem bernstein gelfand gelfand and a diagramatic representation of the set of positive noncompact roots

discrete components in restriction of unitary representations of rank one semisimple lie groups we consider spherical principal series representations of the semisimple lie group of rank one g so n 1 k g so n 1 k k r c hk r c h there is a family of unitarizable representations of g for in an interval on rr the so called complementary series and subquotients or subrepresentations of g for being negative integers we consider the restriction of g g under the subgroup h so n 1 1 k h so n 1 1 k we prove the appearing of discrete components the corresponding results for the exceptional lie group f4 20 f4 20 and its subgroup spin0 8 1 spin0 8 1 are also obtained

coupled kdv equations with multi hamiltonian structures abstract we present a large class of systems of n equations which possess n 1 purely differential compatible hamiltonian structures our generic equations are isospectral to n 1 0 e i 2 i 2 n 1 0 i 2 i 2 n but our class also includes degenerate nondispersive systems which are unrelated to this linear problem embedded in this class for each n there are n distinct coupled kdv systems when n 2 our class includes 3 known equations dispersive water waves ito s equation and reduced benney s equations these equations are thus tri hamiltonian we also present 2 examples of 3 component quadri hamiltonian systems which generalise the above mentioned dispersive and nondispersive water waves

unipotent representations attached to spherical nilpotent orbits a coadjoint nilpotent orbit inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i of a complex semisimple lie group g is said to be spherical if it contains an open orbit of a borel subgroup we determine when and how to attach unitary representations to such an orbit for the real orthogonal and symplectic groups our results actually extend to a larger class of nilpotent coadjoint orbits

split quaternionic analysis and separation of the series for sl 2 r and sl 2 c sl 2 r abstract we extend our previous study of quaternionic analysis based on representation theory to the case of split quaternions h r the special role of the unit sphere in the classical quaternions h identified with the group su 2 is now played by the group sl 2 r realized by the unit quaternions in h r as in the previous work we use an analogue of the cayley transform to relate the analysis on sl 2 r to the analysis on the imaginary lobachevski space sl 2 c sl 2 r identified with the one sheeted hyperboloid in the minkowski space m we study the counterparts of cauchy fueter and poisson formulas on h r and m and show that they solve the problem of separation of the discrete and continuous series the continuous series component on h r gives rise to the minimal representation of the conformal group sl 4 r while the discrete series on m provides its k types realized in a natural polynomial basis we also obtain a surprising formula for the plancherel measure of sl 2 r in terms of the poisson type integral on the split quaternions h r finally we show that the massless singular functions of four dimensional quantum field theory are nothing but the kernels of projectors onto the discrete and continuous series on the imaginary lobachevski space sl 2 c sl 2 r our results once again reveal the central role of the minkowski space in quaternionic and split quaternionic analysis as well as a deep connection between split quaternionic analysis and the four dimensional quantum field theory

on a camassa holm type equation with two dependent variables we consider a generalization of the camassa holm ch equation with two dependent variables called ch2 introduced in a paper by liu and zhang liu s q and zhang y 2005 j geom phys 54 427 53 we briefly provide an alternative derivation of it based on the theory of hamiltonian structures on the dual of a lie algebra the lie algebra involved here is the same algebra as underlies the nls hierarchy we study the structural properties of the hierarchy defined by the ch2 equation within the bi hamiltonian theory of integrable pdes and provide its lax representation then we explicitly discuss how to construct classes of solutions both of peakon and of algebro geometrical type finally we sketch the construction of a class of singular solutions defined by setting to zero one of the two dependent variables

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

generic transfer for general spin groups we prove langlands functoriality for the generic spectrum of general spin groups both odd and even contrary to other recent instances of functoriality our resulting automorphic representations on the general linear group are not self dual together with cases of classical groups this completes the list of cases of split reductive groups whose l groups have classical derived groups the important transfer from gsp4 to gl4 follows from our result as a special case

self dual yang mills equations in split signature abstract we study the self dual yang mills equations in split signature we give a special solution called the basic split instanton and describe the adhm construction in the split signature moreover a split version of t hooft ansatz is described

achieving diagonal dominance there are many occasions when one would like to see whether a transfer function can be made diagonally dominant there is the obvious and important case of system design using rosenbrock s methods but one also wants to use diagonal dominance in a different manner in robustness studies and in several nonlinear system analysis techniques commonly the compensators or similarity transforms used to achieve dominance are designed in an ad hoc way this paper shows how in the particular case of diagonal compensators or similarity transforms there is a very easy way based on perron frobenius theory of obtaining optimal reduction in the off diagonal elements

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

quaternionic analysis representation theory and physics we develop quaternionic analysis using as a guiding principle representation theory of various real forms of the conformal group we first review the cauchy fueter and poisson formulas and explain their representation theoretic meaning the requirement of unitarity of representations leads us to the extensions of these formulas in the minkowski space which can be viewed as another real form of quaternions representation theory also suggests a quaternionic version of the cauchy formula for the second order pole remarkably the derivative appearing in the complex case is replaced by the maxwell equations in the quaternionic counterpart we also uncover the connection between quaternionic analysis and various structures in quantum mechanics and quantum field theory such as the spectrum of the hydrogen atom polarization of vacuum one loop feynman integrals we also make some further conjectures the main goal of this and our subsequent paper is to revive quaternionic analysis and to show profound relations between quaternionic analysis representation theory and four dimensional physics

minimal representations of e6 e7 and e8 and the generalized capelli identity abstract we explicitly construct in a uniform fashion the unique minimal and spherical representation pi0 of the split real lie group of exceptional type e6 e7 or e8 we obtain several algebraic and analytic results about pi0

