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

the principal series of p adic groups with disconnected center let g be a split connected reductive group over a local non archimedean field we classify all irreducible complex g representations in the principal series irrespective of the dis connectedness of the center of g this leads to a local langlands correspondence for principal series representations of g it satisfies all expected properties in particular it is functorial with respect to homomorphisms of reductive groups at the same time we show that every bernstein component s in the principal series has the structure of an extended quotient of bernstein s torus by bernstein s finite group both attached to s

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

coxeter elements and kazhdan lusztig cells abstract by the correspondence between coxeter elements of a coxeter system w s and the acyclic orientations of the coxeter graph we study some properties of elements in the set c0 w we show that when w is of finite affine or hyperbolic type any w c0 w satisfies w lr w j with l wj j m w for some j s now assume that w is of finite or affine type we give an explicit description for all the distinguished involutions d of w with d l w for some w e w which verifies a conjecture proposed in jian yi shi adv sci china math 3 1990 79 98 conjecture 8 10 in our case we show that any left cell of w containing some element of c0 w is left connected which verifies a conjecture of lusztig ryoshi hotta ed in problems from the conference on algebraic groups and representations held at katata august 29 september 3 1983 in our case

balanced projectives and axiom 3 abstract a global definition of k nice subgroup is formulated which generalizes both the notion of a nice subgroup in the context of torsion groups and the notion of a k nice submodule used in a recent study of p local balanced projective groups it is shown analogous to hill s treatment of totally projective p groups that the global balanced projective groups of warfield are precisely those mixed abelian groups that satisfy the third axiom of countability for k nice subgroups a variety of equivalent characterizations of balanced projectives are derived by methods independent of the uniqueness and existence theorems previously established for these groups

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 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

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

canonical left cells in affine weyl groups soit w s un groupe de weyl affine irreductible on montre que chaque cellule a droite et a gauche de w contient une cellule canonique a gauche

the joint relations and the set d1 in certain crystallographic groups abstract for any coxeter group w d kazhdan and g lusztig 1979 invent math 53 165 184 introduced some joint relations among the elements of w by which certain representations of w and its associated hecke algebra h are constructed recently in a letter to lusztig t a springer obtained some interesting results concerning the connection between some non zero structural coefficients of h and the joint relations in w when w is a certain crystallographic group he investigated some properties of a subset d 1 of w springer denoted d 1 by f in his letter which plays an important role for the joint relations in w in the present paper we shall develop some results of springer we shall deduce further joint relations in w from some non zero structural coefficients of h in particular we shall make some detailed discussion for the set d 1 this includes the approach to the proof of a conjecture of lusztig concerning the characterization of the set d 1 the connection between the finiteness of d 1 and some joint relations in w the explicit description for some elements of d 1 and the connectedness of the set d 0 d 2 1 see 3 2 and 5 2 for the notations

the combinatorics of coxeter categories we present an alternative construction of soergel s category of bimodules associated to a reflection faithful representation of a coxeter system we show that its objects can be viewed as sheaves on the associated moment graph we introduce an exact structure and show that soergel s special bimodules are the projective objects then we construct the indecomposable projectives by both a global and a local method discuss a version of the kazhdan lusztig conjecture and prove it for universal coxeter systems

homological invariants in category for the general linear superalgebra we study three related homological properties of modules in the bgg category mathcal o for basic classical lie superalgebras with specific focus on the general linear superalgebra these are the projective dimension associated variety and complexity we demonstrate connections between projective dimension and singularity of modules and blocks similarly we investigate the connection between complexity and atypicality this creates concrete tools to describe singularity and atypicality as homological and hence categorical properties of a block however we also demonstrate how two integral blocks in category mathcal o with identical global categorical characteristics of singularity and atypicality will generally still be inequivalent this principle implies that category mathcal o for mathfrak gl m vert n can contain infinitely many non equivalent blocks which we work out explicitly for mathfrak gl 3 vert 1 all of this is in sharp contrast with category mathcal o for lie algebras but also with the category of finite dimensional modules for superalgebras furthermore we characterise modules with finite projective dimension to be those with trivial associated variety we also study the associated variety of verma modules to do this we also classify the orbits in the cone of self commuting odd elements under the action of an even borel subgroup

cells and the reflection representation of weyl groups and hecke algebras let f be the generic algebra of the finite crystallographic coxeter group w defined over the ring q u1 2 u 2 first the two sided cell corresponding to the reflection representation of f is shown to consist of the nonidentity elements of w having a unique reduced expression next the matrix entries of this representation are computed in terms of certain kazhdanlusztig polynomials finally the kazhdan lusztig polynomials just mentioned are described in case w is of type a1 lor b

