yang s system of particles and hecke algebras the graded hecke algebra has a simple realization as a certain algebra of operators acting on a space of smooth functions this operator algebra arises from the study of the root system analogue of yang s system of n particles on the real line with delta function potential it turns out that the spectral problem for this generalization of yang s system is related to the problem of finding the spherical tempered representations of the graded hecke algebra this observation turns out to be very useful for both these problems application of our technique to affine hecke algebras yields a simple formula for the formal degree of the generic iwahori spherical discrete series representations

on generalized cells in affine weyl groups abstract we determine the lowest generalized two sided cell for affine weyl groups we show that it consists of at most w 0 generalized left cells where w 0 denotes the corresponding finite weyl group for parameters coming from graph automorphisms we prove that this bound is exact for such parameters we also characterize all generalized left cells for finite and affine weyl groups

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

the distinguished involutions with a value n2 3n 3 in the weyl group of type dn let w s be a weyl group and h its associated hecke algebra let a z u u 1 be the laurent polynomial ring kazhdan and lusztig representation of coxeter groups and hecke algebras invent math 53 1979 165 184 introduced two a bases tw w w and cw w w for the hecke algebra h associated to w let yw y wul w l y ty then yw w w is also an a base for the hecke algebra in this paper we give an explicit expression for certain kazhdan lusztig basis elements cw as a linear combination of yx s in the hecke algebra of type dn in fact this gives also an explicit expression for certain kazhdan lusztig basis elements cw as a linear combination of tx s in the hecke algebra of type dn thus we describe also explicitly the kazhdan lusztig polynomials for certain elements of the weyl group we study also the joint relation among some elements in w and some distinguished involutions with a value n2 3n 3 in the weyl group of type dn

the principal series for a reductive symmetric space ii eisenstein integrals in this paper we develop a theory of eisenstein integrals related to the principal series for a reductive symmetric space g h here g is a real reductive group of harish chandra s class an involution of g and h an open subgroup of the group g of xed points for the group g itself is a symmetric space for the left right action of g g we refer to this setting as the group case up to a normalization our eisenstein integrals generalize those of harish chandra 18 associated with a minimal parabolic subgroup in the group case

the based ring of the lowest two sided cell of an affine weyl group we show that lusztig s homomorphism from an affine hecke algebra to the direct summand of its asymptotic hecke algebra corresponding to the lowest two sided cell is related to the homomorphism constructed by chriss and ginzburg using equivariant k theory by a matrix over the representation ring of the associated algebraic group

spectral correspondences for affine hecke algebras we introduce the notion of spectral transfer morphisms between normalized affine hecke algebras and show that such morphisms induce spectral measure preserving correspondences on the level of the tempered spectra of the affine hecke algebras involved we define a partial ordering on the set of isomorphism classes of normalized affine hecke algebras which plays an important role for the langlands parameters of lusztig s unipotent representations

an explicit expression for the harish chandra c function for the homogeneous space so 2 n s o 2 o n n 2 4 6 this paper is devoted to presenting an explicit expression for the harish chandra c function for the lie ball our computation technique is based on the expression for the generalized spherical function and its asymptotic behavior

hecke algebras and harmonic analysis on p adic groups we compute plancherel measures and formal degrees for unipotent representations of p adic groups using hecke algebra isomorphisms the results verify special cases of the conjectural uniformity of formal degrees in an l packet

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

homological algebra for affine hecke algebras abstract in this paper we study homological properties of modules over an affine hecke algebra h in particular we prove a comparison result for higher extensions of tempered modules when passing to the schwartz algebra s a certain topological completion of the affine hecke algebra the proof is self contained and based on a direct construction of a bounded contraction of certain standard resolutions of h modules this construction applies for all positive parameters of the affine hecke algebra this is an important feature since it is an ingredient to analyse how the irreducible discrete series representations of h arise in generic families over the parameter space of h for irreducible non simply laced affine hecke algebras this will enable us to give a complete classification of the discrete series characters for all positive parameters we will report on this application in a separate article