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

normality scaling and eigenvalue inclusion regions one approach to establishing eigenvalue inclusion regions for robust stability analysis of uncertain linear multivariable feedback systems involves replacing the nominal plant matrix by an approximating normal one this approach incurs additional uncertainty but has the advantage that the resulting regions are insensitive to small perturbations these regions are obviously conservative and it is shown in this paper how this conservatism may be reduced by utilizing the eigenvalue preserving operation of similarity scaling

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

self dual representations of division algebras and weil groups a contrast irreducible selfdual representations of any group fall into two classes those which carry a symmetric bilinear form and the others which carry an alternating bilinear form the langlands correspondence which matches the irreducible representations of the weil group of a local field k of dimension n with the irreducible representations of the invertible elements of a division algebra d over k of index n takes selfdual representations to selfdual representations in this paper we use global methods to study how the langlands correspondence behaves relative to this distinction among selfdual representations we prove in particular that for n even is symplectic if and only if is orthogonal more generally we treat the case of gl m b for b a division algebra over k of index r and n mr

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

lifting automorphic representations on the double covers of orthogonal groups suppose that g and h are connected reductive groups over a number field f and that an l homomorphism lg lh is given the langlands functoriality conjecture predicts the existence of a map from the automorphic representations of g a to those of h a if the adelic points of the algebraic groups g h are replaced by their metaplectic covers one may hope to specify an analogue of the lgroup depending on the cover and then one may hope to construct an analogous correspondence in this paper we construct such a correspondence for the double cover of the split special orthogonal groups

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

twisted endoscopy and reducibility of induced representations for p adic groups 1 introduction this is the first in a series of papers in which we study the reducibility of representations induced from discrete series representations of the levi factors ofmaximal parabolic subgroups of p adic groups on the unitary axis the problem is equivalent to determining certain local langlands l functions which are of arithmetic significance by themselves one of the main aims of this paper is to interpret our results in the direction of the parametrization problem by means of the theory of twisted endoscopy for which our method seems to be very suitable when the representation is supercuspidal we also study the reducibility of the induced representations off the unitary axis let f be a p adic field of characteristic zero fix a positive integer n 1 and let g be either of the groups sp2n s02n or s02n l in all three cases there is a conjugacy class of maximal parabolic subgroups which has gl as its levi factor let p mn be the standard parabolic subgroup ofg in this conjugacy class then m gl let tr be a discrete series representation of m gln f and given s c let i s tr be the representation of g g f induced from tr r idet ls let i tr i 0 tr

block compatible metaplectic cocycles has cardinality r 1 let g be the f rational points of a simple chevalley group defined over f in his thesis matsumoto 5 gave a beautiful construction for the metaplectic cover g of g a central extension of g by r f whose existence is intimately connected with the deep properties of the r th order hilbert symbol f f f r f metaplectic groups figure prominently in the study of number theory representation theory and physics arising naturally in the theory of theta functions dual pair correspondences weil representations and spin geometry in this paper we study the class of central extensions of a simple chevalley group over an arbitrary infinite field of which the metaplectic groups form an important subclass metaplectic groups were constructed quite explicitly in weil s memoir 10 in the case that g is symplectic in 3 and 4 kubota gave the construction of the r fold metaplectic cover of gl2 f moreover he described an explicit 2 cocycle k on gl2 f that represents the second cohomology class of the extension cf 3 corollary 8 which makes it possible to deal quite rapidly with many concrete problems in this setting steinberg 9 and moore 7 considered the algebraic problem of determining the central extensions of a simple chevelley group over an arbitrary field they were also led to the metaplectic groups this line of investigation was completed by matsumoto 5 whose work forms the foundation of the present paper to summarize our results let f be an infinite field g the f rational points of a simple chevalley group defined over f a an abelian group and c f f a a steinberg symbol that is bilinear if g is not symplectic cf 1 in this paper we describe an explicit 2 cocycle g in z g a that represents the cohomology class in h g a of the central extension g of g by a constructed by matsumoto 5

factorisation of energy dependent schrodinger operators miura maps and modified systems we consider the energy dependent schrodinger operator mathbb l sum limits i 0 n lambda i varepsilon i partial 2 u i which we have previously shown to be associated with multi hamiltonian structures 2 in this paper we use an unusual form of the lax approach to derive by asingle construction the time evolutions of the eigenfunctions of mathbb l the associated hamiltonian operators and the hamiltonian functionals we then generalise the well known factorisation of standard lax operators to the case of energy dependent operators the simple product of linear factors is replaced by a dependent quadratic form we thus generalise the resulting construction of miura maps and modified equations we show that for some of our systems there exists a sequence ofn such modifications therth modification possessing n r 1 hamiltonian structures

eigenspaces of the laplacian on hyperbolic spaces composition series and integral transforms abstract let x be a projective real complex or quaternion hyperbolic space realized as the pseudo riemannian symmetric space x g h with g o p q u p q or sp p q these are the classical isotropic symmetric spaces let be the g invariant laplace beltrami operator on x a complete description by k types for each c of all closed g invariant subspaces of the eigenspace f c x f f is given the eigenspace representations are compared with principal series representations using poisson transformations similar results are obtained also for the exceptional isotropic symmetric space the langlands parameters of the spherical discrete series representations are determined