the development version of the chevie package of gap3 abstract i describe the current state of the development version of the chevie package which allows to transform the theories of coxeter groups reductive algebraic groups complex reflection groups hecke algebras braid monoids etc into actual computations examples are given showing the code to check some results of lusztig

garside families and garside germs garside families have recently emerged as a relevant context for extending results involving garside monoids and groups which themselves extend the classical theory of generalized braid groups here we establish various characterizations of garside families that is equivalently various criteria for establishing the existence of normal decompositions of a certain type

affineness of deligne lusztig varieties for minimal length elements abstract we prove that the deligne lusztig varieties associated to elements of the weyl group which are of minimal length in their twisted class are affine our proof differs from the proof of he and orlik rapoport and it is inspired by the case of regular elements which correspond to the varieties involved in broue s conjectures

potential theory on lipschitz domains in riemannian manifolds sobolev besov space results and the poisson problem abstract we continue a program to develop layer potential techniques for pde on lipschitz domains in riemannian manifolds building on l p and hardy space estimates established in previous papers here we establish sobolev and besov space estimates on solutions to the dirichlet and neumann problems for the laplace operator plus a potential on a lipschitz domain in a riemannian manifold with a metric tensor smooth of class c 1 for some 0 we treat the inhomogeneous problem and extend it to the setting of manifolds results obtained for the constant coefficient laplace operator on a lipschitz domain in euclidean space with the dirichlet boundary condition by d jerison and c kenig

existence and uniqueness for a semilinear elliptic problem on lipschitz domains in riemannian manifolds ii abstract for a general class of semilinear equations in a lipschitz domain where m is a smooth compact riemannian manifold we establish the existence and uniqueness of solutions to dirichlet and neumann boundary problems the main contribution of this paper is that we consider rough boundary data that are typically just functions results of this type for the linear equation are typically obtained using singular integral techniques

weighted sobolev spaces and regularity for polyhedral domains we prove a regularity result for the poisson problem u f u p g on a polyhedral domain p r3 using the babuska kondratiev spaces kam p these are weighted sobolev spaces in which the weight is given by the distance to the set of edges 4 33 in particular we show that there is no loss of kam regularity for solutions of strongly elliptic systems with smooth coefficients we also establish a trace theorem for the restriction to the boundary of the functions in kam p

coxeter orbits and brauer trees abstract we study the cohomology with modular coefficients of deligne lusztig varieties associated to coxeter elements under some torsion free assumption on the cohomology we derive several results on the principal l block of a finite reductive group g f q when the order of q modulo l is assumed to be the coxeter number these results include the determination of the planar embedded brauer tree of the block as conjectured by hiss lubeck and malle 1995 in 25 and the derived equivalence predicted by the geometric version of broue s conjecture broue and malle 1993 7

highest rank of a polytope for an this research was supported by a marsden grant uoa1218 of the royal society of new zealand and by the portuguese foundation for science and technology fct fundacao para a ciencia e a tecnologia through cidma center for research and development in mathematics and applications within project uid mat 04106 2013

failure of l 2 boundedness of gradients of single layer potentials for measures with zero low density consider a totally irregular measure mu in mathbb r n 1 that is the upper density limsup r rightarrow 0 frac mu b x r 2r n is positive mu a e in mathbb r n 1 and the lower density liminf r rightarrow 0 frac mu b x r 2r n vanishes mu a e in mathbb r n 1 we show that if t mu f x int k x y f y d mu y is an operator whose kernel k cdot cdot is the gradient of the fundamental solution for a uniformly elliptic operator in divergence form associated with a matrix with holder continuous coefficients then t mu is not bounded in l 2 mu this extends a celebrated result proved previously by eiderman nazarov and volberg for the n dimensional riesz transform

spectral theory and complex interpolation abstract some relations between the spectral theory for linear operators and the complex theory of interpolation for families of banach spaces are studied an invertibility theorem is proven and size estimates for the spectra of operators on the interpolation spaces are given

navier stokes equations on lipschitz domains in riemannian manifolds the navier stokes equations are a system of nonlinear evolution equations modeling the flow of a viscous incompressible fluid one ingredient in the analysis of this system is the stationary linear system known as the stokes system a boundary value problem bvp that will be described in detail in the next section layer potential methods in smoothly bounded domains in euclidean space have proven useful in the analysis of the stokes system starting with work of odqvist and lichtenstein and including work of solonnikov and many others see the discussion in chapter iii of 10 and in 17 for the case of flow in regions with smooth boundary a treatment based on the modern language of pseudodifferential operators can be found in 18 in 1988 e fabes c kenig and g verchota 6 extended this classical layer potential approach to cover lipschitz domains in euclidean space in 6 the main result concerning the constant coefficient stokes system on lipschitz domains with connected boundary in euclidean space is the treatment of the l dirichlet boundary value problem and its regular version to achieve this the authors solve certain auxiliary neumann type problems and then exploit the duality between these and the original bvp s at the level of boundary integral operators p deuring and w von wahl 4 made use of the analysis in 6 to demonstrate the short time existence of solutions to the navier stokes equations in bounded lipschitz domains in threedimensional euclidean space it was necessary in 4 to include the hypothesis that the boundary be connected the hypothesis that the boundary be connected pervaded much work on the application of layer potentials to analysis on lipschitz domains it was certainly natural to speculate that this restriction was an artifact of the methods used and not partly supported by nsf grant dms 9870018 partially supported by nsf grant dms 9877077 1991mathematics subject classification primary 35q30 76d05 35j25 secondary 42b20 45e05

