associating vertex algebras with the unitary lie algebra abstract in this paper we associate vertex algebras and their two different kinds of module categories with the unitary lie algebra u n c for n 2 being a positive integer and q n n z where the nonzero complex number q is not a root of unity it is proved that for any complex number l the category of restricted u n c modules of level l is canonically isomorphic to the category of quasi modules for certain vertex algebra and we also prove that the category of restricted u n c modules of level l is isomorphic to the category of equivariant coordinated quasi modules for the same vertex algebra where is an automorphism group of this vertex algebra

g equivariant ϕ coordinated quasi modules for quantum vertex algebras this is a paper in a series to study quantum vertex algebras and their relations with various quantum algebras in this paper we introduce a notion of t type quantum vertex algebra and a notion of g equivariant coordinated quasi module for a t type quantum vertex algebra with an automorphism group g we refine and extend several previous results and we obtain a commutator formula for g equivariant coordinated quasi modules as an illustrating example we study a special case of the deformed virasoro algebra virp q with q 1 to which we associate a clifford vertex superalgebra and its g equivariant coordinated quasi modules

an easy priority free proof of a theorem of friedberg in 1958 friedberg published the now classical result that p 1 the set of all partial recursive one place functions can be enumerated without repetition his proof is based on a priority argument we give a priority free construction that is much simpler than the original this stands is contrast to kucera s recent priority free proof of the friedberg muchnik theorem which is more complicated than the standard finite injury proof

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

kazhdan lusztig representations and whittaker space of some genuine representations we prove a formula for the dimension of whittaker functionals of irreducible constituents of a regular unramified genuine principal series for covering groups the formula explicitly relates such dimension to the kazhdan lusztig representations associated with certain right cells of the weyl group we also state a refined version of the formula which is proved under some natural assumption the refined formula is also verified unconditionally in several important cases

γ conformal algebras conformal algebra is an axiomatic description of the operator product expansion of chiral fields with simple poles at finitely many points we classify these algebras and their representations in terms of lie algebras and their representations with an action of the group to every conformal algebra and a character of we associate a lie algebra generated by fields with the ope with simple poles examples include twisted affine kac moody algebras the sin algebra which is a conformal analogue of the general linear algebra and its analogues the algebra of pseudodifferential operators on the circle etc

toroidal vertex algebras and their modules nsa h98230 11 1 0161 china nsf 11128103 10931006 11001229 phd programs foundation of ministry of education of china 20100121110014 fundamental research funds for the central university 2010121003

associating quantum vertex algebras to lie algebra gl abstract in this paper we present a canonical association of quantum vertex algebras and their coordinated modules to lie algebra gl and its 1 dimensional central extension to this end we construct and make use of another closely related infinite dimensional lie algebra

a metaplectic casselman shalika formula for gl r we provide formulas for various bases of spherical whittaker functions on the n fold metaplectic cover of rm gl r over a p adic field and show that there is a basis of symmetric functions in the complex parameter in addition we relate a specific spherical whittaker function to the p power part of the weyl group multiple dirichlet series for the root system of type a r 1 constructed from n th order gauss sums we also show that the zonal spherical functions can be computed explicitly in terms of hall littlewood polynomials as in macdonald s formula for rm gl r

φ coordinated quasi modules for quantum vertex algebras we develop a theory of phi coordinated quasi modules for a general nonlocal vertex algebra where phi is what we call an associate of the one dimensional additive formal group by specializing phi to a particular associate we obtain a new construction of weak quantum vertex algebras in the sense of li selecta mathematica new series 11 349 397 2005 as an application we associate weak quantum vertex algebras to quantum affine algebras and we also associate quantum vertex algebras and phi coordinated modules to a certain quantum system explicitly

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

towards a classification of global integral constructions and functorial liftings using the small representations method abstract this is a survey article in which we discuss some recent approach toward the classification of global factorizable integrals and constructions of langlands functorial lifting using small representations to do that we use the language of unipotent orbits and their relation to fourier coefficients we also introduce the dimension equation and set our goal to classify all global integrals of a certain form which satisfies this equation