a version of hormander s theorem for the fractional brownian motion it is shown that the law of an sde driven by fractional brownian motion with hurst parameter greater than 1 2 has a smooth density with respect to lebesgue measure provided that the driving vector fields satisfy hormander s condition the main new ingredient of the proof is an extension of norris lemma to this situation

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

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

cellular structures on hecke algebras of type b abstract the aim of this paper is to gather and try to unify several approaches for the modular representation theory of hecke algebras of type b we attempt to explain the connections between geck s cellular structures coming from kazhdan lusztig theory with unequal parameters and ariki s theorem on the canonical basis of the fock spaces

highest weight sl_2 categorifications ii structure theory this paper continues the study of highest weight categorical sl2 actions initi ated in part i we start by refining the definition given there and showing that all examples considered in part i are also highest weight categorifications in the refined sense then we prove that any highest weight sl2 categorification can be filtered in such a way that the suc cessive quotients are so called basic highest weight sl2 categorifications for a basic highest weight categorification we determine minimal projective resolutions of standard objects we use this in particular to examine the structure of tilting objects in basic categorifications and to show that the ringel duality is given by the rickard complex we apply some of these structural results to categories o for cyclotomic rational cherednik algebras we finish by discussing open problems

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

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

rough differential equations driven by signals in besov spaces rough differential equations are solved for signals in general besov spaces unifying in particular the known results in holder and p variation topology to this end the paracontrolled distribution approach which has been introduced by gubinelli imkeller and perkowski 24 to analyze singular stochastic pdes is extended from holder to besov spaces as an application we solve stochastic differential equations driven by random functions in besov spaces and gaussian processes in a pathwise sense

variational solutions for partial differential equations driven by a fractional noise abstract in this article we develop an existence and uniqueness theory of variational solutions for a class of nonautonomous stochastic partial differential equations of parabolic type defined on a bounded open subset d r d and driven by an infinite dimensional multiplicative fractional noise we introduce two notions of such solutions for them and prove their existence and their indistinguishability by assuming that the noise is derived from an l 2 d valued fractional wiener process w h with hurst parameter h 1 1 1 whose covariance operator satisfies appropriate integrability conditions and where 0 1 denotes the holder exponent of the derivative of the nonlinearity in the stochastic term of the equations we also prove the uniqueness of solutions when the stochastic term is an affine function of the unknown random field our existence and uniqueness proofs rest upon the construction and the convergence of a suitable sequence of faedo galerkin approximations while our proof of indistinguishability is based on certain density arguments as well as on new continuity properties of the stochastic integral we define with respect to w h

controlled differential equations as young integrals a simple approach the theory of rough paths allows one to define controlled differential equations driven by a path which is irregular the most simple case is the one where the driving path has finite p variations with 1 p 2 in which case the integrals are interpreted as young integrals the prototypal example is given by stochastic differential equations driven by fractional brownian motion with hurst index greater than 1 2 using simple computations we give the main results regarding this theory existence uniqueness convergence of the euler scheme flow property which are spread out among several articles

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

on the category o for rational cherednik algebras we study the category o of representations of the rational cherednik algebra aw attached to a complex reflection group w w e construct an exact functor called knizhnik zamolodchikov functor o hw mod where hw is the finite iwahori hecke algebra associated to w we prove that the knizhnik zamolodchikov functor induces an equiv alence between o otor the quotient of o by the subcategory of aw modules supported on the discriminant and the category of finite dimensional hw modules the standard aw modules go under this equivalence to certain modules arising in kazhdan lusztig theory of cells provided w is a weyl group and the hecke algebra hw has equal parameters we prove that the category o is equivalent to the module category over a finite dimensional algebra a generalized q schur algebra associated to w

dunkl operators for complex reflection groups dunkl operators for complex reflection groups are defined in this paper these commuting operators give rise to a parameterized family of deformations of the polynomial de rham complex this leads to the study of the polynomial ring as a module over the rational cherednik algebra and a natural contravariant form on this module in the case of the imprimitive complex reflection groups g m p n the set of singular parameters in the parameterized family of these structures is described explicitly using the theory of non symmetric jack polynomials

