exchange dynamical quantum groups for any simple lie algebra and any complex number q which is not zero or a nontrivial root of unity but may be equal to 1 we construct a dynamical quantum group hopf algebroid whose representation theory is essentially the same as the representation theory of the quantum group uq this dynamical quantum group is obtained from the fusion and exchange relations between intertwining operators in representation theory of uq and is an algebraic structure standing behind these relations

hochschild cohomology and atiyah classes abstract in this paper we prove that on a smooth algebraic variety the hkr morphism twisted by the square root of the todd genus gives an isomorphism between the sheaf of poly vector fields and the sheaf of poly differential operators both considered as derived gerstenhaber algebras in particular we obtain an isomorphism between hochschild cohomology and the cohomology of poly vector fields which is compatible with the lie bracket and the cupproduct the latter compatibility is an unpublished result by kontsevich our proof is set in the framework of lie algebroids and so applies without modification in much more general settings as well

component groups of unipotent centralizers in good characteristic abstract let g be a connected reductive group over an algebraically closed field of good characteristic for u g unipotent we describe the conjugacy classes in the component group a u of the centralizer of u our results extend work of the second author done for simple adjoint g over the complex numbers when g is simple and adjoint the previous work of the second author makes our description combinatorial and explicit moreover it turns out that knowledge of the conjugacy classes suffices to determine the group structure of a u thus we obtain the result previously known through case checking that the structure of the component group a u is independent of good characteristic

representations of semisimple lie algebras in prime characteristic and the noncommutative springer resolution we prove most of lusztig s conjectures on the canonical basis in homology of a springer ber the conjectures predict that this basis controls numerics of representations of the lie algebra of a semisimple algebraic group over an algebraically closed eld of positive characteristic we check this for almost all characteristics to this end we construct a noncommutative resolution of the nilpotent cone which is derived equivalent to the springer resolution on the one hand this noncommutative resolution is closely related to the positive characteristic derived localization equivalences obtained earlier by the present authors and rumynin on the other hand it is compatible with the t structure arising from an equivalence with the derived category of perverse sheaves on the ane ag variety of the langlands dual group this equivalence established by arkhipov and the rst author ts the framework of local geometric langlands duality the latter compatibility allows one to apply frobenius purity theorem to deduce the desired properties of the basis we expect the noncommutative counterpart of the springer resolution to be of independent interest from the perspectives of algebraic geometry and geometric langlands duality

character sheaves v this paper is part of a series s 131 devoted to the study of a class g of irreducible perverse sheaves called character sheaves on a connected reductive algebraic group g the numbering of chapters sections and references will continue that of s 131 this paper is a step towards the classification of character sheaves on g one of the main results is the following one under certain assumptions there is a natural surjective map with finite fibers from g to the set of all pairs 9 c up to conjugacy by the weyl group where 9 is a tame local system on the maximal torus and c is a two sided cell in the stabilizer w of 9 in the weyl group the assumptions made on g are

quantum cohomology of the hilbert scheme of points on an resolutions we determine the two point invariants of the equivariant quantum cohomology of the hilbert scheme of points of surface resolutions associated to type an singularities the operators encoding these invariants are expressed in terms of the action of the the affine lie algebra b gl n 1 on its basic representation assuming a certain nondegeneracy conjecture these operators determine the full structure of the quantum cohomology ring a relationship is proven between the quantum cohomology and gromov witten donaldsonthomas theories of an p 1 we close with a discussion of the monodromy properties of the associated quantum differential equation and a generalization to singularities of type d and e

local representation groups group cohomology techniques are used to derive conditions for a group g to be a local splitting group for g as a first step the exactness of the inflation restriction sequence gives a characterisation of factor systems arising in locally operating representations of transitive transformation groups some examples and applications of the theory are also given

koszul duality and modular representations of semi simple lie algebras in this paper we prove that if g is a connected simply connected semi simple algebraic group over an algebraically closed field of sufficiently large characteristic then all the blocks of the restricted enveloping algebra of the lie algebra of g can be endowed with a koszul grading extending results of andersen jantzen soergel we also give information about the koszul dual rings in the case of the block associated to a regular character x of the harish chandra center the dual ring is related to modules over the enveloping algebra specialized at x with generalized trivial frobenius character our main tool is the localization theory developed by bezrukavnikov mirkovic rumynin

construction of central elements in the affine hecke algebra via nearby cycles 0 1 overview let g be a connected reductive group over a finite field fq and let g k be the corresponding group over the local field k fq t let g o g k be a maximal compact subgroup of g k here o fq t and let hsph denote the hecke algebra of g k with respect to g o in other words hsph as a vector space consists of compactly supported bi g o invariant functions g k q and the product is defined by