l2 poisson integral representations of solutions of the hua system on the bounded symmetric domain su n n s u n u n abstract let d z m n c i n zz positive definite be the matrix ball of rank n and let hd be the associated hua operator for a complex number such that rei n 1 we give a necessary and sufficient condition on solutions f of the following hua system of differential equations on d h d f z 2 n 2 f z i n to have an l2 poisson integral representations over the shilov boundary s of d namely f is the poisson integral of an l2 function on the shilov boundary s if and only if its satisfies the following growth condition of hardy type f 2 sup 0 r 1 r 2 n n rei s f ru 2 du in particular for in this shows that every harmonic function f with respect to the hua operator hd has an l2 poisson representation over s if and only if its hardy norm is finite

restriction to compact subgroups in the cyclic homology of reductive p adic groups restriction of functions from a reductive p adic group g to its compact subgroups defines an operator on the hochschild and cyclic homology of the hecke algebra of g we study the commutation relations between this operator and others coming from representation theory jacquet functors idempotents in the bernstein centre and characters of admissible representations

homology of graded hecke algebras let h be a graded hecke algebra with complex deformation parameters and weyl group w we show that the hochschild cyclic and periodic cyclic homologies of h are all independent of the parameters and compute them explicitly we use this to prove that if the deformation parameters are real the collection of irreducible tempered h modules with real central character forms a q basis of the representation ring of w our method involves a new interpretation of the periodic cyclic homology of nite type algebras in terms of the cohomology of a sheaf over the underlying complex ane variety

left cells in the affine weyl group of typec4 we find a representative set of left cells of the affine weyl groupwaof typec4as well as its left cell graphs by applying an algorithm this algorithm was designed in my previous paper tohoku j math 46 1994 105 124 it is reformulated and improved in a more efficient form here these representatives of left cells are presented as the vertices of so called essential graphs so that the generalized invariants of left cells ofwaare actually described explicitly the latter almost characterize the left cells ofwa some comments and conjectures are proposed on cells of affine weyl groups mostly for the case of typecl l 2

analytic r groups of affine hecke algebras we define analytic r groups for affine hecke algebras and prove the analog of the knapp stein dimension theorem as a corollary we prove that the commutant algebra of a unitary principal series representation is isomorphic to the complex group algebra of the r group twisted by a certain 2 cocycle gamma for classical hecke algebras we prove that gamma is always trivial

hua operators on bounded homogeneous domains in cnand alternative reproducing kernels for holomorphic functions let d be a bounded homogeneous domain in cn note that d is not assumed to be hermitian symmetric in this work we are interested in studying various classes of harmonic functions on d and the possibility of representing them as poisson integrals over the bergman shilov boundary one such class of harmonic functions is the hua harmonic functions specifically by forming a contraction of with the holomorphic curvature tensor we define a canonical system of differential operators which generalizes the classical hua system this system is invariant under all bi holomorphisms of d the hua harmonic functions are by definition the nullspace of this system our main result concerning this system is that every bounded hua harmonic function is the poisson integral over the bergman shilov boundary of a uniquel function against the poisson kernel for the laplace beltrami operator we also consider spaces of harmonic functions defined as the kernel of a single real differential operator which is invariant under a particular solvable lie group which acts transitively on d we show that there exists such an operator which a annihilates holomorphic functions b satisfies the hormander condition and c has the bergman shilov boundary as its maximal boundary it follows that the corresponding bounded harmonic functions are in one to one correspondence with thel functions on the bergman shilov boundary under poisson integration

the periodic cyclic homology of crossed products of finite type algebras we study the periodic cyclic homology groups of the cross product of a finite type algebra a by a discrete group in case a is commutative and the group is finite our results are complete and given in terms of the singular cohomology of the sets of fixed points these groups identify our cyclic homology groups with the orbifold cohomology of the underlying algebraic orbifold the proof is based on a careful study of localization at fixed points and of the resulting koszul complexes this is achieved by extending to our class of noncommutative algebras the concept of an infinitesimal neighborhood that plays such an important role in commutative algebra we provide examples of azumaya algebras for which this identification is however no longer valid as an example we discuss some affine weyl groups

