affine yangians and deformed double current algebras in type a we study the structure of yangians of affine type and deformed double current algebras which are deformations of the enveloping algebras of matrix w1 8 algebras we prove that they admit a pbw type basis establish a connection limit construction between these two types of algebras and toroidal quantum algebras and we give three equivalent definitions of deformed double current algebras we construct a schur weyl functor between these algebras and rational cherednik algebras

a differential complex for poisson manifolds construction du complexe canonique d une variete de poisson homologie canonique des varietes symplectiques application a l homologie de hochschild des algebres non commutatives

sugawara operators and kac kazhdan conjecture soit g une algebre de lie de kac moody affine de type x l 1 soit l le module de poids le plus eleve unique irreductible pour chaque poids de kac kazhdan on a chl exp re 1 exp 1

a unification of knizhnik zamolodchikov and dunkl operators via affine hecke algebras some generalizations of the lusztig lascoux schutzenberger operators for affine hecke algebras are considered as corollaries we obtain lusztig s isomorphisms from affine hecke algebras to their degenerate versions a natural interpretation of the dunkl operators and a new class of differential difference operators generalizing dunkl s ones and the knizhnik zamolodchikov operators from the two dimensional conformal field theory

complex reflection groups braid groups hecke algebras presentations a la coxeter are given for all irreducible nite com plex re ection groups they provide presentations for the corresponding generalized braid groups for all but six cases which allow us to generalize some of the known properties of nite coxeter groups and their associated braid groups such as the computation of the center of the braid group and the construction of deformations of the nite group algebra hecke algebras we introduce monodromy representations of the braid groups which factorize through the hecke algebras extending results of cherednik opdam kohno and others summary introduction complex re ection groups and their presentations a background from complex re ection groups b presentations braid groups and their diagrams a generalities about hyperplane complements b generalities about the braid groups c the braid diagrams proofs of the main theorems for the braid groups b de e r a notation and prerequisites b computation of b de e r and of its center for d c computation of b e e r and of its center hecke algebras a background from di erential equations and monodromy b a family of monodromy representations of the braid group c hecke algebras diagrams and tables appendix generators of the monodromy around a divisor appendix tables to mathematics subject classi cation primary g we thank jean michel peter orlik pierre vogel for useful conversations and the isaac newton institute for its hospitality while the last version of this manuscript was written up the second named author gratefully acknowledges nancial support by the fondation alexander von humboldt for his stays in paris michel brou e gunter malle raphael rouquier

symmetric groups and the cup product on the cohomology of hilbert schemes let c sn be the z module of integer valued class functions on the symmetric group sn we introduce a graded version of the con volution product on c sn and show that there is a degree preserving ring isomorphism c sn h hilb n a 2 z to the cohomology of the hilbert scheme of points in the complex affine plane

porosity results on fixed points for nonexpansive set valued maps in hyperbolic spaces abstract we establish the porosity results on the existence and the topological structure of fixed points for nonexpansive set valued maps in hyperbolic spaces which in particular extend and or improve some known results in this direction

newton noda iteration for finding the perron pair of a weakly irreducible nonnegative tensor we present a newton noda iteration nni for computing the perron pair of a weakly irreducible nonnegative mth order tensor mathscr a a by combining the idea of newton s method with the idea of the noda iteration the method requires the selection of a positive parameter theta k k in the kth iteration and produces a scalar sequence approximating the spectral radius of mathscr a a and a positive vector sequence approximating the perron vector we propose a halving procedure to determine the parameters theta k k starting with theta k 1 k 1 for each k such that the scalar sequence is monotonically decreasing convergence of this sequence to the spectral radius of mathscr a a and convergence of the vector sequence to the perron vector is guaranteed for any initial positive unit vector as long as the sequence theta k k so chosen is bounded below by a positive constant in this case we always have theta k 1 k 1 near convergence and the convergence is quadratic very often the halving procedure will return theta k 1 k 1 i e no halving is actually used for each k if the tensor is semisymmetric m ge 4 m 4 and theta k 1 k 1 then the computational work in the kth iteration of nni is roughly the same as that for one iteration of the ng qi zhou algorithm which is linearly convergent for the smaller class of weakly primitive tensors

algebra structure on the hochschild cohomology of the ring of invariants of a weyl algebra under a finite group abstract let a n be the n th weyl algebra and let g sp 2 n c aut a n be a finite group of linear automorphisms of a n in this paper we compute the multiplicative structure on the hochschild cohomology hh a g n of the algebra of invariants of g we prove that as a graded algebra hh a g n is isomorphic to the graded algebra associated to the center of the group algebra c g with respect to a filtration defined in terms of the defining representation of g