cohomology of tilting modules over quantum groups and t structures on derived categories of coherent sheaves the paper is concerned with cohomology of the small quantum group at a root of unity and of its upper triangular subalgebra with coefficients in a tilting module it turns out to be related to irreducible objects in the heart of a certain t structure on the derived category of equivariant coherent sheaves on the springer resolution and to equivariant coherent ic sheaves on the nil cone the support of the cohomology is described in terms of cells in affine weyl groups the basis in the grothendieck group provided by the cohomology modules is shown to coincide with the kazhdan lusztig basis as predicted by j humphreys and v ostrik

homology of the zero set of a nilpotent vector field on a flag manifold 0 1 let x be a linear transformation of a finite dimensional vector space v the configuration of flags in v which are fixed by x has rather remarkable properties when x is unipotent though this case is especially interesting the proper generality in which to study such configurations is in the theory of reductive algebraic groups where their definition can be reformulated in the language of borel subalgebras as follows let g be a connected reductive group over c with lie algebra g and let n e g be a nilpotent element let q be the variety of all borel subalgebras of g and let

hochschild cohomology and deformation quantization of affine toric varieties abstract for an affine toric variety spec a we give a convex geometric description of the hodge decomposition of its hochschild cohomology under certain assumptions we compute the dimensions of the hodge summands t i 1 a generalizing the existing results about the andre quillen cohomology group t 1 1 a we prove that every poisson structure on a possibly singular affine toric variety can be quantized in the sense of deformation quantization

two row w graphs in affine type a abstract for affine symmetric groups we construct finite w graphs corresponding to two row shapes and prove their uniqueness this gives the first non trivial family of purely combinatorial constructions of finite w graphs in an affine type we compare our construction with quotients of periodic w graphs defined by lusztig under certain positivity assumption on the latter the two are shown to be isomorphic

locally operating realizations of transformation lie groups using the mackey theory of induced representations a systematic study of the locally operating multiplier realizations of a connected lie group g that acts transitively on a space time manifold is presented we obtain a mathematical characterization of the locally operating multiplier realizations and a reduction of the problem of multiplier locally operating realizations to linear ones via a splitting group g m for g in this way the locally operating multiplier realizations are obtained by induction from finite dimensional linear representations of a well determined subgroup of g some examples such as the two dimensional euclidean group the galilei group and the one dimensional newton hooke group are given

localization for quantum groups at a root of unity let c be the field of complex numbers and fix q g c let g be a semi simple lie algebra over c and let g be the corresponding simply connected algebraic group let uq be the quantized enveloping algebra of q let oq be the algebra of quantized functions on g let oq b be the quotient hopf algebra of oq corresponding to a borel subgroup b of g in the paper bk we defined categories of equivariant quantum o modules and pg modules on the quantum flag variety of g we proved that the beilinson bernstein localization theorem holds at a generic q namely the global section functor gives an equivalence between categories of modules and vq iaod es on the quantum flag variety thus one can translate questions about the represen tation theory of quantum groups to the study of the noncommutative geometry of the quantum flag variety in this paper we prove that a derived version of this theorem holds at the root of unity case using this equivalence we get that one can understand the representation theory of quantum groups at roots of unity through the now commutative geometry of the springer fibers we now recall the main results in bk the constructions given there are crucial for the present paper and a fairly detailed survey of the material there is given in the next section we defined an equivariant sheaf of quasi coherent modules over the quantum flag variety to be a left oq mod e equipped with a right q b comod ile structure satisfying certain compatibility conditions such objects form a category denoted ai q gq it contains certain line bundles oq for in the weight lattice we proved that oq x is ample for 0 holds for every q this implies that the category m bq gq is a proj category in the sense of serre

the ramification of centres lie algebras in positive characteristic and quantised enveloping algebras let h be a hopf algebra over the field k which is a finite module over a central affine sub hopf algebra r examples include enveloping algebras u mathfrak g of finite dimensional k lie algebras mathfrak g in positive characteristic and quantised enveloping algebras and quantised function algebras at roots of unity the ramification behaviour of the maximal ideals of z h with respect to the subalgebra r is studied and the conclusions are then applied to the cases of classical and quantised enveloping algebras in the case of u mathfrak g for mathfrak g semisimple a conjecture of humphreys 28 on the block structure of u mathfrak g is confirmed in the case of u epsilon mathfrak g for mathfrak g semisimple and epsilon an odd root of unity we obtain a quantum analogue of a result of mirkovic and rumynin 35 and we fully describe the factor algebras lying over the regular sheet 9 the blocks of u epsilon mathfrak g are determined and a necessary condition which may also be sufficient for a baby verma u epsilon mathfrak g module to be simple is obtained

