a chiral borel weil bott theorem abstract we compute the cohomology of modules over the algebra of twisted chiral differential operators over the flag manifold this is applied to 1 finding the character of g integrable irreducible highest weight modules over the affine lie algebra at the critical level and 2 computing a certain elliptic genus of the flag manifold the main tool is a result that interprets the drinfeld sokolov reduction as a derived functor

on the rigidity theorems of witten in this paper we prove the rigidity theorems predicted by witten in 1986 about the index of certain elliptic operators on manifolds with an s1 action w witten s insight was the culmination of an interesting interchange of ideas between him and hopkins landweber ochanine and stong for the detailed history we refer the reader to la the present account is essentially a reinterpretation of the second author s taubes t original proof of the theorem the senior author s contribution was solely to notice that the rather densely written arguments of the original manuscript could be formulated in terms of the well known fixed point formulae of equivariant index theory and equivariant cohomology in this context the final proof then appears in direct lineage of the atiyah hirzebruch theorem concerning the vanishing of the a genus of spin manifolds admitting a circle action a h and of the even older idea of lusztig concerning circle actions in the complex case there remains however the beautiful physics inspired novelty of connecting these techniques with elliptic function theory

a classification of simple lie modules having a 1 dimensional weight space let l denote a simple lie algebra over the complex numbers in this paper we classify and construct all simple l modules which may be infinite dimensional but have at least one 1 dimensional weight space this completes the study begun earlier by the authors for the case of l a the approach presented here relies heavily on the results of suren fernando whose dissertation dealt with simple weight modules and their weight systems 0 introduction let l be a finite dimensional simple lie algebra over the complex field c having a cartan subalgebra h and denote by c l the centralizer of h in the universal enveloping algebra u of l if x h c is a weight function of a 1 dimensional weight space mx in a simple l module m then q c l c defined by q c v cv for v e mx and c e c l is an algebra homomorphism called a mass function of m clearly x restricted to h is equal to x conversely given any algebra homomorphism q c l c one can construct a unique simple l module which admits x as a mass function 4 10 in 4 the authors determined all algebra homomorphisms q c l c for the simple lie algebras l of type an and using these classified all pointed an modules where we call a module pointed if it is simple and has at least one 1 dimensional weight space in this paper we complete the classification of all pointed l modules for arbitrary simple lie algebras the collection of all pointed l modules clearly includes the highest weight l modules and is included in the collection of all harish chandra l modules relative to the cartan subalgebra h i e simple l modules having a weight space decomposition and finite dimensional weight spaces relative to h the latter inclusion is strict since there exist examples of harish chandra a2 modules in which every weight space is two dimensional in the special case of a1 modules every harish chandra a1 module is pointed our approach to this problem makes heavy use of the results from fernando s thesis 8 in particular from fernando s results we see that in place of determining all algebra homomorphisms q c l c it suffices to find only those algebra homomorphisms x which are associated with so called pointed torsion free l received by the editors may 23 1984 this paper was presented at the 1984 canadian mathematical society s summer seminar on lie algebras and related topics held at the university of windsor 1980 mathematics subject classification primary 17b10

the universal regular quotient of the chow group of points on projective varieties let x be a projective variety of dimension n de ned over an algebraically closed eld k for x irreducible and non singular matsusaka ma constructed an abelian variety alb x and a morphism a x alb x called the albanese variety and mapping respectively depending on the choice of a base point on x which is universal among the morphisms to abelian varieties see lang la serre se for other constructions over the eld of complex numbers the existence of alb x and a was known before and has a purely hodge theoretic description see igusa i for the hodge theoretic construction incidentally the terminology albanese variety was introduced by a weil for reasons explained in his commentary on the article 1950a of volume i of his collected works see w one of which is that the paper alb of albanese de nes it for a surface as a quotient of the group of 0 cycles of degree 0 modulo an equivalence relation let chn x deg 0 denote the chow group of 0 cycles of degree 0 on x modulo rational equivalence when x is irreducible and non singular a remarkable feature of the albanese morphism a is that it factors through a regular homomorphism u chn x deg 0 alb x that is a homomorphism which when composed with the cycle map c x chn x deg 0 gives an algebraic morphism this follows immediately from the fact that an abelian variety does not contain any rational curve thus one can reformulate matsusaka s theorem as the invent math 135 595 664 1999 doi 10 1007 s002229900890

fock representations of the affine lie algebra a 1 1 the aim of this note is to show that the affine lie algebraa1 1 has a natural family v of fock representations on the spacec xi yj i andj parametrized by v c2 by corresponding the highest weight of to each the parameter spacec2 forms a double cover of the weight spacec 0 c 1 with singularities at linear forms of level 2 this number is 1 times the dual coxeter number our results contain explicit realizations of irreducible non integrable highest wieghta1 1 modules for generic v

on generalizations of the albanese variety in this note the relation between two generalizations of the classical albanese variety namely the one for smooth non complete varieties by faltings and wustholz and the one for singular varieties by esnault srinivas and viehweg is investigated over the field of complex numbers

