on a theorem of benson and curtis iwahori conjectured 5 that the hecke algebra over q of a chevalley group g over a finite field f with respect to a bore1 subgroup is isomorphic to the group algebra over q of the weyl group the analogous statement with q replaced by its algebraic closure was proved by tits 3 ex 27 p 561 using a deformation argument in 11 benson and curtis proved using a case by case analysis that iwahori s conjecture is true over q whenever g is simple of type ze e with the help of springer they proved that when g is of type e or e there exists irreducible representations of the hecke algebra whose character involve 4 and hence iwahori s conjecture over q is false in general the main result of this paper is an explicit construction of an isomorphism of the hecke algebra over q q with the group algebra over q q of the weyl group the construction makes essential use of the polynomials p w of 8 at some point in the proof i have to appeal to the interpretation of p l in terms of multiplicities in verma modules 8 i 51 which has recently been established by brylinski and kashiwara 4 and independently by belinson and bernstein

weak type 1 1 of some operators for the laplacian with drift let v v 1 ldots v n be a vector in mathbb r n setminus 0 consider the laplacian on mathbb r n with drift delta v sum i 1 n big frac partial 2 partial x i 2 2 v i frac partial partial x i big and the measure d mu x e 2 langle v x rangle dx with respect to which delta v is self adjoint let d and nabla denote the euclidean distance and the gradient operator on mathbb r n consider the space mathbb r n d d mu which has the property of exponential volume growth we obtain weak type 1 1 for the riesz transform nabla delta v frac 1 2 and for the heat maximal operator with respect to d mu further we prove that the uncentered hardy littlewood maximal operator is bounded on l p for 1 p le infty but not of weak type 1 1 if n ge 2

gradient estimates for heat kernels and harmonic functions abstract let x d be a doubling metric measure space endowed with a dirichlet form e deriving from a carre du champ assume that x d e supports a scale invariant l 2 poincare inequality in this article we study the following properties of harmonic functions heat kernels and riesz transforms for p 2 i g p l p estimate for the gradient of the associated heat semigroup ii r h p l p reverse holder inequality for the gradients of harmonic functions iii r p l p boundedness of the riesz transform p iv g b e a generalised bakry emery condition we show that for p 2 i ii iii are equivalent while for p i ii iv are equivalent moreover some of these equivalences still hold under weaker conditions than the l 2 poincare inequality our result gives a characterisation of li yau s gradient estimate of heat kernels for p while for p 2 it is a substantial improvement as well as a generalisation of earlier results by auscher coulhon duong hofmann 7 and auscher coulhon 6 applications to isoperimetric inequalities and sobolev inequalities are given our results apply to riemannian and sub riemannian manifolds as well as to non smooth spaces and to degenerate elliptic parabolic equations in these settings

hecke algebras of finite type are cellular let mathcal h be the one parameter hecke algebra associated to a finite weyl group w defined over a ground ring in which bad primes for w are invertible using deep properties of the kazhdan lusztig basis of mathcal h and lusztig s a function we show that mathcal h has a natural cellular structure in the sense of graham and lehrer thus we obtain a general theory of specht modules for hecke algebras of finite type previously a general cellular structure was only known to exist in types an and bn

positivity conjectures for kazhdan lusztig theory on twisted involutions the finite case abstract let w s be any coxeter system and let w w be an involution of w which preserves the set of simple generators s lusztig and vogan have shown that the corresponding set of twisted involutions i e elements w w with w 1 w naturally generates a module of the hecke algebra of w s with two distinguished bases the transition matrix between these bases defines a family of polynomials p y w which one can view as a twisted analogue of the much studied family of kazhdan lusztig polynomials of w s the polynomials p y w can have negative coefficients but display several conjectural positivity properties of interest which parallel positivity properties of the kazhdan lusztig polynomials this paper reports on some calculations which verify four such positivity conjectures in several finite cases of interest in particular for the non crystallographic coxeter systems of types h 3 and h 4

