description of infinite orbits on multiple projective spaces abstract let g be the general linear group of the degree n 2 over an algebraically closed field k in this article we describe the orbit decomposition of a product of copies of p n 1 k under the diagonal action of g we also construct representatives of orbits if m 4 the number of orbits is infinite and we give a description of them

on special bessel periods and the gross prasad conjecture for mathrm so 2n 1 times mathrm so 2 in this paper we investigate one instance of the global gross prasad conjecture our main result proves that if an irreducible cuspidal automorphic representation pi of an odd dimensional special orthogonal group whose local component pi w at some finite place w is generic admits the special bessel model corresponding to a quadratic extension e over a base field f then the central l value l 1 2 pi l 1 2 pi times chi e does not vanish here chi e denotes the quadratic character of mathbb a f times corresponding to e as an application we obtain the equivalence between the non vanishing of the special bessel period and that of the corresponding central l value when pi is associated to a full modular holomorphic siegel cusp form of degree two which is a hecke eigenform and e is an imaginary quadratic extension of mathbb q

multiplicity one theorems the archimedean case let g be one of the classical lie groups gln 1 r gln 1 c u p q 1 o p q 1 on 1 c so p q 1 son 1 c and let g 0 be respectively the subgroup gln r gln c u p q o p q on c so p q son c embedded in g in the standard way we show that every irreducible casselman wallach representation of g 0 occurs with multiplicity at most one in every irreducible casselman wallach representation of g similar results are proved for the jacobi groups gln r nh2n 1 r gln c n h2n 1 c u p q nh2p 2q 1 r sp2n r nh2n 1 r sp2n c nh2n 1 c with their respective subgroups gln r gln c u p q sp2n r and sp2n c

a class of parabolic k subgroups associated with symmetric k varieties let g be a connected reductive algebraic group defined over a field k of characteristic not 2 an involution of g defined over k h a k open subgroup of the fixed point group of gk resp hk the set of k rational points of g resp h and gk hk the corresponding symmetric k variety a representation induced from a parabolic k subgroup of g generically contributes to the plancherel decomposition of l2 gk hk if and only if the parabolic k subgroup is split so for a study of these induced representations a detailed description of the hk conjucagy classes of these split parabolic k subgroups is needed in this paper we give a description of these conjugacy classes for general symmetric kvarieties this description can be refined to give a more detailed description in a number of cases these results are of importance for studying representations for real and p adic symmetric k varieties

spherical unitary highest weight representations in this paper we give an almost complete classification of the h spherical unitary highest weight representations of a hermitian lie group g where g h is a symmetric space of cayley type

median structures on asymptotic cones and homomorphisms into mapping class groups the main goal of this paper is a detailed study of asymptotic cones of the mapping class groups in particular we prove that every asymptotic cone of a mapping class group has a bi lipschitz equivariant embedding into a product of real trees sending limits of hierarchy paths onto geodesics and with image a median subspace one of the applications is that a group with kazhdan s property t can have only finitely many pairwise non conjugate homomorphisms into a mapping class group we also give a new proof of the rank conjecture of brock and farb previously proved by behrstock and minsky and independently by hamenstaedt

power law decay and the ergodic nonergodic transition in simple fluids it is well known that mode coupling theory mct leads to a two step power law time decay in dense simple fluids we show that much of the mathematical machinery used in the mct analysis can be taken over to the analysis of the systematic theory developed in the fundamental theory of statistical particle dynamics mazenko in phys rev e 81 6 061102 2010 we show how the power law exponents can be computed in the second order approximation where we treat hard sphere fluids with statics described by the percus yevick solution

the schwartz space of a general semisimple lie group i wave packets of eisenstein integrals in previous papers we studied matrix coefficients of relative discrete series representations for a connected simple group with infinite center and showed how to combine them into wave packets along the continuous parameter to construct schwartz functions on the group here we construct schwartz class wave packets of matrix coefficients corresponding to the various induced series of tempered representations for any connected reductive lie group g in the special case that g p m these are exactly the wave packets of relative discrete series matrix coefficients constructed before

