character formula for infinite dimensional unitarizable modules of the general linear superalgebra the fock space of m p bosonic and n q fermionic quantum oscillators forms a unitarizable module of the general linear superalgebra glm p n q its tensor powers decompose into direct sums of infinite dimensional irreducible highest weight glm p n q modules we obtain an explicit decomposition of any tensor power of this fock space into irreducibles and develop a character formula for the irreducible glm p n q modules arising in this way

dimension formula for graded lie algebras and its applications abstract in this paper we investigate the structure of infinite dimensional lie algebras l l l graded by a countable abelian semigroup satisfying a certain finiteness condition the euler poincare principle yields the denominator identities for the graded lie algebras from which we derive a dimension formula for the homogeneous subspaces l our dimension formula enables us to study the structure of the graded lie algebras in a unified way we will discuss some interesting applications of our dimension formula to the various classes of graded lie algebras such as free lie algebras kac moody algebras and generalized kac moody algebras we will also discuss the relation of graded lie algebras and the product identities for formal power series

modular invariant representations of infinite dimensional lie algebras and superalgebras abstract in this paper we launch a program to describe and classify modular invariant representations of infinite dimensional lie algebras and superalgebras we prove a character formula for a large class of highest weight representations l of a kac moody algebra unk with a symmetrizable cartan matrix generalizing the weyl kac character formula kac v g 1974 funct anal appl 8 68 70 in the case of an affine unk this class includes modular invariant representations of arbitrary rational level m t u where t unk z and u unk n are relatively prime and m g g u g is the dual coxeter number we write the characters of these representations in terms of theta functions and calculate their asymptotics generalizing the results of kac and peterson kac v g peterson d h 1984 adv math 53 125 264 and of kac and wakimoto kac v g wakimoto m 1988 adv math 70 156 234 for the u 1 integrable case we work out in detail the case unk a1 1 in particular classifying all its modular invariant representations furthermore we show that the modular invariant representations of the virasoro algebra vir are precisely the minimal series of belavin et al belavin a a polyakov a m zamolodchikov a b 1984 nucl phys b 241 333 380 using the character formulas of feigin and fuchs feigin b l fuchs d b 1984 lect notes math 1060 230 245 we show that tensoring the basic representation and modular invariant representations of a1 1 produces all modular invariant representations of vir generalizing the results of goddard et al goddard p kent a olive d 1986 commun math phys 103 105 119 and of kac and wakimoto kac v g wakimoto m 1986 lect notes phys 261 345 371 in the unitary case we study the general branching functions as well all these results are generalized to the kac moody superalgebras introduced by kac kac v g 1978 adv math 30 85 136 and to n 1 super virasoro algebras we work out in detail the case of the superalgebra b 0 1 1 showing in particular that restricting to its even part produces again all modular invariant representations of vir these results lead to general conjectures about asymptotic behavior of positive energy representations and classification of modular invariant representations

character and dimension formulae for general linear superalgebra the generalized kazhdan lusztig polynomials for the finite dimensional irreducible representations of the general linear superalgebra are computed explicitly the result is then applied to prove a conjectural character formula put forward by van der jeugt et al in the late 80s we simplify this character formula to cast it into the kac weyl form and derive from it a closed formula for the dimension of any finite dimensional irreducible representation of the general linear superalgebra

the uncertainty principle on considere l existence et la regularite des solutions d equations aux derivees partielles la construction de solutions fondamentales explicites et les valeurs propres d operateurs de schrodinger

characters of some unitary highest weight representations via the theta correspondence abstract in this article we consider a dual pair g g in the symplectic group sp w with g compact and let g g be the preimages of g and g in the metaplectic group sp w for every irreducible representation of g appearing in howe correspondence we compute explicitly the restriction of the character of the associated representation of g on the set of regular points on the compact cartan subgroup h of g

stationary navier stokes equations with critically singular external forces existence and stability results abstract we show the unique existence of solutions to stationary navier stokes equations with small singular external forces belonging to a critical space to the best of our knowledge this is the largest critical space that is available up until now for this kind of existence this result can be viewed as the stationary counterpart of the existence result obtained by h koch and d tataru for the free non stationary navier stokes equations with initial data in bmo 1 the stability of the stationary solutions in such spaces is also obtained by a series of sharp estimates for resolvents of a singularly perturbed operator and the corresponding semigroup

howe duality and combinatorial character formula for orthosymplectic lie superalgebras we study the howe dualities involving the reductive dual pairs o d spo 2m 2n and sp d osp 2m 2n on the super symmetric tensor of cd cm n we obtain complete decompositions of this space with respect to their respective joint actions we also use these dualities to derive a character formula for these irreducible representations of spo 2m 2n and osp 2m 2n that appear in these decompositions