primitive ideals and orbital integrals in complex exceptional groups in 2 two related problems were studied for complex classical groups determination of the primitive spectrum of the enveloping algebra as a set and fourier inversion of unipotent orbital integrals the first of these was solved completely extending work of joseph and others in sl n c and small groups although our techniques presumably solve the second as well up to determination of normalizing constants we carried out the calculations only for special unipotent classes see 181 we had of course tried to apply the same methods to the exceptional groups but they seem to be inadequate since that work was done however brylinski and kashiwara and beilinson and bernstein have established the kazhdan lusztig conjecture giving character formulas for irreducible highest weight modules 3 71 because of a conjecture of joseph proved in 24 this determines in principle the primitive ideals with a fixed regular integral infinitesimal character they are in one to one correspondence with what kazhdan and lusztig call left cells in the weyl group however the algorithm given by kazhdan and lusztig to compute these cells is enormously complicated requiring one to compute roughly 1 i polynomials of degrees on the order of half the number of positive roots this is very unsatisfactory however it

relations between young s natural and the kazhdan lusztig representations of sn abstract our main result here is that under a suitable order of standard tableaux the classical representation of sn introduced by young in the collected papers of alfred young 1873 1940 univ of toronto press toronto qsa iv and usually referred as the natural representattion the the more recently discovered invent math 53 1979 165 184 kazhdan lusztig k l representation are related by an upper triangular integral matrix with unit diagonal elements we have been led to this discovery by a numerical exploration we noted it in each of the irreducible representations of sn up to n 6 the calculations in these cases were carried out by constructing the corresponding kazhdan lusztig graphs from tables m goresky tables of kazhdan lusztig polynomials unpublished of k l polynomials to extend the calculations to n 7 we have used graphs obtained by means of an algorithm given by lascoux and schutzenberger polynomes de kazhdan lusztig pour les grassmanniennes preprint remarkably the same property holds also for these graphs these findings appear to confirm the assertion made by these authors that their algorithm does indeed yield k l graphs for the case of hook shapes we have obtained an explicit construction of the transforming matrices a result which was also suggested by our numerical data for general shapes the transforming matrices are less explicit and our proof is based on certain properties of the kazhdan lusztig representations given in their article invent math 53 1979 165 184 and on a purely combinatorial construction of the natural representation

sensitivity of rough differential equations an approach through the omega lemma the ito map assigns the solution of a rough differential equation a generalization of an ordinary differential equation driven by an irregular path when existence and uniqueness hold by studying how a path is transformed through the vector field which is integrated we prove that the ito map is holder or lipschitz continuous with respect to all its parameters this result unifies and weaken the hypotheses of the regularity results already established in the literature

controlling rough paths abstract we formulate indefinite integration with respect to an irregular function as an algebraic problem which has a unique solution under some analytic constraints this allows us to define a good notion of integral with respect to irregular paths with holder exponent greater than 1 3 e g samples of brownian motion and study the problem of the existence uniqueness and continuity of solution of differential equations driven by such paths we recover young s theory of integration and the main results of lyons theory of rough paths in holder topology

frequency hopped spread spectrum transmission with band efficient modulations and simplified noncoherent sequence estimation frequency hopped spread spectrum transmission using band efficient modulations that are phase continuous during each hop is presented a range of system parameters is considered including signaling spectrum reception system complexity and performance in the presence of noise and jamming the particular cases where the hopped modulation is minimum shift keying msk duobinary minimum shift keying dmsk or tamed fm tfm are studied in detail results are presented for various modulation indexes rectangular and raised cosine pulse shapings and a range of hop interval lengths sequence estimation on a hop by hop basis is considered the noncoherent likelihood sequence receiver must keep all possible paths and so the computation and complexity becomes large for system transmitting many bits per hop therefore a simplified noncoherent viterbi like sequence estimation algorithm with reduced complexity is introduced system performance has been evaluated in gaussian noise partial band jamming and multiple tone jamming using bounds and a system simulation the compact nature of the hopped spectrum when a number of bits are transmitted per hop offers greater spectrum utilization and the prospect of improved performance in the presence of multiple tone jamming or interference