relations d orthogonalite de schur generalisees pour les espaces symetriques reductifs resume nous introduisons une semi norme sur un espace de fonctions c sur un espace symetrique reductif g h nous montrons que les fonctions c k finies propres sous l action de l algebre des operateurs differentiels g invariants et pour lesquelles cette semi norme est finie de meme que pour ses derivees par les elements de l algebre enveloppante de l algebre de lie de g sont temperees reciproquement on explicite cette semi norme pour les fonctions temperees k finies qui sont propres pour un caractere regulier de l algebre des operateurs differentiels g invariants l application de ces resultats aux integrales d eisenstein permet d etablir des relations d orthogonalite de schur generalisees notre travail generalise et precise des resultats de h midorikawa 1988 adv stud pure math 14 257 287 pour les groupes copyright 2001 academic press we introduce a seminorm on a space of c functions on a reductive symmetric space g h we show that the c k finite eigenfunctions of the algebra of all g invariant differential operators on g h for which this semi norm is finite as well as for their derivatives from the left are tempered reciprocally we make explicit this seminorm for the k finite tempered eigenfunctions with a regular character applying those result to eisenstein integrals we obtain general schur orthogonality relations our work generalizes and specifies the results of h midorikawa 1988 adv stud pure math 14 257 287 in the group case

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

orbits and invariants associated with a pair of commuting involutions let be commuting involutions of the connected reductive algebraic group g where and g are defined over a usually algebraically closed field k char k 2 we have fixed point groups h g and k g and an action h k g g where h k g hgk 1 h h k k g g let g h k denote spec o g h k the categorical quotient let a be maximal among subtori s of g such that s s s 1 for all s s there is the associated weyl group w wh k a we show the inclusion a g induces an isomorphism a w g h k in particular the closed h k orbits are precisely those which intersect a the fibers of g g h k are the same as those occurring in certain associated symmetric varieties in particular the fibers consist of finitely many orbits we investigate the structure of w and its relation to other naturally occurring weyl groups and to the action of on the a weight spaces of the relation of the orbit type stratifications of a w and g h k along the way we simplify some of richardson s proofs for the symmetric case and at the end we quickly recover results of berger flensted jensen hoogenboom and matsuki ber57 fj78 hoo84 mat97 for the case 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

generalized bernstein reznikov integrals we find a closed formula for the triple integral on spheres in mathbb r 2n times mathbb r 2n times mathbb r 2n whose kernel is given by powers of the standard symplectic form this gives a new proof to the bernstein reznikov integral formula in the n 1 case our method also applies for linear and conformal structures

symmetry breaking operators for real reductive groups of rank one abstract for a pair of real reductive groups g g we consider the space hom g g of intertwining operators between spherical principal series representations of g and of g also called symmetry breaking operators restricting to those pairs g g where dim hom g g and g and g are of real rank one we classify all symmetry breaking operators explicitly in terms of their distribution kernels this generalizes previous work by kobayashi speh for g g o 1 n 1 o 1 n to the reductive pairs g g u 1 n 1 f u 1 m 1 f f with f c h o and f u n m f in most cases all symmetry breaking operators can be constructed using one meromorphic family of distributions whose poles and residues we describe in detail in addition to this family there may occur some sporadic symmetry breaking operators which we determine explicitly

on the classification of k involutions let g be a connected reductive algebraic group defined over a field k of characteristic not 2 an involution of g defined over k h a k open subgroup of the fixed point group of and gk resp hk the set of k rational points of g resp h the variety gk hk is called a symmetric k variety these varieties occur in many problems in representation theory geometry and singularity theory over the last few decades the representation theory of these varieties has been extensively studied for k r and c as most of the work in these two cases was completed the study of the representation theory over other fields like local fields and finite fields began the representations of a homogeneous space usually depend heavily on the fine structure of the homogeneous space like the restricted root systems with weyl groups etc thus it is essential to study first this structure and the related geometry in this paper we give a characterization of the isomorphy classes of these symmetric k varieties together with their fine structure of restricted root systems and also a classification of this fine structure for the real numbers p adic numbers finite fields and number fields