kostant homology formulas for oscillator modules of lie superalgebras abstract we provide a systematic approach to obtain formulas for characters and kostant u homology groups of the oscillator modules of the finite dimensional general linear and ortho symplectic superalgebras via howe dualities for infinite dimensional lie algebras specializing these lie superalgebras to lie algebras we recover in a new way formulas for kostant homology groups of unitarizable highest weight representations of hermitian symmetric pairs in addition two new reductive dual pairs related to the above mentioned u homology computation are worked out

classification of all star irreps of gl m n all the finite dimensional star irreps of gl m n are classified in terms of their highest weights thereby completing the classification of all finite dimensional star irreps of the basic classical lie superalgebras the lowest weights of such irreps are determined explicitly and it is shown that the contravariant and covariant tensor irreps of gl m n are star irreps of type 1 and 2 respectively this explains the applicability of the young diagram method to these two classes of representations however it is shown that one parameter families of typical star irreps occur that are intrinsically different from the tensorial irreps

jacquet modules and dirac cohomology abstract we show that dirac cohomology of the jacquet module of a harish chandra module is a harish chandra module for the corresponding levi subgroup we obtain an explicit formula of dirac cohomology of the jacquet module for most of the principal series based on our determination of dirac cohomology of irreducible generalized verma modules with regular infinitesimal characters

u cohomology formula for unitarizable modules over general linear superalgebras abstract the u cohomology groups with coefficients in infinite dimensional irreducible unitarizable highest weight modules are introduced for the general linear superalgebra the corresponding cohomology groups associated with a large class of infinite dimensional irreducible unitarizable representations known as the oscillator representations are computed explicitly results lead to a more detailed understanding of the character formula for such modules derived in an earlier publication using other methods

free lie algebras generalized witt formula and the denominator identity abstract let be a countable abelian semigroup satisfying a suitable finiteness condition and let l l be the free lie algebra generated by a graded vector space v over c in this paper from the denominator identity we derive a dimension formula for the homogeneous subspaces of the free lie algebra l l and discuss numerous applications of our dimension formula to various interesting cases our dimension formula will be expressed in terms of the witt partition functions various expressions of the witt partition functions will give rise to a number of interesting combinatorial identities as a special case we obtain a recursive relation for the coefficients of the elliptic modular function j

kostant s formula for kac moody lie algebras abstract this paper is a generalization of h garland and j lepowsky s paper of 1976 let g a be the kac moody lie algebra defined by a symmetrizable generalized cartan matrix a m the irreducible highest weight module of g a with a dominant integral highest weight kostant s homology and cohomology formulas for the lie algebra g a and the module m are proved without assuming that the reductive part of the parabolic subalgebra of g a is of finite type a resolution of m is constructed for any j the jth term of the resolution has a filtration such that all the factors of the filtration are generalized verma modules of the form vm w p p where w ranges over a certain subset of the weyl group of g a

resolutions and hilbert series of determinantal varieties and unitary highest weight modules abstract let g k be a hermitian symmetric pair and let g and k denote the corresponding complexified lie algebras let g k p p be the usual decomposition of g as a k module there is a natural correspondence between k c orbits in p and a distinguished family of unitarizable highest weight modules for g called the wallach representations we denote by yk the closure of the k c orbit in p that is associated to the kth wallach representation in this article we give explicit formulas for the numerator polynomials of the hilbert series of the varieties yk by using bgg resolutions of unitarizable highest weight modules a preliminary result gives a new branching formula for a certain two parameter family of finite dimensional representations of the even orthogonal groups our work is an extension of previous work by enright and willenbring

monstrous moonshine and monstrous lie superalgebras we prove conway and norton s moonshine conjectures for the infinite dimensional representation of the monster simple group constructed by frenkel lepowsky and meurman to do this we use the no ghost theorem from string theory to construct a family of generalized kac moody superalgebras of rank 2 which are closely related to the monster and several of the other sporadic simple groups the denominator formulas of these superalgebras imply relations between the thompson functions of elements of the monster i e the traces of elements of the monster on frenkel lepowsky and meurman s representation which are the replication formulas conjectured by conway and norton these replication formulas are strong enough to verify that the thompson functions have most of the moonshine properties conjectured by conway and norton and in particular they are modular functions of genus 0 we also construct a second family of kac moody superalgebras related to elements of conway s sporadic simple group col these superalgebras have even rank between 2 and 26 for example two of the lie algebras we get have ranks 26 and 18 and one of the superalgebras has rank 10 the denominator formulas of these algebras give some new infinite product identities in the same way that the denominator formulas of the a n n e kac moody algebras give the macdonald identities

