exact solution of a 1d asymmetric exclusion model using a matrix formulation several recent works have shown that the one dimensional fully asymmetric exclusion model which describes a system of particles hopping in a preferred direction with hard core interactions can be solved exactly in the case of open boundaries here the authors present a new approach based on representing the weights of each configuration in the steady state as a product of noncommuting matrices with this approach the whole solution of the problem is reduced to finding two matrices and two vectors which satisfy very simple algebraic rules they obtain several explicit forms for these non commuting matrices which are in the general case infinite dimensional their approach allows exact expressions to be derived for the current and density profiles finally they discuss briefly two possible generalizations of their results the problem of partially asymmetric exclusion and the case of a mixture of two kinds of particles

derivation of the matrix product ansatz for the heisenberg chain from the algebraic bethe ansatz we derive a matrix product representation of the bethe ansatz state for the xxx and xxz spin heisenberg chains using the algebraic bethe ansatz in this representation the components of the bethe eigenstates are expressed as traces of products of matrices which act on the tensor product of auxiliary spaces by changing the basis in we derive explicit finite dimensional representations for the matrices these matrices are the same as those appearing in the recently proposed matrix product ansatz by alcaraz and lazo 2006 j phys a math gen 39 11335 apart from normalization factors we also discuss the close relation between the matrix product representation of the bethe eigenstates and the six vertex model with domain wall boundary conditions korepin 1982 commun math phys 86 391 and show that the change of basis corresponds to a mapping from the six vertex model to the five vertex model

on the ramanujan conjecture over number fields we extend to an arbitrary number eld the best known bounds towards ramanujan for the group gln n 2 3 4 in particular we present a technique which overcomes the analytic obstacles posed by the presence of an innite group of units

ricci flow on kahler einstein surfaces in this paper we prove that if m is a kahler einstein surface with positive scalar curvature if the initial metric has nonnegative sectional curvature and the curvature is positive somewhere then the kahler ricci flow converges to a kahler einstein metric with constant bisectional curvature in a subsequent paper 7 we prove the same result for general kahler einstein manifolds in all dimension this gives an affirmative answer to a long standing problem in kahler ricci flow on a compact kahler einstein manifold does the kahler ricci flow converge to a kahler einstein metric if the initial metric has a positive bisectional curvature our main method is to find a set of new functionals which are essentially decreasing under the kahler ricci flow while they have uniform lower bounds this property gives the crucial estimate we need to tackle this problem

c_0 c0 positivity and a classification of closed three dimensional cr torsion solitons a closed cr 3 manifold is said to have c 0 positive pseudohermitian curvature if w c 0 tor x x 0 for any 0 ne x in t 1 0 m we discover an obstruction for a closed cr 3 manifold to possess c 0 positive pseudohermitian curvature we classify closed three dimensional cr yamabe solitons according to c 0 positivity for c 0 1 and the potential function lies in the kernel of paneitz operator moreover we show that any closed three dimensional cr torsion soliton must be the standard sasakian space form at last we discuss the persistence of c 0 positivity along the cr torsion flow starting from a pseudo einstein contact form

boundedness of automorphic l functions in vertical strips the purpose of this paper is to prove the boundedness in vertical strips of finite width for all the l functions that appear in constant terms of eisenstein series under a certain natural assumption on local normalized intertwining operators as a corollary we prove the boundedness for a number of important l functions among them the symmetic cube and triple product l functions attached to cusp forms on gl 2 as well as several rankin selberg product l functions where the local assumption was alredy proved to be valid in each case under the same assumption the finiteness of poles on all of c for all of these l functions is also proved a necessary step in the proof of boundedness in vertical strips

pseudo einstein structures on cr manifolds etant donne une variete cr compacte non degeneree on cherche une structure pseudohermitienne associee pour laquelle le tenseur de ricci pseudohermitien est un multiple scalaire de la forme de levi

