on the ramanujan conjecture over number fields we extend to an arbitrary number eld the best known bounds towards ramanujan for the group gln n 2 3 4 in particular we present a technique which overcomes the analytic obstacles posed by the presence of an innite group of units

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

boundedness of automorphic l functions in vertical strips the purpose of this paper is to prove the boundedness in vertical strips of finite width for all the l functions that appear in constant terms of eisenstein series under a certain natural assumption on local normalized intertwining operators as a corollary we prove the boundedness for a number of important l functions among them the symmetic cube and triple product l functions attached to cusp forms on gl 2 as well as several rankin selberg product l functions where the local assumption was alredy proved to be valid in each case under the same assumption the finiteness of poles on all of c for all of these l functions is also proved a necessary step in the proof of boundedness in vertical strips

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 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

selberg s orthogonality conjecture for automorphic l functions let and be automorphic irreducible unitary cuspidal representations of inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i and inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i respectively assume that either or is self contragredient under the ramanujan conjecture on and we deduce a prime number theorem for l s inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 03i which can be used to asymptotically describe whether or det i 0 for some nonzero 0 inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 04i or det it for any t inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 05i as a consequence we prove the selberg orthogonality conjecture in a more precise form for automorphic l functions l s and l s under the ramanujan conjecture when m m 2 and and are representations corresponding to holomorphic cusp forms our results are unconditional

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

minimization of the ground state for two phase conductors in low contrast regime in this article we consider the problem of the optimal distribution of two conducting materials with given volume inside a fixed domain in order to minimize the first eigenvalue the ground state of a dirichlet operator it is known when the domain is a ball that the solution is radial and it was conjectured that the optimal distribution of the materials consists of putting the material with the highest conductivity in a ball around the center we show that this conjecture is not true in general for this we consider the particular case where the two conductivities are close to each other low contrast regime and we perform an asymptotic expansion with respect to the difference of conductivities we find that the optimal solution is the union of a ball and an outer ring when the amount of the material with the higher density is large enough

symmetry breaking and other phenomena in the optimization of eigenvalues for composite membranes we consider the following eigenvalue optimization problem given a bounded domain and numbers 0 a 0 find a subset d of area a for which the first dirichlet eigenvalue of the operator d is as small as possible

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

une preuve simple des conjectures de langlands pour gl n sur un corps p adique let f be a finite extension of p for each integer n 1 we construct a bijection from the set f 0 n of isomorphism classes of irreducible degree n representations of the absolute weil group of f onto the set f 0 n of isomorphism classes of smooth irreducible supercuspidal representations of gl n f those bijections preserve epsilon factors for pairs and hence we obtain a proof of the langlands conjectures for gl n over f which is more direct than harris and taylor s our approach is global and analogous to the derivation of local class field theory from global class field theory we start with a result of kottwitz and clozel on the good reduction of some shimura varieties and we use a trick of harris who constructs non galois automorphic induction in certain cases

strongly positive representations of gspin_ 2n 1 and the jacquet module method we explicitly construct the structure of jacquet modules of parabolically induced representations of gspin 2n 1 over a p adic field f of any characteristic using this construction of the jacquet module we obtain a classification of strongly positive representations of gspin 2n 1 over f and describe the general discrete series representations of gspin 2n 1 over f assuming the half integer conjecture one application of this paper is the proof of the equality of l functions from the langlands shahidi method and artin l functions through the local langlands correspondence kim in langlands shahidi l functions for gspin groups and the generic arthur packet conjecture preprint

on the generic local langlands correspondence for gspin groups in the case of split gspin groups we prove an equality of l functions between automorphic local l functions defined by the langlands shahidi method and local artin l functions our method of proof is based on previous results of the first author which allow to reduce the problem to supercuspidal representations of levi subgroups of gspin by constructing langlands parameters for general generic irreducible admissible representations of gspin from the one for generic irreducible supercuspidal representations of its levi subgroups