the farahat higman ring of wreath products and hilbert schemes we study the structure constants of the class algebra rz n of the wreath products n associated to an arbitrary finite group with respect to the basis of conjugacy classes we show that a suitable filtration on rz n gives rise to the graded ring g n with non negative integer structure constants independent of n some of which are computed which are then encoded in a farahat higman ring g the real conjugacy classes of come to play a distinguished role and are treated in detail in the case when is a subgroup of sl2 c the above results provide new insight to the cohomology rings of hilbert schemes of points on a quasi projective surface x

rings with auslander dualizing complexes abstract a ring with an auslander dualizing complex is a generalization of an auslander gorenstein ring we show that many results which hold for auslander gorenstein rings also hold in the more general setting on the other hand we give criteria for existence of auslander dualizing complexes which show these occur quite frequently the most powerful tool we use is the local duality theorem for connected graded algebras over a field filtrations allow the transfer of results to nongraded algebras we also prove some results of a categorical nature most notably the functoriality of rigid dualizing complexes

automorphisms and ideals of the weyl algebra let a 1 be the first weyl algebra and let g be its automorphism group we study the natural action of g on the space of isomorphism classes of right ideals of a 1 equivalently of finitely generated rank 1 torsion free right a 1 modules we show that this space breaks up into a countable number of orbits each of which is a finite dimensional algebraic variety our results are strikingly similar to those for the commutative algebra of polynomials in two variables however we do not know of any general principle that would allow us to predict this in advance as a key step in the proof we obtain a new description of the bispectral involution of w1 we also make some comments on the group g from the viewpoint of shafarevich s theory of infinite dimensional algebraic groups

existence theorems for dualizing complexes over non commutative graded and filtered rings abstract in this note we prove existence theorems for dualizing complexes over graded and filtered rings thereby generalizing some results by zhang yekutieli and jorgensen

generalized double affine hecke algebras of rank 1 and quantized del pezzo surfaces abstract let d be a simply laced dynkin diagram of rank r whose affinization has the shape of a star i e d 4 e 6 e 7 e 8 to such a diagram one can attach a group g whose generators correspond to the legs of the affinization have orders equal to the leg lengths plus 1 and the product of the generators is 1 the group g is then a 2 dimensional crystallographic group g z l z 2 where l is 2 3 4 and 6 respectively in this paper we define a flat deformation h t q of the group algebra c g of this group by replacing the relations saying that the generators have prescribed orders by their deformations saying that the generators satisfy monic polynomial equations of these orders with arbitrary roots which are deformation parameters the algebra h t q for d 4 is the cherednik algebra of type c c 1 which was studied by noumi sahi and stokman and controls askey wilson polynomials we prove that h t q is the universal deformation of the twisted group algebra of g and that this deformation is compatible with certain filtrations on c g we also show that if q is a root of unity then for generic t the algebra h t q is an azumaya algebra and its center is the function algebra on an affine del pezzo surface for generic q the spherical subalgebra e h t q e provides a quantization of such surfaces we also discuss connections of h t q with preprojective algebras and painleve vi

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

principal nilpotent pairs in a semisimple lie algebra 1 this is the first of a series of papers devoted to certain pairs of commuting nilpotent elements in a semisimple lie algebra that enjoy quite remarkable properties and which are expected to play a major role in representation theory the properties of these pairs and their role is similar to those of the principal nilpotents each principal nilpotent pair gives rise to a harmonic polynomial on the cartesian square of the cartan subalgebra that transforms under an irreducible representation of the weyl group in the special case of n the conjugacy classes of principal nilpotent pairs and the irreducible representations of the symmetric group s n are both parametrised in a compatible way by young diagrams in general our theory provides a natural generalization to arbitrary weyl groups of the classical construction of simple s n modules in terms of young s symmetrisers first results towards a complete classification of all principal nilpotent pairs in a simple lie algebra are presented at the end of this paper in an appendix written by a elashvili and d panyushev

geometric representations of graded and rational cherednik algebras abstract we provide geometric constructions of modules over the graded cherednik algebra h gr and the rational cherednik algebra h rat attached to a simple algebraic group g together with a pinned automorphism these modules are realized on the cohomology of affine springer fibers of finite type that admit c actions in the rational cherednik algebra case the standard grading on these modules is derived from the perverse filtration on the cohomology of affine springer fibers coming from its global analog hitchin fibers when is trivial we show that our construction gives the irreducible finite dimensional spherical modules l triv of h gr and of h rat we give a formula for the dimension of l triv and give a geometric interpretation of its frobenius algebra structure the rank two cases are studied in further details