on the langlands functoriality for automorphic forms this is the extended notes of my lcctures on the langlands functoriality of automorphic forms which are in particular on the topics closely related to my recent work

on the cr analogue of obata s theorem in a pseudohermitian 3 manifold in this paper we first prove the cr analogue of obata s theorem on a closed pseudohermitian 3 manifold with zero pseudohermitian torsion secondly instead of zero torsion we have the cr analogue of li yau s eigenvalue estimate on the lower bound estimate of positive first eigenvalue of the sub laplacian in a closed pseudohermitian 3 manifold with nonnegative cr paneitz operator p0 finally we have a criterion for the positivity of first eigenvalue of the sub laplacian on a complete noncompact pseudohermitian 3 manifold with nonnegative cr paneitz operator the key step is a discovery of integral cr analogue of bochner formula which involving the cr paneitz operator

the local langlands conjecture for gsp 4 we prove the local langlands conjecture for gsp4 f where f is a nonarchimedean local eld of characteristic zero

upper and lower bounds at s 1 for certain dirichlet series with euler product estimates of the form l j s mathscr a ll epsilon j mathscr d a mathscr r epsilon mathscr a in the range s 1 ll 1 log mathscr r a for general l functions where mathscr r a is a parameter related to the functional equation of l s mathscr a can be quite easily obtained if the ramanujan hypothesis is assumed we prove the same estimates when the l functions have euler product of polynomial type and the ramanujan hypothesis is replaced by a much weaker assumption about the growth of certain elementary symmetrical functions as a consequence we obtain an upper bound of this type for every l s pi where pi is an automorphic cusp form on rm gl mathbf d mathbb a k we employ these results to obtain siegel type lower bounds for twists by dirichlet characters of the third symmetric power of a maass form

the bethe ansatz as a matrix product ansatz the bethe ansatz in its several formulations is a common tool for the exact solution of one dimensional quantum hamiltonians this ansatz asserts that several eigenfunctions of the hamiltonians are given in terms of a sum of permutations of plane waves we present results that induce us to expect that alternatively the eigenfunctions of all the exact integrable quantum chains can also be expressed by a matrix product ansatz in this ansatz several components of the eigenfunctions are obtained through the algebraic properties of properly defined matrices this ansatz allows an unified formulation of several exact integrable hamiltonians we show how to formulate this ansatz for a large family of quantum chains such as the anisotropic heisenberg model fateev zamolodchikov model izergin korepin model sutherland model t j model hubbard model etc

matrix li yau hamilton inequality for the cr heat equation in pseudohermitian 2n 1 manifolds in this paper we first derive the cr analogue of matrix li yau hamilton inequality for the positive solution to the cr heat equation in a closed pseudohermitian 2n 1 manifold with nonnegative bisectional curvature and bitorsional tensor we then obtain the cr li yau gradient estimate in the heisenberg group we apply this cr gradient estimate and extend the cr matrix li yau hamilton inequality to the case of the heisenberg group as a consequence we derive the hessian comparison property for the heisenberg group

rationality and holomorphy of langlands shahidi l functions over function fields we prove that all langlands shahidi automorphic l functions over function fields are rational after twists by highly ramified characters they become polynomials and if pi is a globally generic cuspidal automorphic representation of a split classical group or a unitary group and tau is a cuspidal unitary automorphic representation of a general linear group then l s pi times tau is holomorphic for mathfrak r s 1 and has at most a simple pole at s 1 we also prove the holomorphy and non vanishing of automorphic exterior square symmetric square and asai l functions for mathfrak r s 1 finally we complete previous results on functoriality for the classical groups over function fields with applications to the ramanujan conjecture and riemann hypothesis

smooth solutions of degenerate laplacians on strictly pseudoconvex domains on considere la caracterisation des solutions lisses globales de u 0 sur un domaine general strictement pseudoconvexe ou est le laplacien pour la metrique avec forme de kakler i 2 log 1