on the local equivalence in a unidimensional world the physical meaning of some semiunitary irreducible realizations of the unidimensional poincare and galilei groups including time inversion which are characteristic of a unidimensional world is investigated it is shown that in order to achieve this purpose the local equivalence is the most appropriate and it does not coincide with an equivalence of the global kind in this study use has been made of the theory of locally operating realizations developed by hoogland while dealing with the nontrivial factor systems the representation group technics were used

căldăraru s conjecture and tsygan s formality in this paper we complete the proof of caldararu s conjecture on the compatibility between the module structures on differential forms over poly vector fields and on hochschild homology over hochschild cohomology in fact we show that twisting with the square root of the todd class gives an isomorphism of precalculi between these pairs of objects our methods use formal geometry to globalize the local formality quasi isomorphisms introduced by kontsevich and shoikhet the existence of the latter was conjectured by tsygan we also rely on the fact recently proved by the first two authors that shoikhet s quasi isomorphism is compatible with cap products after twisting with a maurer cartan element

corrigendum to deformation quantization in algebraic geometry adv math 198 2005 383 432 this note contains a correction of the proofs of the main results of 3 namely theorems 0 1 and 0 2 the results are correct as originally stated the mistake in my original proofs was discovered michel van den bergh and i thank him for calling my attention to it the way to fix the proofs is essentially contained in his paper 2 let me begin by explaining the mistake as can be seen in example 0 1 below the mistake itself is of a rather elementary nature but it was obscured by the complicated context suppose k is a field of characteristic 0 and x is a smooth separated n dimensional scheme over k recall that the coordinate bundle coorx is an infinite dimensional bundle over x with free action by the group gln k the quotient bundle is by definition

kirwan surjectivity for quiver varieties for algebraic varieties defined by hyperkahler or more generally algebraic symplectic reduction it is a long standing question whether the hyperkahler kirwan map on cohomology is surjective we resolve this question in the affirmative for nakajima quiver varieties we also establish similar results for other cohomology theories and for the derived category our proofs use only classical topological and geometric arguments

deformations of affine varieties and the deligne crossed groupoid abstract let x be a smooth affine algebraic variety over a field k of characteristic 0 and let r be a complete parameter k algebra e g r k we consider associative resp poisson r deformations of the structure sheaf o x the set of r deformations has a crossed groupoid i e strict 2 groupoid structure our main result is that there is a canonical equivalence of crossed groupoids from the deligne crossed groupoid of normalized polydifferential operators resp polyderivations of x to the crossed groupoid of associative resp poisson r deformations of o x the proof relies on a careful study of adically complete sheaves in the associative case we also have to use ring theory ore localizations and the properties of the hochschild cochain complex the results of this paper extend previous work by various authors they are needed for our work on twisted deformation quantization of algebraic varieties

local realizations of kinematical groups with a constant electromagnetic field i the relativistic case this paper is devoted to the study of the description of elementary physical systems interacting with an external constant electromagnetic field and the construction of their differential wave equations from a group theoretical point of view in this context certain local realizations of the poincare group are studied the linearization of this problem is carried out by building the associated representation group that turns out to be the well known maxwell group in this way the usual method concerning local realizations that has been employed in studying free systems to the interacting case is extended

a morita theorem for modular finite w algebras we consider the lie algebra mathfrak g of a simple simply connected algebraic group over a field of large positive characteristic for each nilpotent orbit mathcal o subseteq mathfrak g we choose a representative e in mathcal o and attach a certain filtered associative algebra widehat u mathfrak g e known as a finite w algebra defined to be the opposite endomorphism ring of the generalised gelfand graev module associated to mathfrak g e this is shown to be morita equivalent to a certain central reduction of the enveloping algebra of u mathfrak g the result may be seen as a modular version of skryabin s equivalence

two dimensional representations of finitely generated free group over an arbitrary field working over an arbitrary field we give equivalent conditions for a representation of a finitely generated free group with given traces and determinants to exist and to be reducible respectively also we classify all two dimensional representations of a finitely generated free group

vector bundle representations of groups in quantum physics the theory of vector bundle representations of lie groups g is developed induced and locally operating linear representations of g are shown to be generated by vector bundle representations and gauge equivalence of locally operating linear representations is shown to correspond to equivalence of vector bundle representations it is pointed out that every vector bundle representation is equivalent to an induced one some applications to the quantum systems associated with monopole and instanton gauge field configurations are also discussed

adelic chern forms and applications let x be a variety over a field of characteristic 0 given a vector bundle e on x we construct chern forms c v t x ax here a x is the sheaf of beilinson adeles and v is an adelic connection when x is smooth wt x a x h r x the algebraic de rham cohomology and c q v are the usual chern classes we include three applications of the construction 1 existence of adelic secondary chern simons characteristic classes on any smooth x and any vector bundle e 2 proof of the bott residue formula for a vector field action and 3 proof of a gauss bonnet formula on the level of differential forms namely in the de rham residue complex 0 introduction let x be a scheme of finite type over a field k according to beilinson be given any quasi coherent x module m and an integer q9 there is a flasque ox module a dcm called the sheaf of adeles this is a gen eralization of the classical adeles of number theory cf example 2 3 moreover there are homomorphisms d aj ed m a m which make ared m into a complex and m ared m is quasi isomorphism now let qx be the algebra of kahler differential forms on x in hy we proved that the sheaf