hardy littlewood theory for semigroups soit t t t 0 un semi groupe symetrique sousmarkovien et soit n 2 alors on a les proprietes equivalentes a dim t t n b il existe c 2 0 s t t t f c 2 t n 2 f 1 f l 1 x t 0 en fait a b meme dans le cas n 2

riesz transforms and spectral multipliers of the hodge laguerre operator abstract on r d endowed with the laguerre probability measure we define a hodge laguerre operator l acting on differential forms here is the laguerre exterior differentiation operator defined as the classical exterior differential except that the partial derivatives x i are replaced by the laguerre derivatives x i x i and is the adjoint of with respect to inner product on forms defined by the euclidean structure and the laguerre measure we prove dimension free bounds on l p 1 p for the riesz transforms l 1 2 and l 1 2 as applications we prove the strong hodge de rahm kodaira decomposition for forms in l p and deduce existence and regularity results for the solutions of the hodge and de rham equations in l p we also prove that for suitable functions m the operator m l is bounded on l p 1 p

a gaussian estimate for the heat kernel on differential forms and application to the riesz transform let m g be a complete riemannian manifold which satisfies a sobolev inequality of dimension n and on which the volume growth is comparable to the one of mathbb r n for big balls if there is no non zero l 2 harmonic 1 form and the ricci tensor is in l frac n 2 varepsilon cap l infty for an varepsilon 0 then we prove a gaussian estimate on the heat kernel of the hodge laplacian acting on 1 forms this allows us to prove that under the same hypotheses the riesz transform d vardelta 1 2 is bounded on l p for all 1 3

erratum to on the weak l p hodge decomposition and beurling ahlfors transforms on complete riemannian manifolds based on a new martingale representation formula we prove some quantitative upper bound estimates of the l p norm of some singular integral operators on complete riemannian manifolds this leads us to establish the weak l p hodge decomposition theorem and to prove the l p boundedness of the beurling ahlfors transforms on complete non compact riemannian manifolds with non negative weitzenbock curvature operator

symbolic functional calculus and n body resolvent estimates abstract we construct a functional calculus for symbolic functions of scattering pseudodifferential operators on manifolds with boundary this is applied to obtain mapping results on the scattering wavefront set of the b operator used by gerard et al 1994 commutator algebra and resolvent estimates advanced studies in pure mathematics vol 23 math soc japan tokyo to obtain n body resolvent estimates as an application we construct the plane waves for an n body short range schrodinger operator and obtain a scattering wavefront set estimate

riesz transforms through reverse holder and poincare inequalities we study the boundedness of riesz transforms in l p for p 2 on a doubling metric measure space endowed with a gradient operator and an injective omega accretive operator l satisfying davies gaffney estimates if l is non negative self adjoint we show that under a reverse holder inequality the riesz transform is always bounded on l p for p in some interval 2 2 epsilon and that l p gradient estimates for the semigroup imply boundedness of the riesz transform in l q for q in 2 p this improves results of auscher et al ann sci ecole norm sup 37 4 911 957 2004 and auscher and coulhon ann scuola norm sup pisa 4 531 555 2005 where the stronger assumption of a poincare inequality and the assumption e tl 1 1 were made the poincare inequality assumption is also weakened in the setting of a sectorial operator l in the last section we study elliptic perturbations of riesz transforms

regularity of the free boundary in parabolic phase transition problems in this paper we start the study of the regularity properties of the free boundary for parabolic two phase free boundary problems may be the best known example of a parabolic two phase free boundary problem is the stefan problem a simplified model describing the melting or solidification of a material with a solid liquid interphase the concept of solution can be stated in several ways classical solution weak so lution on divergence form or viscosity solution and as usual one would like to prove that the weak solutions that may be constructed are in fact as smooth and classical as possible locally a classical solution of the stefan problem may be described as following on the unit cylinder q1 b1 1 1 we have two complementary domains and qi separated by a smooth surface s oi2 nq1 in fl and qi we have two smooth solutions ul and u2 of the heat equations