density matrix formulation for quantum renormalization groups a generalization of the numerical renormalization group procedure used first by wilson for the kondo problem is presented it is shown that this formulation is optimal in a certain sense as a demonstration of the effectiveness of this approach results from numerical real space renormalization group calculations for heisenberg chains are presented

dynamical correlation functions of the xxz model at finite temperature combining a lattice path integral formulation for thermodynamics with the solution of the quantum inverse scattering problem for local spin operators we derive a multiple integral representation for the time dependent longitudinal correlation function of the spin 1 2 heisenberg xxz chain at finite temperature and in an external magnetic field our formula reproduces the previous results in the following three limits the static the zero temperature and the xy limits

hidden grassmann structure in the xxz model ii creation operators in this article we unveil a new structure in the space of operators of the xxz chain for each we consider the space mathcal w alpha of all quasi local operators which are products of the disorder field q alpha sum j infty 0 sigma 3 j with arbitrary local operators in analogy with cft the disorder operator itself is considered as primary field in our previous paper we have introduced the annhilation operators b c which mutually anti commute and kill the primary field here we construct the creation counterpart b c and prove the canonical anti commutation relations with the annihilation operators we conjecture that the creation operators mutually anti commute thereby upgrading the grassmann structure to the fermionic structure the bosonic operator t is the generating function of the adjoint action by local integrals of motion and commutes entirely with the fermionic creation and annihilation operators operators b c t create quasi local operators starting from the primary field we show that the ground state averages of quasi local operators created in this way are given by determinants

une preuve simple des conjectures de langlands pour gl n sur un corps p adique let f be a finite extension of p for each integer n 1 we construct a bijection from the set f 0 n of isomorphism classes of irreducible degree n representations of the absolute weil group of f onto the set f 0 n of isomorphism classes of smooth irreducible supercuspidal representations of gl n f those bijections preserve epsilon factors for pairs and hence we obtain a proof of the langlands conjectures for gl n over f which is more direct than harris and taylor s our approach is global and analogous to the derivation of local class field theory from global class field theory we start with a result of kottwitz and clozel on the good reduction of some shimura varieties and we use a trick of harris who constructs non galois automorphic induction in certain cases

a note on compact gradient yamabe solitons abstract we will give a simple proof that the metric of any compact yamabe gradient soliton m g is a metric of constant scalar curvature when the dimension of the manifold n 3

strongly positive representations of gspin_ 2n 1 and the jacquet module method we explicitly construct the structure of jacquet modules of parabolically induced representations of gspin 2n 1 over a p adic field f of any characteristic using this construction of the jacquet module we obtain a classification of strongly positive representations of gspin 2n 1 over f and describe the general discrete series representations of gspin 2n 1 over f assuming the half integer conjecture one application of this paper is the proof of the equality of l functions from the langlands shahidi method and artin l functions through the local langlands correspondence kim in langlands shahidi l functions for gspin groups and the generic arthur packet conjecture preprint

matrix coordinate bethe ansatz applications to xxz and asep models we present the construction of the full set of eigenvectors of the open asymmetric simple exclusion process asep and xxz models with special constraints on the boundaries the method combines both recent constructions of coordinate bethe ansatz and the old method of matrix ansatz specific to the asep this matrix coordinate bethe ansatz can be viewed as a non commutative coordinate bethe ansatz the non commutative part being related to the algebra appearing in the matrix ansatz

generalization of the matrix product ansatz for integrable chains we present a general formulation of the matrix product ansatz for exactly integrable chains on periodic lattices this new formulation extends the matrix product ansatz present in our previous articles f c alcaraz and m j lazo 2004 j phys a math gen 37 l1 l7 and f c alcaraz and m j lazo 2004 j phys a math gen 37 4149 82