rank 1 phenomena for mapping class groups we prove that every element of the mapping class group g has linear growth confirming a conjecture of n ivanov and that g is not boundedly generated we also provide restrictions on linear representations of g and its finite index subgroups

asymptotic behavior of matrix coefficients of admissible representations introduction 869 1 generalities on reductive groups 873 2 the infinitesimal cartan decomposition 876 3 the t radial components 880 4 differential equations satisfied by sphereical functions 885 5 asymptotic behavior of spherical functions on a 888 6 asymptotic behavior of spherical functions along the walls of a 892 7 leading characters and growth estimates on the group 898 8 admissible representations and their matrix coefficients 906 appendix 917

kinetic equations governing smoluchowski dynamics in equilibrium we continue our study of the statistical properties of particles in equilibrium obeying smoluchowski dynamics we show that the system is governed by a kinetic equation of the memory function form and that the memory function is given by one of the self energies available via perturbation theory as introduced in previous work we determine the memory function explicitly to second order in an expansion in a pseudopotential the method we use allows for a straightforward computation of corrections via a formal expansion and we therefore view it as an improvement over the conventional mode coupling theory mct formalism where it is not clear how to make systematic corrections in addition the formalism we have introduced is flexible enough to allow for a wide array of different approximation schemes including density expansions the convergence criteria for our formal series are not worked out here but the second order equation that we derive is promising in the sense that it leads to analytic and numerical results consistent with expectations from computer simulations of the hard sphere system in addition to replicating the desired features from conventional mct e g a two step decay these particular solutions will be discussed in forthcoming work

representations characters and spherical functions associated to causal symmetric spaces abstract in this paper we study spherical unitary highest weight representations associated to a compactly causal symmetric space and use the results to prove estimates for the corresponding spherical functions of the c dual non compactly causal symmetric space such estimates turn out to be useful in determining the spectra of the associated weighted bergman spaces

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

representations of reductive groups distinguished by symmetric subgroups let g be a complex connected reductive group g theta be its fixed point subgroup under a galois involution theta and h be an open subgroup of g theta we show that any h distinguished representation pi satisfies 1 pi theta simeq tilde pi where tilde pi is the contragredient representation and pi theta is the twist of pi under theta 2 mathrm dim mathbb c left pi right h le left b backslash g h right where b is a borel subgroup of g by proving the first statement we give a partial answer to a conjecture by prasad and lapid

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

existence of klyachko models for gl n r and gl n c we prove that any irreducible unitary representation of gl n r and gl n c admits an equivariant linear form with respect to one of the subgroups considered by klyachko

quelques proprietes des representations spheriques pour les espaces symetriques reductifs let gh be a reductive symmetric space and suppose v is an admissible g k module of finite length possessing a linear functional t vsu which is fixed by h and h k we prove that v can be mapped equivariantly into c gh such that t becomes the pull back of the dirac measure at the origin essential in the proof is the fact that the formal power series of certain matrix coefficients of v satisfy a system of differential equations with regular singularities

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

orbital symmetric spaces and finite multiplicity abstract it is proven that any algebraic symmetric space has finite multiplicity the multiplicity is then shown to be bounded provided that is the case for reductive symmetric spaces a widely believed but as yet unproven fact the method of proof involves an amalgamation of the mackey machine and the orbit method a necessary and sufficient orbital condition is also derived for the quasi regular representation of a symmetric space to be irreducible

espace de schwartz pour la transformation de fourier hypergeometrique resume let r be a reduced root system in an euclidean vector space e w the weyl group of r one determines the hypergeometric fourier transform cf e opdam cuspidal hypergeometric functions preprint of the related schwartz space of w invariant functions on e introduced by tinfou the answer is very natural and quite similar to results of harish chandra the proof requires a theory of the constant term for w invariant functions satisfying the hypergeometric system this is analogous to harish chandra s theory once one has realized that simple difference operators play here the role of some elements of the unipotent radical of a parabolic subalgebra also the fact that the parameter of cuspidal hypergeometric functions might be singular introduces new difficulties this theory being established one can take advantage of the techniques we used for real reductive symmetric spaces cf the introduction to the article by the author formule de plancherel pour les espaces symetriques reductifs ann math 147 1998 417 452 especially the truncation and its consequences