gundy varopoulos martingale transforms and their projection operators on manifolds and vector bundles this paper proves the l p boundedness of generalized first order riesz transforms obtained as conditional expectations of martingale transforms a la gundy varopoulos for quite general diffusions on manifolds and vector bundles several specific examples and applications are presented lie groups of compact type the heisenberg group mathbb su 2 and riesz transforms on forms and spinors

computing kazhdan lusztig cells for unequal parameters abstract following lusztig we consider a coxeter group w together with a weight function l this gives rise to the pre order relation l and the corresponding partition of w into left cells we introduce an equivalence relation on weight functions such that in particular l is constant on equivalent classes we shall work this out explicitly for w of type f 4 and check that several of lusztig s conjectures concerning left cells with unequal parameters hold in this case even for those parameters which do not admit a geometric interpretation the proofs involve some explicit computations using chevie

riesz transform and l p cohomology for manifolds with euclidean ends let m be a smooth riemannian manifold which is the union of a compact part and a finite number of euclidean ends rr n setminus b 0 r for some r 0 each of which carries the standard metric our main result is that the riesz transform on m is bounded from l p m to l p m t m for 1 n the result is new for 2 2 for a more general class of manifolds assume that m is a n dimensional complete manifold satisfying the nash inequality and with an o r n upper bound on the volume growth of geodesic balls we show that boundedness of the riesz transform on l p for some p 2 implies a hodge de rham interpretation of the l p cohomology in degree 1 and that the map from l 2 to l p cohomology in this degree is injective

corner formation for the undercooled stefan problem in this paper a mechanism of corner formation for the one phase undercooled stefan problem in two and three dimensions is described by means of asymptotic methods

lp self improvement of generalized poincare inequalities in spaces of homogeneous type abstract in this paper we study self improving properties in the scale of lebesgue spaces of generalized poincare inequalities in spaces of homogeneous type in contrast with the classical situation the oscillations involve approximation of the identities or semigroups whose kernels decay fast enough and the resulting estimates take into account their lack of localization the techniques used do not involve any classical poincare or sobolev poincare inequalities and therefore they can be used in general settings where these estimates do not hold or are unknown we apply our results to the case of riemannian manifolds with doubling volume form and assuming gaussian upper bounds for the heat kernel of the semigroup e t with being the laplace beltrami operator we obtain generalized poincare inequalities with oscillations that involve the semigroup e t and with right hand sides containing either or 1 2

fully commutative elements in the weyl and affine weyl groups abstract let w be a weyl or an affine weyl group and let w c be the set of fully commutative elements in w we associate each w w c to a digraph g w by using g w we give a graph theoretic description for lusztig s a function on w c and describe explicitly all the distinguished involutions of w c the results verify two conjectures in our case one was proposed by myself in adv sci china math 3 1990 79 98 conjecture 8 10 and the other by lusztig in t asai et al open problems in algebraic groups in r hotta ed problems from the conference on algebraic groups and representations katata august 29 september 3 1983

positivity results for the hecke algebras of noncrystallographic finite coxeter groups abstract this paper is a report on a computer check of some important positivity properties of the hecke algebra in type h 4 including the nonnegativity of the structure constants in the kazhdan lusztig basis this answers a long standing question of lusztig s the same algorithm carried out by hand also allows us to deal with the case of dihedral coxeter groups

on the strong l p hodge decomposition over complete riemannian manifolds we establish the strong lp hodge decomposition theorem and the lp poincare inequalities on differential forms over complete non compact riemannian manifolds as applications we prove some lp estimates and existence theorems for the de rham operator as well as some vanishing theorems for the lp cohomology and the lp torsion over complete non compact riemannian manifolds with suitable geometric conditions

weighted norm inequalities off diagonal estimates and elliptic operators part iv riesz transforms on manifolds and weights this is the fourth article of our series here we study weighted norm inequalities for the riesz transform of the laplace beltrami operator on riemannian manifolds and of subelliptic sum of squares on lie groups under the doubling volume property and gaussian upper bounds