on the ramanujan conjecture and finiteness of poles for certain l functions in this paper we prove a number of general results about automorphic l functions which appear in the constant terms of eisenstein series for an arbitrary quasi split connected reductive algebraic group over a number field these l functions were first introduced by langlands for chevalley groups 20 and through them he was led to make some of his deep conjectures 21 one significance of these l functions is that all the automorphic l functions studied so far are among them there are three general results first we obtain a uniform line of absolute convergence for all of these l functions theorem 5 1 a uniform estimate for the hecke eigenvalues of generic cusp forms on many absolutely simple quasi split connected reductive algebraic groups over number fields corollary 5 4 and an improvement on the best available estimate for the fourier coefficients of maass wave forms corollary 5 5 also follow next we establish a meromorphic continuation and functional equation for each of these l functions theorem 6 1 finally in theorem 6 2 we prove finiteness of poles on the whole complex plane for an important class of these l functions corollaries 6 6 through 6 10 more precisely let g be a quasi split connected reductive algebraic group over a number field f set g g a f where a f is the ring of adeles of f fix a borel subgroup b of g over f and let u be its unipotent radical let p be a maximal f parabolic subgroup of g with p d b in the context of the problems studied here nothing new will be obtained if one drops the maximality condition on p write p mn a levi decomposition n c u and let b u p m and n be the corresponding groups of adelic points for every place v of f let gv g fj similarly we have bv us pv me and nv let x xv be a generic character of u f u cf section 3 then each xv is generic let r ov 7tv be a cusp form on m we shall say 7t is

twisted symmetric square l functions for mathrm gl _n and invariant trilinear forms following the works of bump and ginzburg and of takeda we develop a theory of twisted symmetric square l functions for mathrm gl n we characterize their pole in terms of certain trilinear period integrals determine all irreducible summands of the discrete spectrum of mathrm gl n having nonvanishing trilinear periods and construct nonzero local invariant trilinear forms on a certain family of induced representations

a proof of langland s conjecture on plancherel measures complementary series of p adic groups analysis of p adic reductive groups our first result theorem 7 9 proves a conjecture of langlands on normalization of intertwining operators by means of local langlands root numbers and l functions at least when the group is quasi split and the inducing representation is generic assuming two natural conjectures in harmonic analysis of p adic groups we also prove the validity of the conjecture in general theorem 9 5 as our second result we obtain all the complementary series and special representations of quasi split p adic groups coming from rank one parabolic subgroups and generic supercuspidal represen

combinatorial properties of symmetric polynomials from integrable vertex models in finite lattice we introduce and study several combinatorial properties of a class of symmetric polynomials from the point of view of integrable vertex models in a finite lattice we introduce the l operator related to the uq sl2 r matrix and construct the wavefunctions and their duals we prove the exact correspondence between the wavefunctions and symmetric polynomials which is a quantum group deformation of the grothendieck polynomials this is proved by combining the matrix product method and an analysis on the domain wall boundary partition functions as applications of the correspondence between the wavefunctions and symmetric polynomials we derive several properties of the symmetric polynomials such as the determinant pairing formulas and the branching formulas by analyzing the domain wall boundary partition functions and the matrix elements of the b operators

coefficient of maass forms and the siegel zero with an appendix by dorian goldfeld jeffrey hoffstein and daniel lieman

intertwining operators and residues i weighted characters operateurs d entrelacement normalisation groupes reels groupes p adiques representations standard les distributions j m la distribution j m x residus changements de contour les espaces 7b h ac g f s et 7b f ac g f s l application m

on a certain metaplectic eisenstein series and the twisted symmetric square l function in our earlier paper based on a paper by bump and ginzburg we used an eisenstein series on the double cover of text gl r to obtain an integral representation of the twisted symmetric square l function of text gl r using that we showed that the incomplete twisted symmetric square l function of text gl r is holomorphic for text re s 1 in this paper we will determine the possible poles of this eisenstein series more precisely and show that the incomplete twisted symmetric square l function is entire except possible simple poles at s 0 and s 1