smooth transfer of kloosterman integrals the archimedean case we establish the existence of a transfer which is compatible with kloosterman integrals between schwartz functions on rm gl n bbb r and schwartz functions on the variety of non degenerate hermitian forms namely we consider an integral of a schwartz function on rm gl n bbb r along the orbits of the two sided action of the groups of upper and lower unipotent matrices twisted by a non degenerate character this gives a smooth function on the torus we prove that the space of all functions obtained in such a way coincides with the space that is constructed analogously when rm gl n bbb r is replaced with the variety of non degenerate hermitian forms we also obtain similar results for frak gl n bbb r the non archimedean case was done by h jacquet it duke math j 2003 and our proof is based on the ideas of this work however we have to face additional difficulties that appear only in the archimedean case those results are crucial for the comparison of the kuznetsov trace formula and the relative trace formula of rm gl n with respect to the maximal unipotent subgroup and the unitary group as done by h jacquet and by b feigon e lapid and o offen

field theoretic formulation of kinetic theory basic development we show how kinetic theory the statistics of classical particles obeying newtonian dynamics can be formulated as a field theory the field theory can be organized to produce a self consistent perturbation theory expansion in an effective interaction potential the need for a self consistent approach is suggested by our interest in investigating ergodic nonergodic transitions in dense fluids the formal structure we develop has been implemented in detail for the simpler case of smoluchowski dynamics one aspect of the approach is the identification of a core problem spanned by the variables the number density and b a response density in this paper we set up the perturbation theory expansion with explicit development at zeroth and first order we also determine all of the cumulants in the noninteracting limit among the core variables and b

invariant syzygies and semisimple groups the purpose of this paper is to explore some intriguing connections between lie theory relative homological algebra invariant theory and representations of finite dimensional algebras in the process some results of hilbert 20 cartan and eilenberg 7 hochschild 21 gelfand 171 and others are generalized and a question of zl is partly answered in theorem 1 5 duflo seems to have stronger unpublished results on this problem sections 1 to 3 take place in a fairly general setting whereas section 4 focuses on semisimple lie groups the problem considered here goes back in some sense to hilbert 20 this paper was published 102 years ago to explain this let me start by singling out two out of many fundamental contributions of the great mathematician the first one is invariant theory namely the study of the algebra sv n consisting of those polynomial functions on a finite dimensional vector space v which are invariant under the action of a group h of linear transformations the second one is the so called syzygy theorem which asserts in today s language that the global dimension of the full symmetric algebra sv is dim v this prompts the question is there an algebra d attached to v as naturally as sv is on which h acts by automorphisms such that the global dimension of the algebra b dh of invariants is dim v the most natural candidate is

kazhdan and haagerup properties from the median viewpoint abstract we prove the existence of a close connection between spaces with measured walls and median metric spaces we then relate properties t and haagerup a t menability to actions on median spaces and on spaces with measured walls this allows us to explore the relationship between the classical properties t and haagerup and their versions using affine isometric actions on l p spaces it also allows us to answer an open problem on a dynamical characterization of property t generalizing results of robertson and steger

multiplicity one theorems in the local characteristic 0 non archimedean case we consider distributions on gl n 1 which are invariant under the adjoint action of gl n we prove that such distributions are invariant by transposition this implies multiplicity at most one for restrictions from gl n 1 to gl n similar theorems are obtained for orthogonal or unitary groups