differential operators on quantized flag manifolds at roots of unity abstract the quantized flag manifold which is a q analogue of the ordinary flag manifold is realized as a non commutative scheme and we can define the category of d modules on it using the framework of non commutative algebraic geometry however when the parameter q is a root of unity lusztig s frobenius morphism allows us to handle d modules on the quantized flag manifold through modules over a certain sheaf of rings on the ordinary flag manifold in this paper we will show that this sheaf of rings on the ordinary flag manifold is an azumaya algebra over its center we also show that its restriction to certain subsets are split azumaya algebras these are analogues of some results of bezrukavnikov mirkovic rumynin on d modules on flag manifolds in positive characteristics

equivariant elliptic cohomology and rigidity equivariant elliptic cohomology with complex coefficients was defined axiomatically by ginzburg kapranov and vasserot and constructed by grojnowski we give an invariant definition of complex s 1 equivariant elliptic cohomology and use it to give an entirely cohomological proof of the rigidity theorem of witten for the elliptic genus we also state and prove a rigidity theorem for families of elliptic genera

quantized symplectic actions and w algebras with a nilpotent element in a semisimple lie algebra g one associates a finitely generated associative algebra w called a w algebra of finite type this algebra is obtained from the universal enveloping algebra u g by a certain hamiltonian reduction we observe that w is the invariant algebra for an action of a reductive group g with lie algebra g on a quantized symplectic affine variety and use this observation to study w our results include an alternative definition of w a relation between the sets of prime ideals of w and of the corresponding universal enveloping algebra the existence of a one dimensional representation of w in the case of classical g and the separation of elements of w by finite dimensional representations

fedosov quantization in positive characteristic the so called deformation quantization of symplectic manifolds originally ap peared in c symplectic geometry the crucial breakthrough was independently made in the early 1980s by b fedosov and m de wilde p lecomte the input of a deformation quantization problem is a symplectic or more generally a poisson manifold m the output is a non commutative one parameter deformation of the algebra of functions on m both fedosov and de wilde lecomte provided general procedures which solve the deformation quantization problem for c manifolds recently motivated in part by m kontsevich k there has been much interest in generalizing the deformation quantization procedures to the case of algebraic manifolds equipped with an algebraic symplectic form see e g bk1 y and an earlier paper nt for the holomorphic situation in particular in bk1 it has been shown that under some mild assumptions on the manifold the fedosov quantization can be made to work in the algebraic setting the present paper is a continuation of bk1 to which we refer the reader for a more complete bibliography and historical discussion namely one of the most important assumptions in bk1 as well as in other papers on the subject was that the field of definition for all algebraic manifolds has characteristic 0 in this paper we study what happens in the case of positive characteristic the most obvious new feature of the theory in positive characteristic is the presence of a large poisson center in the sheaf ox of functions on a poisson manifold x since for any two local functions f g ox and any poisson bracket on ox we have fp g 0 the image px c ox of the frobenius map lies in the center of any poisson structure this phenomenon already observed in bmr allows for interesting applications see e g bk2 but makes the quantization procedures more involved in this paper we were not able to prove any meaningful results for general quantizations in positive characteristic and we had to restrict our attention to a special class of them the so called frobenius constant quantization roughly speaking the precise definition is definition 1 4 below a quantization is frobenius constant if the poisson center opx c ox stays central in the quantized algebra oh for quantization of this type we were able to achieve under mild assumptions on the manifold x a reasonably complete classification theorem in particular frobenius constant quantizations do exist

connection problems for quantum affine kz equations and integrable lattice models cherednik attached to an affine hecke algebra module a compatible system of difference equations called quantum affine knizhnik zamolodchikov kz equations in the case of a principal series module we construct a basis of power series solutions of the quantum affine kz equations relating the bases for different asymptotic sectors gives rise to a weyl group cocycle which we compute explicitly in terms of theta functions

on global deformation quantization in the algebraic case abstract we give a proof of yekutieli s global algebraic deformation quantization result which does not rely on the choice of local sections of the bundle of affine coordinate systems instead we use an argument inspired by algebraic de rham cohomology