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

finitely correlated states on quantum spin chains we study a construction that yields a class of translation invariant states on quantum spin chains characterized by the property that the correlations across any bond can be modeled on a finite dimensional vector space these states can be considered as generalized valence bond states and they are dense in the set of all translation invariant states we develop a complete theory of the ergodic decomposition of such states including the decomposition into periodic neel ordered states the ergodic components have exponential decay of correlations all states considered can be obtained as local functions of states of a special kind so called purely generated states which are shown to be ground states for suitably chosen finite range vbs interactions we show that all these generalized vbs models have a spectral gap our theory does not require symmetry of the state with respect to a local gauge group in particular we illustrate our results with a one parameter family of examples which are not isotropic except for one special case this isotropic model coincides with the one dimensional antiferromagnet recently studied by affleck kennedy lieb and tasaki

exact solutions of exactly integrable quantum chains by a matrix product ansatz most of the exact solutions of quantum one dimensional hamiltonians are obtained thanks to the success of the bethe ansatz on its several formulations according to this ansatz the amplitudes of the eigenfunctions of the hamiltonian are given by a sum of permutations of appropriate plane waves in this paper alternatively we present a matrix product ansatz that asserts that those amplitudes are given in terms of a matrix product the eigenvalue equation for the hamiltonian defines the algebraic properties of the matrices defining the amplitudes the consistency of the commutativity relations among the elements of the algebra implies the exact integrability of the model the matrix product ansatz we propose allows an unified and simple formulation of several exact integrable hamiltonians in order to introduce and illustrate this ansatz we present the exact solutions of several quantum chains with one and two global conservation laws and periodic boundaries such as the xxz chain spin 1 fateev zamolodchikov model izergin korepin model sutherland model t j model hubbard model etc formulation of the matrix product ansatz for quantum chains with open ends is also possible as an illustration we present the exact solution of an extended xxz chain with z magnetic fields at the surface and arbitrary hard core exclusion among the spins

higher spin polynomial solutions of quantum knizhnik zamolodchikov equation we provide explicit formulae for highest weight to highest weight correlation functions of perfect vertex operators of u q widehat mathfrak sl 2 in arbitrary integer level l they are given in terms of certain macdonald polynomials we apply this construction to the computation of the ground state of higher spin vertex models spin chains spin l 2 xxz or loop models in the root of unity case q e i pi ell 2

vertex models tasep and grothendieck polynomials we examine the wavefunctions and their scalar products of a one parameter family of integrable five vertex models at a special point of the parameter the model investigated is related to an irreversible interacting stochastic particle system the so called totally asymmetric simple exclusion process tasep by combining the quantum inverse scattering method with a matrix product representation of the wavefunctions the on off shell wavefunctions of the five vertex models are represented as a certain determinant form up to some normalization factors we find that the wavefunctions are given by grothendieck polynomials which are a one parameter deformation of schur polynomials introducing a dual version of the grothendieck polynomials and utilizing the determinant representation for the scalar products of the wavefunctions we derive a generalized cauchy identity satisfied by the grothendieck polynomials and their duals several representation theoretical formulae for the grothendieck polynomials are also presented as a byproduct the relaxation dynamics such as green functions for the periodic tasep are found to be described in terms of the grothendieck polynomials

the fefferman metric and pseudo hermitian invariants c fefferman has shown that a real strictly pseudoconvex hypersurface in complex n space carries a natural conformai lorentz metric on a circle bundle over the manifold this paper presents two intrinsic constructions of the metric valid on an abstract cr manifold one is in terms of tautologous differential forms on a natural circle bundle the other is in terms of webster s pseudohermitian invariants these results are applied to compute the connection and curvature forms of the fefferman metric explicitly