a generalization of borcherds algebra and denominator formula abstract we study generalized lie superalgebras an extension of kac s generalized lie superalgebras and show that they are transformed forms ofl graded lie superalgebras for some abelian groupl we then introduce a generalized lie superalgebra version of the generalized kac moody algebra borcherds algebra since it is a transformed borcherds superalgebra it has several properties similar to those of borcherds superalgebra for example it is defined by similar relations and has a similar denominator formula and character formulas

a character formula for generalized kac moody superalgebras abstract in 1 borcherds defined generalized kac moody denoted by gkm for short algebras and gave a character formula for lowest weight modules of these algebras in this paper we give a character formula for lowest weight modules of gkm superalgebras for any standard result about contragredient superalgebras that have an immediate generalization to gkm superalgebras see 3 and 4 all vector spaces are assumed to be over the real field

hilbert series howe duality and branching rules abstract let be a partition with l parts and let f be the irreducible finite dimensional representation of gl m associated to when l m the littlewood restriction rule describes how f decomposes when restricted to the orthogonal group o m or to the symplectic group sp m 2 under the condition that l m 2 in this paper this result is extended to all partitions our method combines resolutions of unitary highest weight modules by generalized verma modules with reciprocity laws from the theory of dual pairs in classical invariant theory corollaries include determination of the gelfand kirillov dimension of any unitary highest weight representation occurring in a dual pair setting and the determination of their hilbert series as a graded module for p let l be a unitary highest weight representation of sp n r so 2n or u p q when the highest weight of l plus satisfies a partial dominance condition called quasi dominance we associate to l a reductive lie algebra gl and a graded finite dimensional representation bl of gl the representation bl will have a hilbert series p q that is a polynomial in q with positive integer coefficients let l be the gelfand kirillov dimension of l and set cl equal to the ratio of the dimensions of the zeroth levels in the gradings of l and bl then the hilbert series of l may be expressed in the form in the easiest example of the correspondence l bl the two components of the weil representation of the symplectic group correspond to the two spin representations of an orthogonal group

decomposition of free lie algebras into irreducible components letvbe a representation of a finite groupg by using the trace formula for the free lie algebra generated byv we compute the growth of multiplicities of irreducible components in ln v asnvaries we also decompose l v1 vk and ln v1 vk into irreduciblegl v1 gl vk modules

siegel automorphic form corrections of some lorentzian kac moody lie algebras we find automorphic form corrections which are generalized lorentzian kac moody superalgebras without odd real simple roots for two elliptic lorentzian kac moody algebras of rank 3 with a lattice weyl vector and calculate multiplicities of their simple and arbitrary roots these kac moody algebras are defined by hyperbolic symmetrized generalized cartan matrices inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i of rank 3 both these algebras have elliptic type i e their weyl groups have fundamental polyhedra of finite volume in corresponding hyperbolic spaces and have a lattice weyl vector the correcting automorphic forms are siegel modular forms the form corresponding to g 1 is the classical siegel cusp form of weight 5 which is the product of ten even theta constants in particular we find an infinite product formula for this modular form

cohomology of lie superalgebras and their generalizations the cohomology groups of lie superalgebras and more generally of e lie algebras are introduced and investigated the main emphasis is on the case where the module of coefficients is nontrivial two general propositions are proved which help to calculate the cohomology groups several examples are included to show the peculiarities of the super case for l sl 1 2 the cohomology groups h1 l v and h2 l v with v a finite dimensional simple graded l module are determined and the result is used to show that h2 l u l with u l the enveloping algebra of l is trivial this implies that the superalgebra u l does not admit any nontrivial formal deformations in the sense of gerstenhaber garland s theory of universal central extensions of lie algebras is generalized to the case of e lie algebras

bernstein gelfand gelfand resolutions for basic classical lie superalgebras abstract we study kostant cohomology and bernstein gelfand gelfand resolutions for finite dimensional representations of basic classical lie superalgebras for each choice of parabolic subalgebra and representation of such a lie superalgebra there is a natural definition of the boundary and coboundary operators which define co homology of the nilradical of the parabolic subalgebra we prove that complete reducibility of the homology groups is a necessary condition to have a resolution of an irreducible module in terms of generalised verma modules every such a resolution is then given by modules induced by these homology groups we also prove that if these homology groups are completely reducible a sufficient condition for the existence of this resolution is the property that these groups are isomorphic to the kernel of the kostant quabla operator which is equivalent with disjointness of the boundary and coboundary operators then we use these results to derive very explicit conditions under which bgg resolutions exist which are particularly useful for the superalgebras of type i for the unitarisable representations of gl m n and osp 2 2 n we derive conditions on the parabolic subalgebra under which the bgg resolutions exist we also apply the obtained theory to construct specific examples of bgg resolutions for osp m 2 n finally we state some results for typical modules of gl m n and osp 2 2 n