koszul duality patterns in representation theory the aim of this paper is to work out a concrete example as well as to provide the general pattern of applications of koszul duality to repre sentation theory the paper consists of three parts relatively independent of each other the first part gives a reasonably selfcontained introduction to koszul rings and koszul duality koszul rings are certain z graded rings with particularly nice homological properties which involve a kind of duality thus to a koszul ring one associates naturally the dual koszul ring the second part is devoted to an application to representation theory of semisimple lie algebras we show license or copyright restrictions may apply to redistribution see http www ams org journal terms of use koszul duality patterns 527 that the block of the bernstein gelfand gelfand category o that corresponds to any fixed central character is governed by the koszul ring moreover the dual of that ring governs a certain subcategory of the category o again this generalizes the selfduality theorem conjectured by beilinson and ginsburg in 1986 and proved by soergel in 1990 in the third part we study certain cate gories of mixed perverse sheaves on a variety stratified by affine linear spaces we provide a general criterion for such a category to be governed by a koszul ring in the flag variety case this reduces to the setup of part two in the more general case of affine flag manifolds and affine grassmannians the criterion should yield interesting results about representations of quantum groups and affine lie algebras department of mathematics massachusetts institute of technology cambridge massachusetts 02139 e mail address sasha math mit edu department of mathematics the university of chicago chicago illinois 60637 e mail address ginzburg math uchicago edu max planck institut fur mathematik gottfried claren strase 26 d 53 bonn 3 germany current address mathematisches institut universitat freiburg albertstrase 23b d 79104 freiburg germany e mail address soergel sun1 mathematik uni freiburg de license or copyright restrictions may apply to redistribution see http www ams org journal terms of use

deformation quantization of gerbes this is the first in a series of articles devoted to deformation quantization of gerbes we introduce basic definitions interpret deformations of a given stack as maurer cartan elements of a differential graded lie algebra dgla and classify deformations of a given gerbe in terms of maurer cartan elements of the dgla of hochschild cochains twisted by the cohomology class of the gerbe we also classify all deformations of a given gerbe on a symplectic manifold as well as provide a deformation theoretic interpretation of the first rozansky witten class

products of conjugacy classes of two by two matrices abstract we compute the covering number and the extended covering number for the group psl 2 f with a field f similar numbers are defined for any group and they are computed for the groups gl 2 f pgl 2 f and sl 2 f

locally operating realizations of nonconnected transformation lie groups a systematic study of locally operating multiplier realizations of nonconnected lie groups of transformations is presented that generalizes previous results on connected groups the semilinear locally operating multiplier realizations of a nonconnected group g are those obtained through an induction process from the finite dimensional semilinear representations of a given subgroup of a representation group g for g

dimensions of irreducible modules over w algebras and goldie ranks the main goal of this paper is to compute two related numerical invariants of a primitive ideal in the universal enveloping algebra of a semisimple lie algebra the first one very classical is the goldie rank of an ideal the second one is the dimension of an irreducible module corresponding to this ideal over an appropriate finite w algebra we concentrate on the integral central character case we prove modulo a conjecture that in this case the two are equal our conjecture asserts that there is a one dimensional module over the w algebra with certain additional properties the conjecture is proved for the classical types also modulo the same conjecture we compute certain scale factors introduced by joseph this allows to compute the goldie ranks of the algebras of locally finite endomorphisms of simples in the bgg category mathcal o this completes a program of computing goldie ranks proposed by joseph in the 80 s for integral central characters and modulo our conjecture we also provide an essentially kazhdan lusztig type formula for computing the characters of the irreducibles in the brundan goodwin kleshchev category mathcal o for a w algebra again under the assumption that the central character is integral in particular this allows to compute the dimensions of the finite dimensional irreducible modules the formula is based on a certain functor from an appropriate parabolic category mathcal o to the w algebra category mathcal o this functor can be regarded as a generalization of functors previously constructed by soergel and by brundan kleshchev we prove a number of properties of this functor including the quotient property and the double centralizer property we develop several side topics related to our generalized soergel functor for example we discuss its analog for the category of harish chandra modules we also discuss generalizations to the case of categories mathcal o over dixmier algebras the most interesting example of this situation comes from the theory of quantum groups we prove that an algebra that is a mild quotient of luszitg s form of a quantum group at a root of unity is a dixmier algebra for this we check that the quantum frobenius epimorphism splits

green polynomials of weyl groups elliptic pairings and the extended dirac index abstract in this paper we give a uniform construction of irreducible genuine characters of the pin cover w of a weyl group w and put them into the context of theory of springer representations in the process we provide a direct connection between springer theory via green polynomials the irreducible representations of w and an extended dirac operator for graded hecke algebras we also introduce a q elliptic pairing for w with respect to the reflection representation v these constructions are of independent interest the q elliptic pairing is a generalization of the elliptic pairing of w introduced by reeder and it is also related to s kato s notion of graded kostka systems for the semidirect product a w c w s v