gerbes of chiral differential operators ii vertex algebroids introduction 605 0 recollections and notation 612 1 vertex algebroids 615 2 from vertex algebras to vertex algebroids 622 3 category of vertex algebroids 628 4 cofibered structure 633 5 computation of a cocycle 637 6 characteristic classes of vector bundles 643 7 torseurs and gerbes 647 8 gerbes of vertex algebroids 653 9 chiral differential operators on curves and sheaves of virasoro algebras 657 10 enveloping algebra of a vertex algebroid 664 11 calabi yau structures 668 12 virasoro and calabi yau structures 671 references 680

d modules on the affine grassmannian and representations of affine kac moody algebras let mathfrak g be a simple lie algebra for a level thought of as a symmetric mathfrak g invariant form of mathfrak g let hat mathfrak g kappa be the corresponding affine kac moody algebra let grg be the affine grassmannian of mathfrak g and let d grg mod be the category of twisted right d modules on grg by taking global sections of a d module we obtain a functor d grg mod mathfrak g mod it is known that this functor is exact and faithful when is less than critical or irrational in this paper we show that the functor is also exact and faithful when is the critical level

chiral de rham complex we define natural sheaves of vertex algebras over smooth manifolds which may be regarded as semi infinite de rham complexes of certain d modules over the loop spaces for calabi yau manifolds they admit n 2 supersymmetry connection with wakimoto modules is discussed

sharp de rham realization we introduce the sharp universal extension of a 1 motive with additive factors and torsion over a field of characteristic zero we define the sharp de rham realization t by passing to the lie algebra over the complex numbers we then show a sharp de rham comparison theorem in the category of formal hodge structures for a free 1 motive along with its cartier dual we get a canonical connection on their sharp extensions yielding a perfect pairing on sharp realizations thus we show how to provide one dimensional sharp de rham cohomology h dr1 of algebraic varieties

z graded oscillator representations of symplectic lie algebras abstract in this paper we study two parameter z graded oscillator representations of symplectic lie algebras obtained from the natural oscillator representation by partially swapping differential operators and multiplication operators we prove that the homogeneous subspaces with nonzero degree or unequal parameters are infinite dimensional irreducible weight submodules with finite dimensional weight subspaces when the two parameters are equal the homogeneous subspace of degree zero is exactly a direct sum of two infinite dimensional irreducible weight submodules with finite dimensional weight subspaces in generic case these irreducible modules are non unitary modules without highest weight vectors our results are extensions of howe s oscillator construction of infinite dimensional multiplicity free irreducible representations for special linear lie algebras

on modules of bounded multiplicities for the symplectic algebras simple infinite dimensional highest weight modules having bounded weight multipicities are classified as submodules of a tensor product also it is shown that a simple torsion free module of finite degree tensored with a finite dimensional module is completely reducible

localization of affine w algebras we introduce the notion of an asymptotic algebra of chiral differential operators we then construct via a chiral hamiltonian reduction one such algebra over a resolution of the intersection of the slodowy slice with the nilpotent cone we compute the space of global sections of this algebra thereby proving a localization theorem for affine w algebras at the critical level

lie algebra modules with finite dimensional weight spaces i let g denote a reductive lie algebra over an algebraically closed field of characteristic zero and let 4 denote a cartan subalgebra of g in this paper we study finitely generated g modules that decompose into direct sums of finite dimensional 4 weight spaces we show that the classification of irreducible modules in this category can be reduced to the classification of a certain class of irreducible modules those we call torsion free modules we also show that if g is a simple lie algebra that admits a torsion free module then g is of type a or c

semi infinite induction and wakimoto modules the purpose of this paper is to suggest the construction and study properties of semi infinite induction which relates to semi infinite cohomology the same way induction relates to homology and coinduction to cohomology we prove a version of the shapiro lemma showing that the semi infinite cohomology of a module is isomorphic to that of the semi infinitely induced module a practical outcome of our construction is a simple construction of the wakimoto modules highest weight modules used in double sided bgg resolutions of irreducible modules

some remarks concerning the grothendieck period conjecture we discuss various results and questions around the grothendieck period conjecture which is a counterpart concerning the de rham betti realization of algebraic vari eties over number fields of the classical conjectures of hodge and tate these results give new evidence towards the conjectures of grothendieck and kontsevich zagier concerning transcen dence properties of the torsors of periods of varieties over number fields let q be the algebraic closure of q in c let x be a smooth projective variety over q and let x an c denote the compact complex analytic manifold that it defines the grothendieck period conjecture in codimension k on x denoted gpc k x asserts that any class in the

elliptic genera of toric varieties and applications to mirror symmetry the paper contains a proof that elliptic genus of a calabi yau manifold is a jacobi form finds in which dimensions the elliptic genus is determined by the hodge numbers and shows that elliptic genera of a calabi yau hypersurface in a toric variety and its mirror coincide up to sign the proof of the mirror property is based on the extension of elliptic genus to calabi yau hypersurfaces in toric varieties with gorenstein singularities

wakimoto modules opers and the center at the critical level abstract wakimoto modules are representations of affine kac moody algebras in fock modules over infinite dimensional heisenberg algebras in this paper we present the construction of the wakimoto modules from the point of view of the vertex algebra theory we then use wakimoto modules to identify the center of the completed universal enveloping algebra of an affine kac moody algebra at the critical level with the algebra of functions on the space of opers for the langlands dual group on the punctured disc giving another proof of the theorem of b feigin and the author