oscillatory behaviour in functional differential systems of neutral type in this paper are obtained several criteria for oscillations of a class of autonomous functional differential systems of neutral type

the lowest two sided cell of a coxeter group with complete graph abstract in this paper we give a description of the structure of the based ring of the lowest two sided cell for a weighted coxeter group with complete graph it is proved that this ring is generated in a simple way by some subrings which are isomorphic to the based rings of the lowest two sided cells for affine weyl groups of type a 1 or a 2

on certain distinguished involutions in the weyl group of type dn let w s be a weyl group let a z u u 1 be the laurent polynomial ring in an indeterminate u kazhdan and lusztig invent math 33 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 basis for the hecke algebra in this paper we assume w of type dn and we express certain kazhdan lusztig basis elements cw as a linear combination of yx s this in turn gives an explicit expression for certain kazhdan lusztig basis elements cw as a linear combination of tx s thus we describe explicitly the kazhdan lusztig polynomials for certain pairs of elements of w we also study the joint relation among some elements in w in particular we find certain distinguished involutions in the two sided cell t of w with a value 12 n2 n 4t2 2t for 1 2t n and n even 12 n2 n 4t2 2t for n odd where the two sided cell t does not contain the longest element w0 j in subgroup wj of w for any j s

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

left connectedness of some left cells in certain coxeter groups of simply laced type abstract let w be an irreducible finite or affine weyl group of simply laced type we show that any w w with a w 6 satisfies condition c w x w j y for some x y w and some j s with w j finite and l w j a w see 0 1 0 2 for the notation w j w j l w and a w we also show that if l is a left cell of w all of whose elements satisfy condition c then the distinguished involution d l of w in l satisfies d l z 1 z z 1 w j z for any z w j z e min l with j l z see 1 6 for the notation z 1 z and 0 3 for l z e min l and e l verifying a conjecture of mine in j y shi a survey on the cell theory of affine weyl groups adv sci china math 3 1990 79 98 conjecture 8 10 in our case if e l e min l then we show that the left cell l is left connected verifying a conjecture of lusztig in our case

simple transitive 2 representations of small quotients of soergel bimodules in all finite coxeter types but i2 12 i2 18 and i2 30 we classify simple transitive 2 representations for the quotient of the 2 category of soergel bimodules over the coinvariant algebra which

oscillatory behavior in linear retarded functional differential equations abstract this paper is devoted to the investigation of when all solutions of autonomous and nonautonomous linear retarded functional differential equations are oscillatory

torsion free groups in this paper we introduce the class of torsion free k groups and the notion of a knice subgroup torsion free k groups form a class of groups more extensive than the separable groups of baer but they enjoy many of the same closure properties we establish a role for knice subgroups of torsion free groups analogous to that played by nice subgroups in the study of torsion groups for example among the torsion free groups the balanced projectives are characterized by the fact that they satisfy the third axiom of countability with respect to knice subgroups separable groups are characterized as those torsion free k groups with the property that all finite rank pure knice subgroups are direct summands the introduction of these new classes of groups and subgroups is based on a preliminary study of the interplay between primitive elements and valuated coproducts as a by product of our investigation new proofs are obtained for many classical results on separable groups our techniques lead naturally to the discovery that a balanced subgroup of a completely decomposable group is itself completely decomposable provided the corresponding quotient is a separable group of cardinality not exceeding 81 that is separable groups of cardinality 81 have balanced projective dimension

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

oscillation for systems of functional differential equations abstract this paper is concerned with the oscillation problem of functional differential systems of the form formula whereq0and are in r n n based on its characteristic equation new explicit conditions for oscillation are obtained by utilizing the lozinskii measures of matrices

affine cellularity of affine yokonuma hecke algebras abstract we establish an explicit algebra isomorphism between the affine yokonuma hecke algebra y r n q and a direct sum of matrix algebras with coefficients in tensor products of affine hecke algebras of type a as an application of this result we show that y r n q is affine cellular in the sense of koenig and xi and further prove that it has finite global dimension when the parameter q is not a root of the poincare polynomial as another application we also recover the modular representation theory of y r n q previously obtained in 7

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

oscillations of differential difference systems of neutral type this note is concerned with the oscillatory behavior of a linear differential difference system of neutral type criteria for oscillations are obtained for autonomous and nonautonomous systems