a sum rule for scale factors in the goldie rank polynomials this paper is a corollary to s and we adopt the same notation which is also that of s there we showed how two deep questions concerning the enveloping algebra of a semisimple lie algebra could be reduced to a purely combinatorial analysis of the kazhdan lusztig polynomials the first concerns the computation of sot l l wl l yl apart from its own intrinsic interest this question has deep interconnections with geometry via the component group a of the associated nilpotent orbit the second concerns the coefficients z which are ratios of goldie ranks these are central to the determination of the scale factors in the goldie rank polynomials which in turn relate to complete primeness and hence to unitarity see 7 1 4 16 7 121 it first seemed that the results in s did not give any easily extractable information however in 13 lusztig shows that for some canonical quotient a of a there exists a left cell of the weyl group w such that the character table of a can be read off in a well defined manner from data involving the hecke algebra module structure of the cell which we shall call a lusztig cell combined with s this gives several remarkable results first when rl z2 are involutions in a lusztig cell sot l l r a l 12 is determined by multiplying appropriate irreducible characters of a and identifying coefficients this implies that when a e s which occurs in type e it can happen that l l zl l r l is not multiplicity free providing thereby a counterexample to conjecture of 5 5 111 even in the diagonal case second if r is an involution in a lusztig cell then z is the degree of an irreducible character of a third we prove that z divides ial for every element w of a lusztig cell in partial answer to a conjecture in 9 3 231

cells in coxeter groups i abstract the purpose of this article is to shed new light on the combinatorial structure of kazhdan lusztig cells in infinite coxeter groups w our main focus is the set d of distinguished involutions in w which was introduced by lusztig in one of his first papers on cells in affine weyl groups we conjecture that the set d has a simple recursive structure and can be enumerated algorithmically starting from the distinguished involutions of finite coxeter groups moreover to each element of d we assign an explicitly defined set of equivalence relations on w that altogether conjecturally determine the partition of w into left right cells we are able to prove these conjectures only in a special case but even from these partial results we can deduce some interesting corollaries

hochschild homology of affine hecke algebras abstract let h h r q be an affine hecke algebra with complex possibly unequal parameters q which are not roots of unity we compute the hochschild and the cyclic homology of h it turns out that these are independent of q and that they admit an easy description in terms of the extended quotient of a torus by a weyl group both of which are canonically associated to the root datum r for positive q we also prove that the representations of the family of algebras h r q c come in families which depend analytically on analogous results are obtained for graded hecke algebras and for schwartz completions of affine hecke algebras

some homological properties of category o iii we prove that thick category o associated to a semi simple complex finite dimensional lie algebra is extension full in the category of all modules we also prove the weak alexandru conjecture both for regular blocks of thick category o and the associated categories of harish chandra bimodules but disprove it for singular blocks

lusztigʼs a function for coxeter groups of rank 3 abstract we show that lusztig s a function of a coxeter group is bounded if the rank of the coxeter group is 3

strongly self dual graphs we present a class of graphs whose adjacency matrices are nonsingular with integral inverses denoted h graphs if the h graphs g and h with adjacency matrices m g and m h satisfy m g 1 sm h s where s is a signature matrix we refer to h as the dual of g the dual is a type of graph inverse if the h graph g is isomorphic to its dual via a particular isomorphism we refer to g as strongly self dual we investigate the structural and spectral properties of strongly self dual graphs with a particular emphasis on identifying when such a graph has 1 as an eigenvalue

affine cellular algebras abstract graham and lehrer have defined cellular algebras and developed a theory that allows in particular to classify simple representations of finite dimensional cellular algebras many classes of finite dimensional algebras including various hecke algebras and diagram algebras have been shown to be cellular and the theory due to graham and lehrer successfully has been applied to these algebras we will extend the framework of cellular algebras to algebras that need not be finite dimensional over a field affine hecke algebras of type a and infinite dimensional diagram algebras like the affine temperley lieb algebras are shown to be examples of our definition the isomorphism classes of simple representations of affine cellular algebras are shown to be parameterised by the complement of finitely many subvarieties in a finite disjoint union of affine varieties in this way representation theory of non commutative algebras is linked with commutative algebra moreover conditions on the cell chain are identified that force the algebra to have finite global cohomological dimension and its derived category to admit a stratification these conditions are shown to be satisfied for the affine hecke algebra of type a if the quantum parameter is not a root of the poincare polynomial