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

the solution to a conjecture of tits on the subgroup generated by the squares of the generators of an artin group let a be an artin group with standard generating set s s s tits conjectured that the only relations in a amongst the squares of the generators are consequences of the obvious ones namely that s 2 and t 2 commute whenever s and t commute for s t s in this paper we prove tits conjecture for all artin groups in fact given a number m s 2 for each s s we show that the elements t s s ms s s generate a subgroup that has a finite presentation in which the only defining relations are that t s and t t commute if s and t commute

fundamental theory of statistical particle dynamics we present a fundamental theory for the kinetics of systems of classical particles the theory represents a unification of kinetic theory brownian motion and field theory it is self consistent and is the dynamic generalization of the functional theory of fluids in equilibrium this gives one a powerful tool for investigating the existence of ergodic nonergodic transitions near the liquid glass transition

superrigidity of actions on finite rank median spaces abstract finite rank median spaces are a simultaneous generalisation of finite dimensional cat 0 cube complexes and real trees if is an irreducible lattice in a product of rank one simple lie groups we show that every action of on a complete finite rank median space has a global fixed point this is in sharp contrast with the behaviour of actions on infinite rank median spaces where even proper cocompact actions can arise the fixed point property is obtained as corollary to a superrigidity result the latter holds for irreducible lattices in arbitrary products of compactly generated topological groups we exploit roller compactifications of median spaces these were introduced in 48 and generalise a well known construction in the case of cube complexes we show that the haagerup cocycle provides a reduced 1 cohomology class that detects group actions with a finite orbit in the roller compactification this is new even for cat 0 cube complexes and has interesting consequences involving shalom s property h f d for instance in gromov s density model random groups at low density do not have h f d

tight geodesics in the curve complex the curve graph mathcal g associated to a compact surface is the 1 skeleton of the curve complex defined by harvey masur and minsky showed that this graph is hyperbolic and defined the notion of a tight geodesic therein we prove some finiteness results for such geodesics for example we show that a slice of the union of tight geodesics between any pair of points has cardinality bounded purely in terms of the topological type of we deduce some consequences for the action of the mapping class group on mathcal g in particular we show that it satisfies an acylindricity condition and that the stable lengths of pseudoanosov elements are rational with bounded denominator

normalizations of eisenstein integrals for reductive symmetric spaces abstract we construct minimal eisenstein integrals for a reductive symmetric space g h as matrix coefficients of the minimal principal series of g the eisenstein integrals thus obtained include those from the minimal principal series in addition we obtain related eisenstein integrals but with different normalizations specialized to the case of the group this wider class includes harish chandra s minimal eisenstein integrals

the spectrum of kleinian manifolds abstract we obtain the plancherel theorem for l 2 g where g is a classical simple lie group of real rank one and g is convex cocompact discrete subgroup and deduce its consequences for the spectrum of locally invariant differential operators on bundles over kleinian manifolds as the main tool we develop a geometric version of scattering theory which in particular contains the meromorphic continuation of the eisenstein series for this situation the central role played by invariant distribution sections supported on the limit set is emphasized

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

thick metric spaces relative hyperbolicity and quasi isometric rigidity we study the geometry of non relatively hyperbolic groups generalizing a result of schwartz any quasi isometric image of a non relatively hyperbolic space in a relatively hyperbolic space is contained in a bounded neighborhood of a single peripheral subgroup this implies that a group being relatively hyperbolic with non relatively hyperbolic peripheral subgroups is a quasi isometry invariant as an application artin groups are relatively hyperbolic if and only if freely decomposable we also introduce a new quasi isometry invariant of metric spaces called metrically thick which is sufficient for a metric space to be non hyperbolic relative to any non trivial collection of subsets thick finitely generated groups include mapping class groups of most surfaces outer automorphism groups of most free groups certain artin groups and others non uniform lattices in higher rank semisimple lie groups are thick and hence non relatively hyperbolic in contrast with rank one which provided the motivating examples of relatively hyperbolic groups mapping class groups are the first examples of non relatively hyperbolic groups having cut points in any asymptotic cone resolving several questions of drutu and sapir about the structure of relatively hyperbolic groups outside of group theory teichmuller spaces for surfaces of sufficiently large complexity are thick with respect to the weil peterson metric in contrast with brock farb s hyperbolicity result in low complexity