layer potential methods for elliptic homogenization problems in this paper we use the method of layer potentials to study l 2 boundary value problems in a bounded lipschitz domain for a family of second order elliptic systems with rapidly oscillating periodic coefficients defining l e div a e 1 x under the assumption that a x is elliptic symmetric periodic and holder continuous we establish the solvability of the l 2 dirichlet regularity and neumann problems for l e u e 0 in with optimal estimates uniform in e 0

wave diffraction by wedges having arbitrary aperture angle abstract the problem of plane wave diffraction by a wedge sector having arbitrary aperture angle has a very long and interesting research background in fact we may recognize significant research on this topic for more than one century despite this fact up to now no clear unified approach was implemented to treat such a problem from a rigourous mathematical way and in a consequent appropriate sobolev space setting in the present paper we are considering the corresponding boundary value problems for the helmholtz equation with complex wave number admitting combinations of dirichlet and neumann boundary conditions the main ideas are based on a convenient combination of potential representation formulas associated with weighted mellin pseudo differential operators in appropriate sobolev spaces and a detailed fredholm analysis thus we prove that the problems have unique solutions with continuous dependence on the data which are represented by the single and double layer potentials where the densities are solutions of derived pseudo differential equations on the half line

finite complex reflection arrangements are k pi 1 let v be a finite dimensional complex vector space and w gl v be a finite complex reflection group let v reg be the complement in v of the reflecting hyperplanes we prove that v reg is a k 1 space this was predicted by a classical conjecture originally stated by brieskorn for complexified real reflection groups the complexified real case follows from a theorem of deligne and after contributions by nakamura and orliksolomon only six exceptional cases remained open in addition to solving these six cases our approach is applicable to most previously known cases including complexified real groups for which we obtain a new proof based on new geometric objects we also address a number of questions about 1 w v the braid group of w this includes a description of periodic elements in terms of a braid analog of springer s theory of regular elements

cauchy integrals calderon projectors and toeplitz operators on uniformly rectifiable domains abstract we develop properties of cauchy integrals associated to a general class of first order elliptic systems of differential operators d on a bounded uniformly rectifiable ur domain in a riemannian manifold m we show that associated to such cauchy integrals are analogues of hardy spaces of functions on annihilated by d and we produce projections of calderon type onto subspaces of l p consisting of boundary values of elements of such hardy spaces we consider toeplitz operators associated to such projections and study their index properties of particular interest is a cobordism argument which often enables one to identify the index of a toeplitz operator on a rough ur domain with that of one on a smoothly bounded domain

cohomologie des varietes de deligne lusztig abstract we study the cohomology of deligne lusztig varieties with aim the construction of actions of hecke algebras on such cohomologies as predicted by the conjectures of broue malle and michel ultimately aimed at providing an explicit version of the abelian defect conjecture we develop the theory for varieties associated to elements of the braid monoid and partial compactifications of them we are able to compute the cohomology of varieties associated to possibly twisted rank 2 groups and powers of the longest element w 0 some indeterminacies remain for g 2 we use this to construct hecke algebra actions on the cohomology of varieties associated to w 0 or its square for groups of arbitrary rank in the subsequent work f digne j michel endomorphisms of deligne lusztig varieties nagoya j math 183 2006 we construct actions associated to more general regular elements and we study their traces on cohomology

quotient of deligne lusztig varieties abstract we study the quotient of parabolic deligne lusztig varieties by a finite unipotent group u f where u is the unipotent radical of a rational parabolic subgroup p lu we show that in some particular cases the cohomology of this quotient can be expressed in terms of smaller parabolic deligne lusztig varieties associated to the levi subgroup l