k types for the tempered components of a p adic general linear group with an appendix by p schneider u stuhler the definition of the tempered category

directed intervals and the dual of a graph abstract we present a class of graphs whose adjacency matrices are nonsingular with integral inverses denoted h graphs for the h graph g with adjacency matrix m we consider the problem of identifying exactly when m 1 is signature similar to the adjacency matrix of another h graph g when this holds g is a type of graph inverse of g which is known as the dual of g we present necessary and sufficient conditions for the existence of g as an application we provide a characterization of all dual pairs g and g where both graphs are unicyclic this characterization allows us to identify those h graphs g that are unicyclic and self dual

self inverse unicyclic graphs and strong reciprocal eigenvalue property abstract we consider only simple graphs a graph g is said to be nonsingular if its adjacency matrix a g is nonsingular the inverse of a nonsingular graph g is the unique weighted graph whose adjacency matrix is similar to the inverse of the adjacency matrix a g via a diagonal matrix of 1s an invertible graph g is said to be a self inverse graph if g is isomorphic to its inverse let h be the class of connected bipartite graphs with unique perfect matchings we consider the problem of characterizing the graphs in h which are self inverse graphs in this article we answer this problem for the class of unicyclic graphs the study of the strong reciprocal eigenvalue property is closely related to the study of graphs which are isomorphic to their inverse graphs a nonsingular graph g is said to satisfy the property sr if the reciprocal of each eigenvalue of the adjacency matrix a g is also an eigenvalue of a g and they both have the same multiplicities in this article we show that if a unicyclic graph g in h has property sr then g is invertible and the inverse graph of g is unicyclic as an application we show that a noncorona unicyclic graph in h with property sr can have one of the five specified structures finally our discussions lead to the following problem does there exist a unicyclic graph g h which has property sr but g is not self inverse

the second lowest two sided cell in an affine weyl group abstract let w a be an irreducible affine weyl group with w 0 the associated weyl group the present paper is to study the second lowest two sided cell qr of w a let n qr be the number of left cells of w a in qr we conjecture that the equality n qr 1 2 w 0 should always hold when w a is either a n 1 n 2 or of rank 4 this equality can be verified by the existing data see 0 3 then the main result of the paper is to prove the inequality n qr 1 2 w 0 in all cases

reduction to real infinitesimal character in affine hecke algebras the main result of this paper is to show that the problem of the determination of the unramified unitary dual of a split p adic group is equivalent to the problem of determining the unitary dual of the corresponding graded hecke algebra in bm the authors established this equivalence in the case of iwahori spherical representations under a certain restriction namely it was essential for the infinitesimal character to be real in the terminology of bmj in terms of the langlands deligne lusztig parameters s u p kul the restriction is that s e lg be a purely hyperbolic element the technique used in bm was to combine the notion of the signature of a k character in v with some facts which follow from kl namely that the 2w characters of tempered representations are linearly independent this is essentially true precisely when the infinitesimal character is real for if not s has an elliptic part se such that

cocenters of hecke clifford and spin hecke algebras abstract we determine a basis of the cocenter i e the trace or zeroth hochschild homology of the degenerate affine hecke clifford and spin hecke algebras in classical types

the leading coefficient of certain kazhdan lusztig polynomials of the permutation group sn abstract in this paper we show that the leading coefficient y w of certain kazhdan lusztig polynomials p y w of the permutation group s n of 1 2 n are not greater than 1 more precisely we show that the leading coefficients y w are not greater than 1 whenever a y a w where a s n n is the function defined in g lusztig cells in affine weyl groups in algebraic groups and related topics in adv stud pure math vol 6 kinokunia north holland 1985 pp 255 287 see 1 5 for a simple interpretation of the function a