homologie des invariants d une algebre de weyl sous l action d un groupe fini resume soit g un sous groupe fini de sp 2 n c operant par automorphismes dans l algebre de weyl a n c nous calculons les groupes d homologie et de cohomologie de hochschild de l algebre d invariants a n c g let g be a finite subgroup of sp 2 n c acting by automorphisms in the weyl algebra a n c we compute the hochschild homology and cohomology groups of the invariant algebra a n c g

orthogonal functions generalizing jack polynomials the rational cherednik algebra is a certain algebra of differential reflection operators attached to a complex reflection group w and depending on a set of central parameters each irreducible representation s of w corresponds to a standard module m for this paper deals with the infinite family g r 1 n of complex reflection groups our goal is to study the standard modules using a commutative subalgebra t of h discovered by dunkl and opdam in this case the irreducible w modules are indexed by certain sequences of partitions we first show that t acts in an upper triangular fashion on each standard module m with eigenvalues determined by the combinatorics of the set of standard tableaux on as a consequence we construct a basis for m consisting of orthogonal functions on n with values in the representation s for g 1 1 n with n these functions are the non symmetric jack polynomials we use intertwining operators to deduce a norm formula for our orthogonal functions and give an explicit combinatorial description of the lattice of submodules of m in the case in which the orthogonal functions are all well defined a consequence of our results is the construction of a number of interesting finite dimensional modules with intricate structure finally we show that for a certain choice of parameters there is a cyclic group of automorphisms of so that the rational cherednik algebra for g r p n is the fixed subalgebra our results therefore descend to the rational cherednik algebra for g r p n by clifford theory

quantum integrable systems and representations of lie algebras in this paper the quantum integrals of the hamiltonian of the quantum many body problem with the interaction potential k sinh2 x sutherland operator are constructed as images of higher casimirs of the lie algebra gl n under a certain homomorphism from the center of u gl n to the algebra of differential operators in n variables a similar construction applied to the affine gl n at the critical level k n defines a correspondence between higher sugawara operators and quantum integrals of the hamiltonian of the quantum many body problem with the potential equal to constant times the weierstrass function this allows one to give a new proof of the olshanetsky perelomov theorem stating that this hamiltonian defines a completely integrable quantum system a new expression is also given for eigenfunctions of the quantum integrals of the sutherland operator as traces of intertwining operators between certain representations of gl n

group actions on algebras and the graded lie structure of hochschild cohomology hochschild cohomology governs deformations of algebras and its graded lie structure plays a vital role we study this structure for the hochschild cohomology of the skew group algebra formed by a nite group acting on an al gebra by automorphisms we examine the gerstenhaber bracket with a view toward deformations and developing bracket formulas we then focus on the linear group actions and polynomial algebras that arise in orbifold theory and representation theory deformations in this context include graded hecke alge bras and symplectic reection algebras we give some general results describing when brackets are zero for polynomial skew group algebras which allow us in particular to nd noncommutative poisson structures for abelian groups we express the bracket using inner products of group characters lastly we inter pret results for graded hecke algebras

quantized multiplicative quiver varieties abstract beginning with the data of a quiver q and its dimension vector d we construct an algebra d q d q mat d q which is a flat q deformation of the algebra of differential operators on the affine space mat d q the algebra d q is equivariant for an action by a product of quantum general linear groups acting by conjugation at each vertex we construct a quantum moment map for this action and subsequently define the hamiltonian reduction a d q of d q with moment parameter we show that a d q is a flat formal deformation of lusztig s quiver varieties and their multiplicative counterparts for all dimension vectors satisfying a flatness condition of crawley boevey indeed the product on a d q yields a fedosov quantization the of symplectic structure on multiplicative quiver varieties as an application we give a description of the category of representations of the spherical double affine hecke algebra of type a n 1 and its generalization constructed by etingof oblomkov and rains in terms of a quotient of the category of equivariant d q modules by a serre subcategory of aspherical modules

