a quantization of conjugacy classes of matrices abstract we construct a generator system of the annihilator of the generalized verma module of g l n c induced from any character of any parabolic subalgebra as an analogue of minors and elementary divisors the generator system has a quantization parameter e and it generates the defining ideal of the conjugacy class of square matrices at the classical limit e 0

generalized hua operators and parabolic subgroups let g be a connected noncompact semisimple group with finite center fix g kan an iwasawa decomposition of g that is k is a maximal compact subgroup of g a a maximal vector subgroup of g with ad a consisting of semisimple elements normalizing n a maximal simply connected nilpotent subgroup of g the space 52 g k is a riemannian symmetric space and if m is the centralizer of a in k the group b man is a minimal parabolic subgroup of g and the space g b k m is called the maximal or furstenberg boundary of u2 in general a boundary of 52 will be a space of the form g p where p is some parabolic subgroup of g which we may assume contains b fixing a parabolic subgroup p of g there is a poisson kernel

middle convolution and harnad duality dettweiler reiter s additive description of katz s middle convolution can be interpreted in terms of the harnad duality using this interpretation together with mumford s geometric invariant theory we generalize the additive middle convolution to have a multi parameter and act on systems of linear ordinary differential equations with irregular singularities we show that the generalized operation holds basic properties of the original one and additionally show that katz s algorithm involving our generalization works well in generic case

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

parabolic projective functors in type a abstract we classify projective functors on the regular block of rocha caridi s parabolic version of the bgg category o in type a in fact we show that in type a the restriction of an indecomposable projective functor from o to the parabolic category is either indecomposable or zero as a consequence we obtain that projective functors on the parabolic category o in type a are completely determined up to isomorphism by the linear transformations they induce on the level of the grothendieck group which was conjectured by stroppel in 39

matrix model approach to minimal liouville gravity revisited using the connection with the frobenius manifold fm structure we study the matrix model description of minimal liouville gravity mlg based on the douglas string equation our goal is to find an exact discrete formulation of the mlg model that intrinsically contains information about the conformal selection rules we discuss how to modify the fm structure appropriately for this purposes we propose a modification of the construction for lee yang series involving the algebra instead of the previously used a1 algebra with the new prescription we calculate correlators on the sphere up to four points and find full agreement with the continuous approach without using resonance transformations

translation and shuffling of projectively presentable modules and a categorification of a parabolic hecke module translation and shuffling of projectively presentable modules and a categorification of the parabolic hecke module

on functors associated to a simple root abstract associated to a simple root of a finite dimensional complex semisimple lie algebra there are several endofunctors defined by arkhipov enright frenkel irving jantzen joseph mathieu vogan and zuckerman on the bgg category o we study their relations compute cohomologies of their derived functors and describe the monoid generated by arkhipov s and joseph s functors and the monoid generated by irving s functors it turns out that the endomorphism rings of all elements in these monoids are isomorphic we prove that the functors give rise to an action of the singular braid monoid on the bounded derived category of o 0 we also use arkhipov s joseph s and irving s functors to produce new generalized tilting modules

categorified skew howe duality and comparison of knot homologies abstract in this paper we show an isomorphism of homological knot invariants categorifying the reshetikhin turaev invariants for sl n over the past decade such invariants have been constructed in a variety of different ways using matrix factorizations category o affine grassmannians and diagrammatic categorifications of tensor products while the definitions of these theories are quite different there is a key commonality between them which makes it possible to prove that they are all isomorphic they arise from a skew howe dual action of gl l for some l in this paper we show that the construction of knot homology based on categorifying tensor products from earlier work of the second author fits into this framework and thus agrees with other such homologies such as khovanov rozansky homology we accomplish this by categorifying the action of gl l gl n on p c l c n using diagrammatic bimodules in this action the functors corresponding to gl l and gl n are quite different in nature but they will switch roles under koszul duality