the frobenius morphism in invariant theory abstract let r be the homogeneous coordinate ring of the grassmannian g gr 2 n defined over an algebraically closed field of characteristic p 0 in this paper we give a completely characteristic free description of the decomposition of r considered as a graded r p module into indecomposables frobenius summands as a corollary we obtain a similar decomposition for the frobenius pushforward of the structure sheaf of g and we obtain in particular that this pushforward is almost never a tilting bundle on the other hand we show that r provides a noncommutative resolution for r p when p n 2 generalizing a result known to be true for toric varieties in both the invariant theory and the geometric setting we observe that if the characteristic is not too small the frobenius summands do not depend on the characteristic in a suitable sense in the geometric setting this is an explicit version of a general result by bezrukavnikov and mirkovic on frobenius decompositions for partial flag varieties we are hopeful that it is an instance of a more general p uniformity principle

quantum cohomology of the hilbert scheme of points in the plane we determine the ring structure of the equivariant quantum cohomology of the hilbert scheme of points of 2 the operator of quantum multiplication by the divisor class is a nonstationary deformation of the quantum calogero sutherland many body system a relationship between the quantum cohomology of the hilbert scheme and the gromov witten donaldson thomas correspondence for local curves is proven

irreducible representations of lie algebras of reductive groups and the kac weisfeiler conjecture letg be the lie algebra of a connected reductive groupg over an algebraically closed field of characteristicp 0 suppose thatg 1 is simply connected andp is good for the root system ofg ifp 2 suppose in addition thatg admits a nondegenerateg invariant trace form letv be an irreducible and faithfulg module withp character g it is proved in the paper that dimv is divisible byp1 2dim where stands for the orbit of under the coadjoint action ofg

2 groups 2 characters and burnside rings abstract we study 2 representations i e actions of 2 groups on 2 vector spaces our main focus is character theory for 2 representations to this end we employ the technique of extended burnside rings our main theorem is that the ganter kapranov 2 character is a particular mark homomorphism of the burnside ring as an application we give a new proof of osorno s formula for the ganter kapranov 2 character of a finite group

relative formality theorem and quantisation of coisotropic submanifolds we prove a relative version of kontsevich s formality theorem this theorem involves a manifold m and a submanifold c and reduces to kontsevich s theorem if c m it states that the dgla of multivector fields on an infinitesimal neighbourhood of c is l quasiisomorphic to the dgla of multidifferential operators acting on sections of the exterior algebra of the conormal bundle applications to the deformation quantisation of coisotropic submanifolds are given the proof uses a duality transformation to reduce the theorem to a version of kontsevich s theorem for supermanifolds which we also discuss in physical language the result states that there is a duality between the poisson sigma model on a manifold with a d brane and the poisson sigma model on a supermanifold without branes or more properly with a brane which extends over the whole supermanifold

quantum k theory on flag manifolds finite difference toda lattices and quantum groups let x y be homogeneous coordinates on cp a degree d holomorphic map cp 1 cp is uniquely determined up to a constant scalar factor by n 1 relatively prime degree d binary forms f0 x y fn x y omitting the condition that the forms are relatively prime we compactify the space of degree d holomorphic maps cp 1 cp to a complex projective space of dimension n 1 d 1 1 we denote this compactification of the space of maps by cp d this construction defines the compactification d r i 1cp ni 1 di of the space of degree d d1 dr maps from cp 1 to composing degree d holomorphic maps from cp 1 to the flag manifold x with the plucker embedding we embed the space of such maps into d the closure qmd of this space in d is often referred to as the drinfeld s compactification of the space of degree d maps from cp 1 to x and will be called the space of quasimaps following 14 it is a generally speaking singular irreducible projective variety of complex dimension dimx 2d1 2dr the flag manifold is a homogeneous space of the group slr 1 c and of its maximal compact subgroup sur 1 the action of these groups on x lifts naturally to the spaces d and qmd in addition to this action the spaces d and the subspaces qmd carry the circle action induced by the rotation of cp 1 defined by x y 7 x ey thus the product group g s sur 1 and its complex version gc c slr 1 c act on the quasimap spaces we will see later that qmd have g equivariant desingularizations qmd

elliptic stable envelopes we construct stable envelopes in equivariant elliptic cohomology of nakajima quiver varieties in particular this gives an elliptic generalization of the results of arxiv 1211 1287 we apply them to the computation of the monodromy of q difference equations arising the enumerative k theory of rational curves in nakajima varieties including the quantum knizhnik zamolodchikov equations

quantization of complex lagrangian submanifolds let be a smooth lagrangian submanifold of a complex symplectic manifold x we construct twisted simple holonomic modules along in the stack of deformationquantization modules on x