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

representations of affine hecke algebras and based rings of affine weyl groups it is known that an interesting part of the study of the representation theory of p adic groups can be reduced to the study of the representation theory of affine hecke algebras b v let w s be an extended affine weyl group and hk qo the corresponding hecke algebra over a field k with a nonzero parameter qo k when k is the complex numbers field and qo is not a root of unity a classification of simple representations of hk qo was established in kl2 deligne langlands lusztig classification for affine type a a classification of simple representations of hk qo was obtained in am for any qo and arbitrary sufficiently large k when k is algebraically closed and has positive characteristic the representations of hk qo were studied by vign ras as part of her study of modular representations of p adic groups v in this paper we shall verify a conjecture of lusztig l6 7 a by means of the based ring of an extended affine weyl group theorem 3 3 the conjecture says that if the parameter qo is not a root of the corresponding poincar polynomial then the classification established in kl2 remains valid the restriction is necessary for the classification see remark 3 4 a

cells of harish chandra modules for real classical groups we compute the w module structure of a large number of cells of harish chandra modules for a classical group g where w is the complex weyl group

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

isotype submodules of p local balanced projective groups by giving necessary and sufficient conditions for two isotype submodules of a p local balanced projective group to be equivalent we are able to introduce a general theory of isotype submodules of p local balanced projective groups or ib modules numerous applications of the above result are available particularly for the special class of ib modules introduced by wick known as skt modules we first show that the class of skt modules is closed under direct summands and then we are able to show that if h appears as an isotype submodule of the p local balanced projective group g such that g h is the coproduct of countably generated torsion groups then h is an skt module finally we show that ib modules satisfy general structural properties such as transitivity full transitivity and the equivalence of p high

a proof of lusztig s conjectures for affine type g2 with arbitrary parameters we prove lusztig s conjectures bf p1 bf p15 for the affine weyl group of type tilde g 2 for all choices of parameters our approach to compute lusztig s mathbf a function is based on the notion of a balanced system of cell representations for the hecke algebra we show that for arbitrary coxeter type the existence of balanced system of cell representations is sufficient to compute the mathbf a function and we explicitly construct such a system in type tilde g 2 for arbitrary parameters we then investigate the connection between kazhdan lusztig cells and the plancherel theorem in type tilde g 2 allowing us to prove bf p1 and determine the set of duflo involutions from there the proof of the remaining conjectures follows very naturally essentially from the combinatorics of weyl characters of types g 2 and a 1 along with some explicit computations for the finite cells

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

motives with exceptional galois groups and the inverse galois problem we construct motivic l adic representations of textup gal overline mathbb q mathbb q into exceptional groups of type e 7 e 8 and g 2 whose image is zariski dense this answers a question of serre the construction is uniform for these groups and is inspired by the langlands correspondence for function fields as an application we solve new cases of the inverse galois problem the finite simple groups e 8 mathbb f ell are galois groups over mathbb q for large enough primes l

oscillatory retarded functional systems this note is concerned with the oscillatory behavior of a linear retarded system several criteria are obtained for having the system oscillatory conditions regarding the existence of nonoscillatory solutions are also given

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

a unipotent support for irreducible representations 1 1 let g f be the group of fq rational points of a connected reductive algebraic group g defined over a finite field f throughout this paper we shall assume that the characteristic of the ground field is large the following is part of problem ii in l9 let p be an irreducible complex character of g f show that there is a unique umpotent class c in g which has the property that cga c cfqi p g 0 and has maximal dimension among classes with this property one of the results of this paper is a solution of this problem we also prove the following refinement if g e gf is such that tr g p 0 then the unipotent part of g lies in c or in a conjugacy class of dimension dim c we call c the unipotent support of p another map from the set of irreducible representations of gf to unipotent classes in g has been defined in l8 13 43 this was the composition of two maps one from irreducible representations of gf to a set of parameters in the langlands dual of g and one from this set of parameters to unipotent classes in g another conjectural definition of this map which did not pass through the langlands dual has been proposed by kawanaka kl k2 k3 in terms of generalized gelfand graev representations in analogy with real lie groups he uses the name wave front set for the unipotent class attached to a representation one of the results of this paper is a proof see section 11 of kawanaka s conjecture that his wave front set is well defined and that it coincides with the one defined in l8 this has been known earlier from the work of kawanaka in certain special cases we also show that the unipotent support of an irreducible representation coincides with the wave front set of the dual representation

the relation lr on some elements of the affine weyl group c n abstract let w s be the affine weyl group of type c n with s its coxeter generator set let 2 n 1 be the set of all partitions 1 r of 2 n 1 such that j 1 2 k 1 j is odd for any k n with 2 k 1 r for any j s let w j be the longest element in the parabolic subgroup of w generated by j we define a map w j j s 2 n 1 and study the preorder lr on the set w j j s and its relation with the partial order on the set w j j s where iterating star operations and primitive pairs play an important role