isomorphisms between affine hecke algebras of type a2 abstract we prove that two affine hecke algebras h p and h q of type a 2 with nonzero parameters p q c are not isomorphic whenever p q and p q 1 theorem 3 1 however they are isomorphic locally more precisely we prove that the quotient algebras h s q and h t p are isomorphic provided that n s q n t p for type a 2 theorem 2 4

boundary value problems for the shilov boundary of a bounded symmetric domain of tube type abstract in this article the images of the poisson transform on the degenerate series representations attached to the shilov boundary of a bounded symmetric domain of tube type are considered we characterize the images by means of second order differential equations

the based ring of the lowest generalized two sided cell of an extended affine weyl group abstract let c 0 be the lowest generalized two sided cell of an extended affine weyl group w with unequal parameters we first prove that certain conjectures of lusztig called p1 p15 hold for c 0 which implies the existence of the based ring of c 0 and then we determine the structure of the based ring as an application we use the structure of the based ring to study certain simple modules of the hecke algebra with unequal parameters associated to w namely those attached to c 0 further we give a set of prime ideals p of the center z of the affine hecke algebra h such that the reduced algebra k p h is simple over k p where k p frac z p is the residue field of z at p in particular we show that the algebra h z frac z is split simple over the field frac z

discrete series characters for affine hecke algebras and their formal degrees we introduce the generic central character of an irreducible discrete series representation of an affine hecke algebra using this invariant we give a new classification of the irreducible discrete series characters for all abstract affine hecke algebras except for the types e n 1 n 6 7 8 with arbitrary positive parameters and we prove an explicit product formula for their formal degrees in all cases

kazhdan lusztig cells and the frobenius schur indicator abstract let w be a finite coxeter group it is well known that the number of involutions in w is equal to the sum of the degrees of the irreducible characters of w following a suggestion of lusztig we show that this equality is compatible with the decomposition of w into kazhdan lusztig cells the proof uses a generalisation of the frobenius schur indicator to symmetric algebras which may be of independent interest

left cells in the weyl group of type e8 abstract the aim of the present paper is to give an explicit description for all the left cells of the weyl group w of type e 8 we first find a representative set for the left cells of w and then use it to describe these cells our results show that most of the left cells in w can be characterized by their generalized invariants alone then the remaining ones which are called exceptional left cells are characterized by the generalized invariants together with the a values

paquets d ondes dans l espace de schwartz d un espace symetrique reductif resume we study holomorphic families of k finite eigenfunctions on symmetric spaces g h called functions ii hol by analogy with hc eisenstein integrals cf b3 d suitably normalized by a polynomial factor provide examples of such families a function ii hol is said ii hol if roughly speaking its constant term along any stable parabolic subgroup is a finite sum of functions ii hol s where s varies in a determined finite set we prove that for a function ii hol one can form wave packets in the schwartz space we prove also a criterion for a function ii hol to be ii hol an important fact is that for minimal stable parabolic subgroups our criterion implies with the help of the maas selberg relations cf b2 b3 a normalization of eisenstein integrals all the article relies on the theory of the constant term cf c

on the determination of kazhdan lusztig cells for affine weyl groups with unequal parameters abstract let w be a coxeter group and l be a weight function on w following lusztig we have a corresponding decomposition of w into left cells which have important applications in representation theory we study the case where w is an affine weyl group of type g 2 using explicit computation with coxeter and chevie we show that 1 there are only finitely many possible decompositions into left cells and 2 the number of left cells is finite in each case thus confirming some of lusztig s conjectures in this case a key ingredient of the proof is a general result which shows that the kazhdan lusztig polynomials of affine weyl group are invariant under large enough translations