local coefficients as artin factors for real groups introduction the purpose of this paper is to prove the equality of certain local coefficients of arithmetic significance which were attached to representations of quasi split real reductive algebraic groups in 27 with their corresponding artin factors attached by local class field theory 21 as a consequence we establish an identity satisfied by certain normalized intertwining operators it seems to be useful in applications of the trace formula 1 29 more precisely let g be the group of real points of a quasi split reductive algebraic group over r let a be the set of simple roots defined by a fixed minimal parabolic subgroup po m0a0u of g fix 6 c a and let p p9 be the corresponding standard parabolic subgroup of g and write p man for its langlands decomposition fix a nondegenerate character x f u i a h o be an irreducible admissible x gic banach in particular x generic unitary representation of m cf section 1 given v g a the complex dual of the lie algebra of a let i v o 0 be the continuously quasi unitarily if o is unitary induced representation indp ga e v 9 and let v v o 0 be its space section 0 then f a vfoo o now let w be the weyl group of ao in g choose w g w such that w 0 c a let n be the unipotent group opposite to n define n u c wn w where w is a representative of w in g for g f a 00 define

doubling constructions and tensor product l functions the linear case we present an integral representation for the tensor product l function of a pair of automorphic cuspidal representations one of a classical group the other of a general linear group our construction is uniform over all classical groups and is applicable to all cuspidal representations it does not require genericity the main new ideas of the construction are the use of generalized speh representations as inducing data for the eisenstein series and the introduction of a new global and local model which generalizes the whittaker model here we consider linear groups but our construction also extends to arbitrary degree metaplectic coverings this will be the topic of an upcoming work

symmetric functions and wavefunctions of xxz type six vertex models and elliptic felderhof models by izergin korepin analysis we present a method to analyze the wavefunctions of six vertex models by extending the izergin korepin analysis originally developed for domain wall boundary partition functions first we apply the method to the case of the basic wavefunctions of the xxz type six vertex model by giving the izergin korepin characterization of the wavefunctions we show that these wavefunctions can be expressed as multiparameter deformations of the quantum group deformed grothendieck polynomials as a second example we show that the izergin korepin analysis is effective for analysis of the wavefunctions for a triangular boundary and present the explicit forms of the symmetric functions representing these wavefunctions as a third example we apply the method to the elliptic felderhof model which is a face type version and an elliptic extension of the trigonometric felderhof model we show that the wavefunctions can be expressed as one parameter deformations of an elliptic analog of the vandermonde determinant and elliptic symmetric functions we present a method to analyze the wavefunctions of six vertex models by extending the izergin korepin analysis originally developed for domain wall boundary partition functions first we apply the method to the case of the basic wavefunctions of the xxz type six vertex model by giving the izergin korepin characterization of the wavefunctions we show that these wavefunctions can be expressed as multiparameter deformations of the quantum group deformed grothendieck polynomials as a second example we show that the izergin korepin analysis is effective for analysis of the wavefunctions for a triangular boundary and present the explicit forms of the symmetric functions representing these wavefunctions as a third example we apply the method to the elliptic felderhof model which is a face type version and an

derivation of a matrix product representation for the asymmetric exclusion process from the algebraic bethe ansatz we derive using the algebraic bethe ansatz a generalized matrix product ansatz for the asymmetric exclusion process asep on a one dimensional periodic lattice in this matrix product ansatz the components of the eigenvectors of the asep markov matrix can be expressed as traces of products of non commuting operators we derive the relations between the operators involved and show that they generate a quadratic algebra our construction provides explicit finite dimensional representations for the generators of this algebra

calculation of norms of bethe wave functions a class of two dimensional completely integrable models of statistical mechanics and quantum field theory is considered eigenfunctions of the hamiltonians are known for these models norms of these eigenfunctions in the finite box are calculated in the present paper these models include in particular the quantum nonlinear schrodinger equation and the heisenbergxxz model

hidden grassmann structure in the xxz model for the critical xxz model we consider the space mathcal w alpha of operators which are products of local operators with a disorder operator we introduce two anti commutative families of operators bf b zeta bf c zeta which act on mathcal w alpha these operators are constructed as traces over representations of the q oscillator algebra in close analogy with baxter s q operators we show that the vacuum expectation values of operators in mathcal w alpha can be expressed in terms of an exponential of a quadratic form of bf b zeta bf c zeta