cohomologically induced distinguished representations and cohomological test vectors let g be a real reductive group and let be a character of a reductive subgroup h of g we construct invariant linear functionals on certain cohomologically induced representations of g and we show that these linear functionals do not vanish on the bottom layers applying this construction we prove two archimedean nonvanishing hypotheses which are vital to the arithmetic study of special values of certain l functions via modular symbols

volume growth temperedness and integrability of matrix coefficients on a real spherical space abstract we apply the local structure theorem from 13 and the polar decomposition of 12 to a real spherical space z g h and control the volume growth on z we define the harish chandra schwartz space on z we give a geometric criterion to ensure l p integrability of matrix coefficients on z

dimension and rank for mapping class groups we study the large scale geometry of the mapping class group mcg s our main result is that for any asymptotic cone of mcg s the maximal dimension of locally compact subsets coincides with the maximal rank of free abelian subgroups of mcg s an application is a proof of brock farb s rank conjecture which asserts that mcg s has quasi flats of dimension n if and only if it has a rank n free abelian subgroup hamenstadt has also given a proof of this conjecture using different methods we also compute the maximum dimension of quasi flats in teichmuller space with the weil petersson metric

matrix coefficients of the middle discrete series of su 2 2 ii the explicit asymptotic expansion abstract we obtain the explicit asymptotic expansion of the matrix coefficients of the middle discrete series representation of su 2 2 by means of degenerate gaussian hypergeometric series

groups acting on tree graded spaces and splittings of relatively hyperbolic groups abstract tree graded spaces are generalizations of r trees they appear as asymptotic cones of groups when the cones have cut points since many questions about endomorphisms and automorphisms of groups solving equations over groups studying embeddings of a group into another group etc lead to actions of groups on the asymptotic cones it is natural to consider actions of groups on tree graded spaces we develop a theory of such actions which generalizes the well known theory of groups acting on r trees as applications of our theory we describe in particular relatively hyperbolic groups with infinite groups of outer automorphisms and co hopfian relatively hyperbolic groups

the classification of kleinian surface groups i models and bounds we give the first part of a proof of thurston s ending lamination conjecture in this part we show how to construct from the end invariants of a kleinian surface group a lipschitz model for the thick part of the corresponding hyperbolic manifold this enables us to describe the topological structure of the thick part and to give a priori geometric bounds

quasi flats and rigidity in higher rank symmetric spaces in this paper we use elementary geometrical and topological methods to study some questions about the coarse geometry of symmetric spaces our results are powerful enough to apply to noncocompact lattices in higher rank symmetric spaces such as sl n 2 n 3 theorem 8 1 is a major step towards the proof of quasiisometric rigidity of such lattices e we also give a different and effective proof of the theorem of kleiner leeb on the quasi isometric rigidity of higher rank symmetric spaces kl

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

divergence in lattices in semisimple lie groups and graphs of groups divergence functions of a metric space estimate the length of a path connecting two points a b at distance 3 s is a finite set of valuations of a number field k including all infinite valuations and o s is the corresponding ring of s integers

kloosterman identities over a quadratic extension we prove an identity of kloosterman integrals which is the fundamental lemma of a relative trace formula for the general linear group in n variables

holomorphic h spherical distribution vectors in principal series representations let g h be a semisimple symmetric space the main tool to embed a principal series representation of g into l2 g h are the h invariant distribution vectors if g h is a non compactly causal symmetric space then g h can be realized as a boundary component of the complex crown in this article we construct a minimal g invariant subdomain h of with g h as shilov boundary let be a spherical principal series representation of g we show that the space of h invariant distribution vectors of which admit a holomorphic extension to h is one dimensional furthermore we give a spectral definition of a hardy space corresponding to those distribution vectors in particular we achieve a geometric realization of a multiplicity free subspace of l2 g h mc in a space of holomorphic functions