parabolic subgroups of artin groups abstract let a be an artin system for x we denote by a x the subgroup of a generated by x such a group is called a parabolic subgroup of a we reprove van der lek s theorem a parabolic subgroup of an artin group is an artin group we give an algorithm which decides whether two parabolic subgroups of an artin group are conjugate let a be a finite type artin group and let a x be a parabolic subgroup with connected associated coxeter graph the quasi centralizer of a x in a is the set of in a such that x 1 x we prove that the commensurator of a x in a is equal to the normalizer of a x in a and that this group is generated by a x and the quasi centralizer of a x in a moreover if a x is not of type d l l 4 and l even then this group is generated by a x and the centralizer of a x in a

action du groupe des tresses sur une categorie we describe how to define by generators and relations an action of the braid group on a category more generally how to define a functor from a generalized positive braids monoid corresponding to a finite coxeter group to a monoidal category applications to constructions of bondal exceptional systems and of broue michel correspondences on flag manifolds are given

sharp hodge decompositions maxwell s equations and vector poisson problems on nonsmooth three dimensional riemannian manifolds we solve three basic potential theoretic problems hodge decompositions for vector fields poisson problems for the hodge laplacian and inhomogeneous maxwell equations in arbitrary lipschitz subdomains of a smooth compact three dimensional riemannian manifold in each case we derive sharp estimates on sobolev besov scales and establish integral representation formulas for the solution the proofs rely on tools from harmonic analysis and algebraic topology such as calderon zygmund theory and de rham theory

large solutions for yamabe and similar problems on domains in riemannian manifolds we present a unifled approach to study large positive solutions i e u x 1 as x of the equation u hu i k u if in an arbitrary domain we assume u is convex and grows su ciently fast as u 1 equations of this type arise in geometry yamabe problem two dimensional curvature equation probability superdifiusion we prove that both existence and uniqueness are local properties of points of the boundary i e they depend only on properties of in arbitrary small neighborhood of each boundary point we also flnd several new necessary and su cient conditions for existence and uniqueness of large solutions including an existence theorem on domains with fractal boundaries

normalizers of parabolic subgroups in coxeter groups we shall describe a simple procedure for obtaining presentations of normalizers of parabolic subgroups of arbitrary coxeter groups in each case the normalizer is the semidirect product of the parabolic subgroup by a complement which acts by permuting the simple re ections this complement may be viewed in a natural way as a maximal subgroup of a certain groupoid for which we obtain a set of generators and de ning relations moreover the de ning relations are all derived from subgroupoids of ranks one and two so that the groupoid can be viewed as a generalization of a coxeter group

normalisateur et groupe d artin de type spherique abstract let a s be an artin group and x a subset of s denote by ax the subgroup of a generated by x when a is of spherical type we prove that the normalizer and the commensurator of ax in a are equal and are the product of ax by the quasi centralizer of ax in a looking the associated monoids a and ax we described the quasi centralizer of ax in a thanks to results in coxeter groups these two results generalize earlier results of paris j algebra 196 1997 369 399 finally we compare in the spherical case the normalizer of a parabolic subgroup in the artin group and in the coxeter group

the helmholtz equation in heterogeneous media a priori bounds well posedness and resonances abstract we consider the exterior dirichlet problem for the heterogeneous helmholtz equation i e the equation a u k 2 n u f where both a and n are functions of position we prove new a priori bounds on the solution under conditions on a n and the domain that ensure nontrapping of rays the novelty is that these bounds are explicit in k a n and geometric parameters of the domain we then show that these a priori bounds hold when a and n are l and satisfy certain monotonicity conditions and thereby obtain new results both about the well posedness of such problems and about the resonances of acoustic transmission problems i e a and n discontinuous where the transmission interfaces are only assumed to be c 0 and star shaped the novelty of this latter result is that until recently the only known results about resonances of acoustic transmission problems were for c convex interfaces with strictly positive curvature

boundary value problems in spaces of distributions on smooth and polygonal domains we study boundary value problems of the form email protected f on w and bu g on the boundary email protected with either dirichlet or neumann boundary conditions where w is a smooth bounded domain in r n and the data f g are distributions this problem has to be first properly reformulated and for practical applications it is of crucial importance to obtain the continuity of the solution u in terms of f and g for f 0 taking advantage of the fact that u is harmonic on w we provide four formulations of this boundary value problem one using nontangential limits of harmonic functions one using green functions one using the dirichlet to neumann map and a variational one we show that these four formulations are equivalent we provide a similar analysis for f 0 and discuss the roles of f and g which turn to be somewhat interchangeable in the low regularity case the weak formulation is more convenient for numerical approximation whereas the nontangential limits definition is closer to the intuition and easier to check in concrete situations we extend the weak formulation to polygonal domains using weighted sobolev spaces we also point out some new phenomena for the concentrated loads at the vertices in the polygonal case