function spaces and reproducing kernels on bounded symmetric domains there are several natural hilbert spaces of holomorphic functions on a bounded symmetric domain such are the bergman type spaces on which the holomorphic discrete series operates and the hardy type spaces which are related to the analytic continuation of the holomorphic discrete series also closely related is the bergmann space of entire functions on the ambient which arises as the closure of the polynomials with respect to a natural inner product the space of holomorphic polynomials decomposes into irreducible subspaces under the action of the isotropy group k of the domain the main facts about this decomposition were proved by schmid 21 for another proof see 22 each irreducible subspace contains a unique normalized l invariant spherical polynomial where l is the isotropy group of the shilov boundary in k our first main result is the explicit computation of the norms of the spherical polynomials with respect to each of the hilbert spaces considered for the domains of classical type they were considered by hua 9 and for some of the hilbert spaces this was done before by upmeier 24 using different methods for certain others there are partial results in 22 we are able to do this in a fairly simple unified way by making strong use of gindikin s generalized gamma function s next we obtain a description of the reproducing kernels of the k irreducible subspaces in each of our hilbert spaces and an expansion in terms of these for

l p poisson integral representations of solutions of the hua system on hermitian symmetric spaces of tube type abstract in this paper we give a necessary and sufficient condition on eigenfunctions of the hua operator on a hermitian symmetric space of tube type x g k to have an l p poisson integral representations over the shilov boundary of x more precisely let c such that r 1 2 being the genus of x and let f be a c valued function on x satisfying the following hua system of second order differential equations h q f 2 2 32 2 f z then f has an l p poisson integral representation 1 p over the shilov boundary of x if and only if it satisfies the following growth condition of hardy type sup t 0 e r r t k f k a t p d k 1 p in particular for we obtain that a hua harmonic function on x has an l p poisson integral representation over the shilov boundary of x if and only if its hardy norm is finite

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

joint relations on elements of the symmetric group abstract we introduce two methods which reduce the checking of the joint relations on element pairs x y i e checking if x y 0 see 1 2 inside a two sided cell of the symmetric group s n to some special cases up to star operations one involves the involutions in s n and the other involves the set for n see 3 1 and 5 1 we give a detailed investigation for the properties of the set in terms of standard tableaux also we study the sets u s and u s and give an affirmative answer to a question of n h xi see proposition 2 9

on unimodular graphs abstract we study graphs whose adjacency matrices have determinant equal to 1 or 1 and characterize certain subclasses of these graphs graphs whose adjacency matrices are totally unimodular are also characterized for bipartite graphs having a unique perfect matching we provide a formula for the inverse of the corresponding adjacency matrix and address the problem of when that inverse is diagonally similar to a nonnegative matrix special attention is paid to the case that such a graph is unicyclic

generalized springer correspondence and green functions for type b c graded hecke algebras this paper presents a combinatorial approach to the tempered representation theory of a graded hecke algebra h of classical type b or c with arbitrary parameters we present various combinatorial results which together give a uniform combinatorial description of what becomes the springer correspondence in the classical situation of equal parameters more precisely by using a general version of lusztig s symbols which describe the classical springer correspondence we associate to a discrete series representation of h with central character w0c a set w0c of w0 characters where w0 is the weyl group this set w0c is shown to parametrize the central characters of the generic algebra which specialize into w0c using the parabolic classification of the central characters of h t r on the one hand and a truncated induction of weyl group characters on the other hand we define a set w0c for any central character w0c of h t r and show that this property is preserved we show that in the equal parameter situation we retrieve the classical springer correspondence by considering a set u of partitions which replaces the unipotent classes of so2n 1 c and sp2n c and a bijection between u and the central characters of h t r we end with a conjecture which basically states that our generalized springer correspondence determines h t r exactly as the classical springer correspondence does in the equal label case in particular we conjecture that w0c indexes the modules in h t r with central character w0c in the following way a module m in h t r has a natural grading for the action of w0 and the w0 representation m in its top degree is irreducible when m runs through the modules in h t r with central character w0c m runs through w0c moreover still in analogy with the equal parameter case we conjecture that the w0 structure of the modules in h t r can be computed using generalized green functions

a counter example to a conjecture of lusztig abstract in lusztig 2003 7 lusztig defined two functions a and a on a coxeter group w and conjectured that they are the same in the present paper we give a counter example to lusztig s conjecture then we propose a new conjecture for the description of two sided cells of w which can be regarded as an improvement of lusztig s conjecture some examples are given to support our conjecture