quadratic duals koszul dual functors and applications this paper studies quadratic and koszul duality for modules over positively graded categories typical examples are modules over a path algebra which is graded by the path length of a not necessarily finite quiver with relations we present a very general definition of quadratic and koszul duality functors backed up by explicit examples this generalizes the work of beilinson ginzburg and soergel 1996 in two substantial ways we work in the setup of graded categories i e we allow infinitely many idempotents and also de ne a koszul duality functor for not necessarily koszul categories as an illustration of the techniques we reprove the koszul duality ryom hansen 2004 of translation and zuckerman functors for the classical category o in a quite elementary and explicit way from this we deduce a conjecture of bernstein frenkel and khovanov 1999 as applications we propose a definition of a koszul dual category for integral blocks of harish chandra bimodules and for blocks outside the critical hyperplanes for the kac moody category o

multiplicity of the adjoint representation in simple quotients of the enveloping algebra of a simple lie algebra let g be a complex simple lie algebra r a cartan subalgebra and u g the enveloping algebra of g we calculate for each maximal two sided ideal jmax a a e r of u g the number of times the adjoint representation occurs in u g jmax a this is achieved by reduction via the kazhdan lusztig polynomials to the case when a lies on a corner i e is a multiple of a fundamental weight remarkably in this case one can always present u g jmax a as a generalized principal series module and here we also calculate its goldie rank as a ring which is a question of independent interest for some of the more intransigent cases it was necessary to use recent very precise results of lusztig on left cells the results are used to show how a recent theorem of gupta established for nonspecial a can fail if a is singular finally we give a quite efficient procedure for testing if an induced ideal is maximal

the dispersionless lax equations and topological minimal models it is shown that perturbed rings of the primary chiral fields of the topological minimal models coincide with some particular solutions of the dispersionless lax equations the exact formulae for the tree level partition functions of an topological minimal models are found the virasoro constraints for the analogue of the function of the dispersionless lax equation correspond ing to these models are proved

a combinatorial approach to functorial quantum sl k knot invariants this paper contains a categorification of the frak sl k link invariant using parabolic singular blocks of category cal o our approach is intended to be as elementary as possible providing essentially combinatorial arguments for the main results of sussan the justification that our combinatorial arguments and steps are correct uses non combinatorial geometric and representation theoretic results e g the kazhdan lusztig and soergel s theorems we take these results as granted and use them like axioms called it facts in the text we first construct an exact functor valued invariant of webs or special trivalent graphs labelled with 1 2 k 1 k satisfying the moy relations afterwards we extend it to the frak sl k invariant of links by passing to the derived categories the approach of khovanov using foams appears naturally in this context more generally we expect that our approach provides a representation theoretic interpretation of the frak sl k homology based on foams and the kapustin li formula from mackaay stosic and vaz conjecturally this implies that the khovanov rozansky link homology is obtained from our invariant by restriction

category o gradings and translation functors abstract in this article we consider a graded version of category o we reprove some results of beilinson et al j amer math soc 9 1996 473 527 using a different approach furthermore we define a graded version of translation functors and duality this provides the construction of various graded modules on the other hand we describe how to get modules which are not gradable

categorification of the temperley lieb category tangles and cobordisms via projective functors to each generic tangle projection from the three dimensional real vector space onto the plane we associate a derived endofunctor on a graded parabolic version of the bernstein gel fand category mathcal o we show that this assignment is up to shifts invariant under tangle isotopies and reidemeister moves and defines therefore invariants of tangles the occurring functors are defined via so called projective functors the first part of the paper deals with the indecomposability of such functors and their connection with generalised temperley lieb algebras which are known to have a realisation via decorated tangles the second part of the paper describes a categorification of the temperley lieb category and proves the main conjectures of bfk moreover we describe a functor from the category of 2 cobordisms into a category of projective functors

a criterion for an ideal to be induced 1 1 let g be a complex semisimple lie algebra in the study of the primitive spectrum prim u g of the enveloping algebra u g of g an important question which remains open is to determine those primitive ideals which are completely prime one such family is provided by those ideals which are annihilators of modules induced from a one dimensional module of a parabolic subalgebra p for the purposes of this article we call such ideals induced with respect to p