estimates for the deviation of solutions and eigenfunctions of second order elliptic dirichlet boundary value problems under domain perturbation estimates in suitable lebesgue or sobolev norms for the deviation of solutions and eigenfunctions of second order uniformly elliptic dirichlet boundary value problems subject to domain perturbation in terms of natural distances between the domains are given the main estimates are formulated via certain natural and easily computable atlas distances for domains with lipschitz continuous boundaries as a corollary similar estimates in terms of more classical distances such as the hausdorff distance or the lebesgue measure of the symmetric difference of domains are derived sharper estimates are also proved to hold in smoother classes of domains

the stationary navier stokes system in nonsmooth manifolds the poisson problem in lipschitz and c 1 domains we consider the linearized version of the stationary navier stokes equations on a subdomain of a smooth compact riemannian manifold m the emphasis is on regularity the boundary of is assumed to be only c1 and even lipschitz and the data are selected from appropriate sobolev besov scales our approach relies on the method of boundary integral equations suitably adapted to the variable coefficient setting we are considering here applications to the stationary nonlinear navier stokes equations in this context are also discussed

well posedness and regularity for the elasticity equation with mixed boundary conditions on polyhedral domains and domains with cracks we prove a regularity result for the anisotropic linear elasticity equation p u rm div left boldmath mathsf c cdot nabla u right f with mixed displacement and traction boundary conditions on a curved polyhedral domain omega subset mathbb r 3 in weighted sobolev spaces mathcal k m 1 a 1 omega for which the weight is given by the distance to the set of edges in particular we show that there is no loss of mathcal k m a regularity our curved polyhedral domains are allowed to have cracks we establish a well posedness result when there are no neighboring traction boundary conditions and a 0 that depends on p on the boundary conditions and on the domain our results extend to other strongly elliptic systems and higher dimensions

boutet de monvel operators on lie manifolds with boundary abstract we introduce and study a general pseudodifferential calculus for boundary value problems on a class of non compact manifolds with boundary so called lie manifolds with boundary this is accomplished by constructing a suitable generalization of the boutet de monvel calculus for boundary value problems the data consists of a compact manifold with corners m that is endowed with a lie structure of vector fields 2 v a so called lie manifold the manifold m is split into two equal parts x and x which intersect in an embedded hypersurface y x our goal is to describe a transmission boutet de monvel calculus for boundary value problems compatible with the structure of lie manifolds starting with the example of b vector fields we show that there are two groupoids integrating the lie structures on m and on y respectively these two groupoids form a bibundle or a groupoid correspondence and under some mild assumptions these groupoids are morita equivalent with the help of the bibundle structure and canonically defined manifolds with corners which are blow ups in particular cases we define a class of boutet de monvel type operators we then define the representation homomorphism for these operators and show closedness under composition with the help of a representation theorem finally we consider appropriate fredholm conditions and construct the parametrices for elliptic operators in the calculus

potential theory on lipschitz domains in riemannian manifolds the case of dini metric tensors we study the applicability of the method of layer potentials in the treatment of boundary value problems for the laplace beltrami operator on lipschitz sub domains of riemannian manifolds in the case when the metric tensor g jk dx j desk has low regularity under the assumption that g jk x g jk y c x y where the modulus of continuity satisfies a dini type condition we prove the well posedness of the classical dirichlet and neumann problems with l p boundary data for sharp ranges of p s and with optimal nontangential maximal function estimates

estimates of first and second order shape derivatives in nonsmooth multidimensional domains and applications in this paper we investigate continuity properties of first and second order shape derivatives of functionals depending on second order elliptic pde s around nonsmooth domains essentially either lipschitz or convex or satisfying a uniform exterior ball condition we prove rather sharp continuity results for these shape derivatives with respect to sobolev norms of the boundary traces of the displacements with respect to previous results of this kind the approach is quite different and is valid in any dimension n geq 2 it is based on sharp regularity results for poisson type equations in such nonsmooth domains we also enlarge the class of functionals and pdes for which these estimates apply applications are given to qualitative properties of shape optimization problems under convexity constraints for the variable domains or their complement

boundary layer methods for lipschitz domains in riemannian manifolds abstract we extend to the variable coefficient case boundary layer techniques that have been successful in the treatment of the laplace equation and certain other constant coefficient elliptic partial differential equations on lipschitz domains in euclidean space we treat the laplace operator on lipschitz domains in a manifold with c 1 metric tensor and study the dirichlet neumann and oblique derivative boundary problems

parabolic deligne lusztig varieties abstract motivated by the broue conjecture on blocks with abelian defect groups for finite reductive groups we study parabolic deligne lusztig varieties and construct on those which occur in the broue conjecture an action of a braid monoid whose action on their l adic cohomology will conjecturally factor through a cyclotomic hecke algebra in order to construct this action we need to enlarge the set of varieties we consider to varieties attached to a ribbon category this category has a garside family which plays an important role in our constructions so we devote the first part of our paper to the necessary background on categories with garside families