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

ricci flow on kahler einstein manifolds this is the continuation of our earlier article 10 for any kahler einstein surfaces with positive scalar curvature if the initial metric has positive bisectional curvature then we have proved see 10 that the kahler ricci flow converges exponentially to a unique kahler einstein metric in the end this partially answers a long standing problem in ricci flow on a compact kahler einstein manifold does the kahler ricci flow converge to a kahler einstein metric if the initial metric has positive bisectional curvature in this article we give a complete affirmative answer to this problem

standard zero free regions for rankin selberg l functions via sieve theory we give a simple proof of a standard zero free region in the t aspect for the rankin selberg l function l s pi times widetilde pi for any unitary cuspidal automorphic representation pi of mathrm gl n mathbb a f that is tempered at every nonarchimedean place outside a set of dirichlet density zero

matrix product solution of the stationary state of two species open zero range processes using the matrix product ansatz we obtain solutions of the steady state distribution of the two species open one dimensional zero range process our solution is based on a conventionally employed constraint on the hop rates which eventually allows us to simplify the constituent matrices of the ansatz it is shown that the matrix at each site is given by the tensor product of two sets of matrices and the steady state distribution assumes an inhomogeneous factorized form our method can be generalized to the cases of more than two species of particles

on the arithmetic of shalika models and the critical values of l functions for gl2n let pi be a cohomological cuspidal automorphic representation of rm gl 2n bbb a over a totally real number field f suppose that pi has a shalika model we define a rational structure on the shalika model of pi f comparing it with a rational structure on a realization of pi f in cuspidal cohomology in top degree we define certain periods omega epsilon pi f we describe the behavior of such top degree periods upon twisting pi by algebraic hecke characters chi of f then we prove an algebraicity result for all the critical values of the standard l functions l s pi otimes chi here we use the recent work of b sun on the non vanishing of a certain quantity attached to pi infty as applications we obtain algebraicity results in the following cases firstly for the symmetric cube l functions attached to holomorphic hilbert modular cusp forms we also discuss the situation for higher symmetric powers secondly for certain self dual of symplectic type rankin selberg l functions for rm gl 3 times rm gl 2 assuming langlands functoriality this generalizes to certain rankin selberg l functions of rm gl n times rm gl n 1 thirdly for the degree four l functions attached to siegel modular forms of genus 2 and full level moreover we compare our top degree periods with periods defined by other authors we also show that our main theorem is compatible with conjectures of deligne and gross

thermodynamic limit of the six vertex model with domain wall boundary conditions we address the question of the dependence of the bulk free energy on boundary conditions for the six vertex model here we compare the bulk free energy for periodic and domain wall boundary conditions using a determinant representation for the partition function with domain wall boundary conditions we derive toda differential equations and solve them asymptotically in order to extract the bulk free energy we find that it is different and bears no simple relation to the free energy for periodic boundary conditions the six vertex model with domain wall boundary conditions is closely related to algebraic combinatorics alternating sign matrices this implies new results for the weighted counting for large size alternating sign matrices finally we comment on the interpretation of our results in particular in connection with domino tilings dimers on a square lattice

li yau gradient estimate and entropy formulae for the cr heat equation in a closed pseudohermitian 3 manifold in this paper we derive two subgradient estimates of the cr heat equation in a closed pseudohermitian 3 manifold which are served as the cr version of the li yau gradient estimate with its applications we first get a subgradient estimate of the logarithm of the positive solution of the cr heat equation secondly we have the harnack inequality and upper bound estimate for the heat kernel finally we obtain perelman type entropy formulae for the cr heat equation

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

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