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

erratum to hausdorffness for lie algebra homology of schwartz spaces and applications to the comparison conjecture let h be a real algebraic group acting equivariantly with finitely many orbits on a real algebraic manifold x and a real algebraic bundle mathcal e on x let mathfrak h be the lie algebra of h let mathcal s x mathcal e be the space of schwartz sections of mathcal e we prove that mathfrak h mathcal s x mathcal e is a closed subspace of mathcal s x mathcal e of finite codimension we give an application of this result in the case when h is a real spherical subgroup of a real reductive group g we deduce an equivalence of two old conjectures due to casselman the automatic continuity and the comparison conjecture for zero homology namely let pi be a casselman wallach representation of g and v be the corresponding harish chandra module then the natural morphism of coinvariants v mathfrak h rightarrow pi mathfrak h is an isomorphism if and only if any linear mathfrak h invariant functional on v is continuous in the topology induced from pi the latter statement is known to hold in two important special cases if h includes a symmetric subgroup and if h includes the nilradical of a minimal parabolic subgroup of g

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

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

generalized harish chandra descent gelfand pairs and an archimedean analog of jacquet rallis s theorem in the first part of the paper we generalize a descent technique due to harish chandra to the case of a reductive group acting on a smooth affine variety both defined over an arbitrary local field f of characteristic zero our main tool is the luna slice theorem in the second part of the paper we apply this technique to symmetric pairs in particular we prove that the pairs gln k f gln f glk f and gln e gln f are gelfand pairs for any local field f and its quadratic extension e in the non archimedean case the first result was proven earlier by jacquet and rallis and the second by flicker we also prove that any conjugation invariant distribution on gln f is invariant with respect to transposition for non archimedean f the latter is a classical theorem of gelfand and kazhdan

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

decay of matrix coefficients on reductive homogeneous spaces of spherical type let z be a homogeneous space z g h of a real reductive lie group g with a reductive subgroup h the investigation concerns the quantitative decay of matrix coefficients on z under the assumption that z is of spherical type that is minimal parabolic subgroups have open orbits on z

knapp stein type intertwining operators for symmetric pairs abstract for a symmetric pair g h of reductive groups we construct a family of intertwining operators between spherical principal series representations of g and h that are induced from parabolic subgroups satisfying certain compatibility conditions the operators are given explicitly in terms of their integral kernels and we prove convergence of the integrals for an open set of parameters and meromorphic continuation in particular this establishes lower bounds for multiplicities of intertwining operators we further discuss uniqueness of intertwining operators and for the rank one cases g h su 1 n f s u 1 m f u n m f with f r c h o and for the pair g h gl 4 n r gl 2 n c we show that for a certain choice of maximal parabolic subgroups our operators generically span the space of intertwiners

constant term of smooth h_psi spherical functions on a reductive p adic group let be a smooth character of a closed subgroup h of a reductive p adic group g if p is a parabolic subgroup of g such that ph is open in g we define the constant term of every smooth function on g which transforms by under the right action of g the example of mixed models is given it includes symmetric spaces and whittaker models in this case a notion of cuspidal function is defined and studied it leads to finiteness theorems

smoluchowski dynamics and the ergodic nonergodic transition we use the recently introduced theory for the kinetics of systems of classical particles to investigate systems driven by smoluchowski dynamics we investigate the existence of ergodic nonergodic ene transitions near the liquid glass transition we develop a self consistent perturbation theory in terms of an effective two body potential and work to second order in this potential at second order we have an explicit relationship between the static structure factor and the effective potential and choose the static structure factor in the case of hard spheres to be given by the solution of the percus yevick approximation for hard spheres then using the analytically determined ene equation for the ergodicity function we find an ene transition for packing fraction greater than a critical value 0 76 which is physically unaccessible the existence of a linear fluctuation dissipation theorem in the problem is shown and used to great advantage

real double flag varieties for the symplectic group abstract in this paper we study a key example of a hermitian symmetric space and a natural associated double flag variety namely for the real symplectic group g and the symmetric subgroup l the levi part of the siegel parabolic p s we give a detailed treatment of the case of the maximal parabolic subgroups q of l corresponding to grassmannians and the product variety of g p s and l q in particular we classify the l orbits here and find natural explicit integral transforms between degenerate principal series of l and g