covariant and equivariant formality theorems abstract we give a proof of kontsevich s formality theorem for a general manifold using fedosov resolutions of algebras of polydifferential operators and polyvector fields the main advantage of our construction of the formality quasi isomorphism is that it is based on the use of covariant tensors unlike kontsevich s original proof which is based on jets of polydifferential operators and polyvector fields using our construction we prove that if a group g acts smoothly on a manifold m and m admits a g invariant affine connection then there exists a g equivariant quasi isomorphism of formality this result implies that if a manifold m is equipped with a smooth action of a finite or compact group g or equipped with a free action of a lie group g then m admits a g equivariant formality quasi isomorphism in particular this gives a solution of the deformation quantization problem for an arbitrary poisson orbifold

knot homology via derived categories of coherent sheaves i the mathfrak sl 2 case using derived categories of equivariant coherent sheaves we construct a knot homology theory which categorifies the quantum mathfrak sl m knot polynomial our knot homology naturally satisfies the categorified moy relations and is conjecturally isomorphic to khovanov rozansky homology our construction is motivated by the geometric satake correspondence and is related to manolescu s by homological mirror symmetry

from local to global deformation quantization of poisson manifolds we give an explicit construction of a deformation quantization of the algebra of functions on a poisson manifold based on m kontsevich s local formula the deformed algebra of functions is realized as the algebra of horizontal sections of a vector bundle with flat connection

two dimensional representations of the free group in two generators over an arbitrary field abstract working over an arbitrary field we classify all two dimensional representations of the free group with two generators

localization of modules for a semisimple lie algebra in prime characteristic with an appendix by r bezrukavnikov and s riche computation for sl 3 we show that on the level of derived categories representations of the lie algebra of a semisimple algebraic group over a field of finite characteristic with a given generalized regular central character are the same as coherent sheaves on the formal neighborhood of the corresponding generalized springer fiber the first step is to observe that the derived functor of global sections provides an equivalence between the derived category of d modules with no divided powers on the flag variety and the appropriate derived category of modules over the corresponding lie algebra thus the derived version of the beilinson bernstein localization theorem holds in sufficiently large positive characteristic next one finds that for any smooth variety this algebra of differential operators is an azumaya algebra on the cotangent bundle in the case of the flag variety it splits on springer fibers and this allows us to pass from d modules to coherent sheaves the argument also generalizes to twisted d modules as an application we prove lusztig s conjecture on the number of irreducible modules with a fixed central character we also give a formula for behavior of dimension of a module under translation functors and reprove the kac weisfeiler conjecture the sequel to this paper bmr2 treats singular infinitesimal characters

on quantizations of complex contact manifolds abstract a holomorphic quantization of a complex contact manifold is a filtered algebroid stack which is locally equivalent to the ring e of microdifferential operators and which has trivial graded the existence of a canonical quantization has been proved by kashiwara in this paper we consider the classification problem showing that the above quantizations are classified by the first cohomology group with values in a certain sheaf of homogeneous forms secondly we consider the problem of existence and classification for quantizations given by algebras

formality theorems for hochschild chains in the lie algebroid setting in this paper we prove lie algebroid versions of tsygan s formality conjecture for hochschild chains both in the smooth and holomorphic settings in the holomorphic setting our result implies a version of tsygan s formality conjecture for hochschild chains of the structure sheaf of any complex manifold and in the smooth setting this result allows us to describe quantum traces for an arbitrary poisson lie algebroid the proofs are based on the use of kontsevich s quasi isomorphism for hochschild cochains of r y 1 y d shoikhet s quasi isomorphism for hochschild chains of r y 1 y d and fedosov s resolutions of the natural analogues of hochschild co chain complexes associated with a lie algebroid

special transverse slices and their enveloping algebras let g be a simple simply connected algebraic group over image image lie g image image the nilpotent cone of image and e h f an imageimage2 triple in image let s e ker ad f the special transverse slice to the adjoint orbit of e and s0 s image image the coordinate ring image s0 is naturally graded see slodowy simple singularities and simple algebraic groups lecture notes in mathematics vol 815 springer verlag berlin heidelberg new york 1980 let z image be the centre of the enveloping algebra u image and z image image an algebra homomorphism identify image with image via a killing isomorphism and let denote the linear function on image corresponding to e following kawanaka generalized gelfand graev representations and ennola duality in algebraic groups and related topics advanced studies in pure mathematics vol 6 pp 175 206 north holland amsterdam new york oxford 1985 moeglin c r acad sci paris ser i 303 no 17 1986 845 848 and premet invent math 121 1995 79 117 we attach to a nilpotent subalgebra image subset ofimage of dimension dim 2 and a 1 dimensional image module image let imageh height 14 width 14 denote the algebra opposite to endimage u image multiply sign in circleu image image and imageh height 14 width 14 imageh height 14 width 14 multiply sign in circlez image image it is proved in the paper that the algebra imageh height 14 width 14 has a natural filtration such that gr imageh height 14 width 14 the associated graded algebra is isomorphic to image s0 this construction yields natural noncommutative deformations of all singularities associated with the adjoint quotient map of image