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

the cells in the weighted coxeter group c n l m abstract the affine weyl group c n s can be realized as the fixed point set of the affine weyl group a m s m m 2 n 1 2 n 2 n 1 under a certain group automorphism m n let l m be the length function of a m the aim of the present paper is to give a combinatorial description for all the left cells of a m which have non empty intersection with c n then we use this description to deduce some formulae for the number of left cells of the weighted coxeter group c n l m in the set e associated to any partition of m 1

kazhdan lusztig coefficients for an affine weyl group of type b 2 in this paper we compute the leading coefficients u w of the kazhdan lusztig polynomials pu w for an affine weyl group of type b 2 by using the a function of a coxeter group defined by lusztig we compute most u w explicitly with part of these values u w we show that a conjecture of lusztig on distinguished involutions is true for an affine weyl group of type b 2 we also show that the conjectural formula concerning some semilinear equations needs a modification

the balanced projective dimension of abelian p groups the balanced projective dimension of every abelian p group is de termined by means of a structural property that generalizes the third axiom of countability as a corollary to our general structure theorem we show for a wn that every pa high subgroup of a p group g has balanced projective dimension exactly n whenever g has cardinality nn but pag o our characterization of balanced projective dimension also leads to new classes of groups where different infinite dimensions are distinguished

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

nonoscillations in retarded systems this note is concerned with the existence of nonoscillatory solutions of a linear retarded system several criteria for nonoscillations are obtained some of them regarding specific classes of continuous and differentiable delay functions

tabular algebras and their asymptotic versions we introduce tabular algebras which are simultaneous generalizations of cellular algebras in the sense of graham lehrer and table algebras in the sense of arad blau we show that if a tabular algebra is equipped with a certain kind of trace map then the algebra has a corresponding asymptotic version whose structure can be explicitly determined we also study various natural examples of tabular algebras

comparison ofa functions abstract we show agreement of lusztig sa function with thea function previously given by c k fan j amer math soc 10 no 1 1997 139 167

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

nonoscillations for odd dimensional systems of linear retarded functional differential equations this paper is concerned with the nonoscillatory problems of odd dimensional systems of linear retarded functional differential equations based upon the corresponding characteristic equations we get some criteria for nonoscillations by utilizing the matrix measures

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

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

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

characterizations of warfield groups abstract warfield groups are defined to be direct summands of simply presented abelian groups by introducing a global definition of quasi sequentially nice subgroups we prove some characterizations of warfield groups involving these groups knice subgroups and valuated coproducts furthermore we exhibit a new proof of a characterization of warfield groups in terms of nice subgroups and valuated coproducts

some examples of square integrable representations of semisimple p adic groups we construct irreducible representations of the hecke algebra of an affine weyl group analogous to kilmoyer s reflection representation corresponding to finite weyl groups and we show that in many cases they correspond to a square integrable representation of a simple p adic group

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

new bases of some hecke algebras via soergel bimodules abstract for extra large coxeter systems m s r 3 we construct a natural and explicit set of soergel bimodules d d w w w such that each d w contains as a direct summand or is equal to the indecomposable soergel bimodule b w when decategorified we prove that d gives rise to a set d w w w that is actually a basis of the hecke algebra this basis is close to the kazhdan lusztig basis and satisfies a positivity condition

axiom 3 modules by introducing the concept of a knice submodule a refinement of the notion of nice subgroup we are able to formulate a version of the third axiom of countability appropriate to the study of plocal mixed groups in the spirit of the well known characterization of totally projective pgroups our axiom 3 modules in fact form a class of zp modules encompassing the totally projectives in the torsion case for which we prove a uniqueness theorem and establish closure under direct summands indeed axiom 3 modules turn out to be precisely the previously classified warfield modules but with the added power of the third axiom of countability characterization we derive numerous new results including the resolution of a long standing problem of warfield and theorems in the vein of familiar criteria due to kulikov and pontryagin

the steinberg variety and representations of reductive groups abstract we give an overview of some of the main results in geometric representation theory that have been proved by means of the steinberg variety steinberg s insight was to use such a variety of triples in order to prove a conjectured formula by grothendieck the steinberg variety was later used to give an alternative approach to springer s representations and played a central role in the proof of the deligne langlands conjecture for hecke algebras by kazhdan and lusztig

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