complete intersection varieties with ample cotangent bundles any smooth projective variety contains many complete intersection subvarieties with ample cotangent bundles of each dimension up to half its own dimension

testing sets for properness of polynomial mappings in this paper we characterize testing sets for properness of polynomial mappings f mathbb c n rightarrow mathbb c m we also study the set of points at which such mappings are not proper as the first application we give a proof of the complementary conjecture of mckay and wang conjecture 12 in 16 the second application is an answer to the problem of kraft russell about a characterization of mathbb c n the third application is given in 13 a solution of the problem of van den essen and shpilrain about endomorphism of the ring mathbb c x 1 x n the fourth application is a theorem about extensions of affine varieties

uniqueness for the navier stokes equations and multipliers between sobolev spaces abstract we re prove various uniqueness theorems for the navier stokes equations stating the assumptions in terms of multipliers between sobolev spaces instead of lebesgue or lorentz spaces

from dynkin diagram symmetries to fixed point structures any automorphism of the dynkin diagram of a symmetrizable kac moody algebra g induces an automorphism of g and a mapping between highest weight modules of g for a large class of such dynkin diagram automorphisms we can describe various aspects of these maps in terms of another kac moody algebra the orbit lie algebra g in particular the generating function for the trace of over weight spaces which we call the twining character of g with respect to the automorphism is equal to a character of g the orbit lie algebras of untwisted affine lie algebras turn out to be closely related to the fixed point theories that have been introduced in conformal field theory orbit lie algebras and twining characters constitute a crucial step towards solving the fixed point resolution problem in conformal field theory

hilbert series howe duality and branching for classical groups an extension of the littlewood restriction rule is given that covers all pertinent parameters and simplifies to the original under littlewood s hypotheses two formulas are derived for the gelfand kirillov dimension of any unitary highest weight representation occurring in a dual pair setting one in terms of the dual pair index and the other in terms of the highest weight for a fixed dual pair setting all the irreducible highest weight representations which occur have the same gelfand kirillov dimension we define a class of unitary highest weight representations and show that each of these representations l has a hilbert series hl q of the form

generalized kac moody algebras on etudie une classe d algebres de lie qui ont une forme bilineaire contravariante qui est presque definie positive ces algebres generalisent celles de kac moody et peuvent etre considerees comme des algebres de kac moody avec des racines simples imaginaires

generalized kac moody algebras and the modular function j the generalized kac moody algebras were introdcued by borcherds in his study of the vertex algebras and monstrous moonshine b1 b3 the structure and the representation theory of generalized kac moody algebras are very similar to those of kac moody algebras and a lot of facts about kac moody algebras can be generalized to generalized kac moody algebras the main difference is that the generalized kac moody algebras may have imaginm 3 simple roots with norms 1 for unitarizable irreducible highest weight modules over symmetrizable generalized kac moody algebras borcherds proved a character formula b2 called the weyl kac borcherds formula which generalizes the weylkac formula for integrable highest weight modules over symmetrizable kac moody algebras k as in the case of kac moody algebras the weyl kac borcherds formula when applied to the l dimensional trivial representation yields the denominator identio in this paper using the weyl kac borcherds formula and the denominator identity we obtain a closed form root multiplicity formula for all symmetrizable generalized kac moody algebras theorem 2 1 our formula enables us to study the structure of a symmetrizable generalized kac moody algebra 9 as a representation of a kac moody algebra 0o contained in the choice of 0 gives rise to various expressions of the root multiplicities of 13 which would yield some interesting combinatorial identities in sect 3 we apply this principle to a rank 2 generalized kac moody algebra to obtain a combinatorial identity for the pascal numbers in b3 borcherds proved the moonshine conjecture cni by constructing the monster lie algebra using the monster vertex algebra flm and the vertex algebra associated with the 2 dimensional even lorentzian lattice iil b 1 the monster lie

dirac cohomology k characters and branching laws inspired by work of enright and willenbring ew we prove a generalized littlewood s restriction formula in terms of dirac cohomology our approach is to use a character formula of irreducible unitary lowest weight modules instead of the bernstein gelfand gelfand resolution and the proof is much simpler we also show that our branching formula is equivalent to the formula of enright and willenbring in terms of nilpotent lie algebra cohomology this follows from the close relationship between the dirac cohomology and the corresponding nilpotent lie algebra cohomology for unitary representations of semisimple lie groups of hermitian type which was established in hpr

hook young diagrams with applications to combinatorics and to representations of lie superalgebras definition et etude des tableaux k l semi standards applications a la combinatoire et aux representations des superalgebres de lie des diagrammes de young a crochets