light leaves and lusztig s conjecture we define a map f with domain a certain subset of the set of light leaves certain morphisms between soergel bimodules introduced by the author in an earlier paper and range the set of prime numbers using results of soergel we prove the following property of f if the image p f l of some light leaf l under f is bigger than the coxeter number of the corresponding weyl group then there is a counterexample to lusztig s conjecture in characteristic p we also introduce the double leaves basis which is an improvement of the light leaves basis that has already found interesting applications in particular it forms a cellular basis of soergel bimodules that allows us to produce an algorithm to find the bad primes for lusztig s conjecture

reductive groups the loop grassmannian and the springer resolution in this paper we prove equivalences of categories relating the derived category of a block of the category of representations of a connected reductive algebraic group over an algebraically closed field of characteristic p bigger than the coxeter number and a derived category of equivariant coherent sheaves on the springer resolution or a parabolic counterpart in the case of the principal block combined with previous results this provides a modular version of celebrated constructions due to arkhipov bezrukavnikov ginzburg for lusztig s quantum groups at a root of unity as an application we prove a graded version of a conjecture of finkelberg mirkovic describing the principal block in terms of mixed perverse sheaves on the dual affine grassmannian and deduce a new proof of lusztig s conjecture in large characteristic

deformation quantization in algebraic geometry abstract we study deformation quantizations of the structure sheaf o x of a smooth algebraic variety x in characteristic 0 our main result is that when x is d affine any formal poisson structure on x determines a deformation quantization of o x canonically up to gauge equivalence this is an algebro geometric analogue of kontsevich s celebrated result

twisted deformation quantization of algebraic varieties abstract let x be a smooth algebraic variety over a field k containing the real numbers we introduce the notion of twisted associative resp poisson deformation of the structure sheaf o x these are stack like versions of usual deformations we prove that there is a twisted quantization operation from twisted poisson deformations to twisted associative deformations which is canonical and bijective on gauge equivalence classes this result extends work of kontsevich and our own earlier work on deformation quantization of algebraic varieties

schubert calculus and torsion explosion the author observes that certain numbers occurring in schubert calculus for sl also occur as entries in intersection forms controlling decompositions of soergel bimodules in higher rank these numbers grow exponentially this observation gives many counter examples to the expected bounds in lusztig s conjecture on the characters of simple rational modules for sl over fields of positive characteristic the examples also give counter examples to the james conjecture on decomposition numbers for symmetric groups

nilpotent orbits in good characteristic and the kempf rousseau theory let g be a connected reductive algebraic group over an algebraically closed field k of characteristic p 0 image and suppose that p is a good prime for the root system of g in this paper we give a fairly short conceptual proof of pommerening s theorem pommerening j algebra 49 1977 525 536 j algebra 65 1980 373 398 which states that any nilpotent element in image is richardson in a distinguished parabolic subalgebra of the lie algebra of a levi subgroup of g as a by product we obtain a short noncomputational proof of the existence theorem for good transverse slices to the nilpotent g orbits in image for earlier proofs of this theorem see kawanaka invent math 84 1986 575 616 premet trans amer math soc 347 1995 2961 2988 spaltenstein j fac sci univ tokyo sect ia math 31 1984 283 286 we extend recent results of sommers internal math res notices 11 1998 539 562 to reductive lie algebras of good characteristic thus providing a satisfactory approach to computing the component groups of the centralisers of nilpotent elements in image and unipotent elements in g earlier computations of these groups in positive characteristics relied mostly on work of mizuno j fac sci univ tokyo sect ia math 24 1977 525 563 tokyo j math 3 1980 391 459 our approach is based on the theory of optimal parabolic subgroups for g unstable vectors also known as the kempf rousseau theory which provides a good substitute for the image theory prominent in the characteristic zero case

semiunitary projective representations of the complete galilei group a systematic study of the semiunitary projective representations of the galilei group including reflections is presented they are found by means of the semiunitary representations of a representation group

algebroid prestacks and deformations of ringed spaces for a ringed space x o we show that the deformations of the abelian category mod o of sheaves of o modules lowen and van den bergh 2006 are obtained from algebroid prestacks as introduced by kontsevich in case x is a quasi compact separated scheme the same is true for qch o the category of quasi coherent sheaves on x it follows in particular that there is a deformation equivalence between mod o and qch o

locally operating realizations of groups and superequivalence of factor systems the unitary multiplier locally operating realizations of a transitive group of transformations g are identified with those obtained by an induction process from linear finite dimensional representations of a group related with the isotopy group of the given action in this induction process each pseudoequivalence class of linear representations of splits into several gauge equivalence classes of unitary multiplier locally operating realizations of g the concept of superequivalence of factor systems appears here in a natural way