a lie algebra generalization of the amitsur levitski theorem 0 1 if r is any algebra over a field f and xi e r i l k let iix 1 y 1 e r denote the alternating sum of products z sg 0 x x over all permutations 0 of l k one says that r satisfies the k fold standard identity if ix xk 0 for any choice xi e r of course if pc r is any subspace of r one can restrict one s attention to p and say that p satisfies the k fold standard identity if 1 x1 x 0 for any choice xi f p now the amitsur levitski theorem asserts that the n x n matrix algebra m n f satisfies the 2n fold standard identity for a survey and history of this theorem see 9 1 see also 17 i it is of course enough to consider the case where f c some time ago see 3 we came upon a proof of the amitsur levitski theorem from the point of lie algebra cohomology the cohomology of m n c as a lie algebra is an exterior algebra with primitive generators having degrees 1 2 2n 1 the failure of m n cc to have a primitive class in degree 2n 1 leads to the trace relation tr x x 0 for any xi e m n c for i i 2n 1 but this quickly implies the amitsur levitski theorem in fact the argument can be pushed further if d g m n cc denotes the lie algebra of all n x n skew symmetric matrices one knows that d has no primitive class in degree 2n 1 in case n is even this led to a new identity in 3 1 for iz even

differential difference operators associated to reflection groups there is a theory of spherical harmonics for measures invariant under a finite reflection group the measures are products of powers of linear functions whose zero sets are the mirrors of the reflections in the group times the rotation invariant measure on the unit sphere in rn a commutative set of differential difference operators each homogeneous of degree 1 is the analogue of the set of first order partial derivatives in the ordinary theory of spherical harmonics in the case of r2 and dihedral groups there are analogues of the cauchy riemann equations which apply to gegenbauer and jacobi polynomial expansions the analysis of orthogonality structures for polynomials in several variables is a problem of vast dimensions this paper is part of an ongoing program to establish a workable theory for one particular class the underlying structure is based on finite coxeter groups these are finite groups acting on euclidean space generated by reflections in the zero sets of a collection of linear functions the roots the weight functions for the orthogonality are products of powers of these linear functions restricted to the surface of the unit sphere in addition the weight function is required to be invariant under the action of the group the resulting theory has strong similarities to the theory of spherical harmonics this was established in previous papers of the author 3 4 5 most notably a homogeneous polynomial is orthogonal to all polynomials of lower degree if and only if it is annihilated by a certain second order differential difference operator ordinary partial differentiation acts as an endomorphism on ordinary harmonic functions the use of such operators and their adjoints leads to recurrence formulas and orthogonal decompositions for harmonic polynomials in this paper we construct a commutative set of first order differentialdifference operators associated to the second order operator previously mentioned received by the editors june 1 1987 1980 mathematics subject classification 1985 revision primary 33a45 33a65 20h15 secondary 20c30 42c10 51f15

symplectic reflection algebras calogero moser space and deformed harish chandra homomorphism to any finite group sp v of automorphisms of a symplectic vector space v we associate a new multi parameter deformation h of the algebra v smash product of with the polynomial algebra on v the parameter runs over points of r where r number of conjugacy classes of symplectic reflections in the algebra h called a symplectic reflection algebra is related to the coordinate ring of a poisson deformation of the quotient singularity v this leads to a symplectic analogue of mckay correspondence which is most complete in case of wreath products if is the weyl group of a root system in a vector space and v then the algebras h are certain rational degenerations of the double affine hecke algebra introduced earlier by cherednik

cyclotomic double affine hecke algebras and affine parabolic category o using the orbifold kz connection we construct a functor from an affine parabolic category o of type a to the category o of a cyclotomic rational double affine hecke algebra h then we give several results concerning this functor

on symmetric smooth and calabi yau algebras one possible definition for a calabi yau algebra is a symmetric smooth pi algebra our main purpose here is to prove some necessary and sufficient criteria for verifying the local symmetric property in smooth pi algebras many known smooth pi algebras are shown to have this property in particular quantum enveloping algebras of complex semi simple lie algebras in the root of unity case and the enveloping algebra of sl n with p n 1 in characteristic p are typical examples a surprising result is that the inj dimt is finite where t is the trace ring of m n n generic matrices over a field of zero characteristic

irreducible representations of the rational cherednik algebra associated to the coxeter group h3 abstract this paper describes irreducible representations in category o of the rational cherednik algebra h c h 3 h associated to the exceptional coxeter group h 3 and any complex parameter c we compute the characters of all these representations explicitly as a consequence we classify all the finite dimensional irreducible representations of h c h 3 h

invariant differential operators on a reductive lie algebra and weyl group representations let v be a finite dimensional vector space and let g be a subgroup of gl v set d v equal to the algebra of differential operators on v with polynomial coefficients and d v g equal to the g invariants in d v if g is a reductive lie algebra over c then t c g is a cartan subgroup of g and if g is the adjoint group of g then w is the weyl group of g harish chandra introduced an algebra homomorphism 5 of d g g to d t w h3 a is given by the obvious restriction mapping on the subalgebra of invariant polynomials and on the invariant constant coefficient differential operators and ker is