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

maximisation and minimisation on classes of rearrangements maximisation and minimisation of the dirichlet integral of a function vanishing on the boundary of a bounded domain are studied subject to the constraint that the laplacean be a rearrangement of a given function when the laplacean is two signed non existence of minimisers is proved and some information on the limits of minimising sequences obtained this contrasts with the known existence of minimisers in the one signed case when the domain is a ball and the laplacean is one signed maximisers and minimisers are shown to be radial and monotone existence of maximisers is proved subject additionally to a finite number of linear constraints with particular reference to ideal fluid flows of prescribed angular momentum in a disc

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

distinguished representations shintani base change and a finite field analogue of a conjecture of prasad abstract let e f be a quadratic extension of fields and g a connected quasi split reductive group over f let g o p be the opposition group obtained by twisting g by the duality involution considered by prasad assume that the field f is finite let be an irreducible generic representation of g e when is a shintani base change lift of some representation of g o p f we give an explicit nonzero g f invariant vector in terms of the whittaker vector of this shows particularly that is g f distinguished when the field f is p adic the paper also proves that the duality involution takes an irreducible admissible generic representation of g f to its contragredient as a special case of this result all generic representations of g 2 f 4 or e 8 are self dual

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

a proof of langland s conjecture on plancherel measures complementary series of p adic groups analysis of p adic reductive groups our first result theorem 7 9 proves a conjecture of langlands on normalization of intertwining operators by means of local langlands root numbers and l functions at least when the group is quasi split and the inducing representation is generic assuming two natural conjectures in harmonic analysis of p adic groups we also prove the validity of the conjecture in general theorem 9 5 as our second result we obtain all the complementary series and special representations of quasi split p adic groups coming from rank one parabolic subgroups and generic supercuspidal represen

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

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

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

applications of langlands functorial lift of odd orthogonal groups together with cogdell piatetski shapiro and shahidi we proved earlier the existence of a weak functorial lift of a generic cuspidal representation of so 2n 1 to gl 2n recently ginzburg rallis and soudry obtained a more precise form of the lift using their integral representation technique namely the lift is an isobaric sum of cuspidal representations of gln i more precisely cuspidal representations of gl2n i such that the exterior square l functions have a pole at s 1 one purpose of this paper is to give a simpler proof of this fact in the case that a cuspidal representation has one supercuspidal component in a separate paper we prove it without any condition using a result on spherical unitary dual due to barbasch and moy we give several applications of the functorial lift first we parametrize square integrable representations with generic supercuspidal support which have been classified by moeglin and tadic second we give a criterion for cuspidal reducibility of supercuspidal representations of gl m x so 2n 1 third we obtain a functorial lift from generic cuspidal representations of so 5 to automorphic representations of gl 5 corresponding to the l group homomorphism sp 4 c gl 5 c given by the second fundamental weight

intertwining operators and residues i weighted characters operateurs d entrelacement normalisation groupes reels groupes p adiques representations standard les distributions j m la distribution j m x residus changements de contour les espaces 7b h ac g f s et 7b f ac g f s l application m

minimization of eigenvalues for a quasilinear elliptic neumann problem with indefinite weight we investigate the minimization of the positive principal eigenvalue of the problem pu m u p 2u in u 0 on over a class of sign changing weights m with m 0 it is proved that minimizers exist and satisfy a bang bang type property in dimension one we obtain a complete description of the minimizers this problem is motivated by applications from population dynamics

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

level set methods for optimization problems involving geometry and constraints i frequencies of a two density inhomogeneous drum many problems in engineering design involve optimizing the geometry to maximize a certain design objective geometrical constraints are often imposed in this paper we use the level set method devised in osher and sethian j comput phys 79 12 1988 the variational level set calculus presented in zhao et al j comput phys 127 179 1996 and the projected gradient method as in rudin et al physica d 60 259 1992 to construct a simple numerical approach for problems of this type we apply this technique to a model problem involving a vibrating system whose resonant frequency or whose spectral gap is to be optimized subject to constraints on geometry our numerical results are quite promising we expect to use this approach to deal with a wide class of optimal design problems in the future

local coefficients as artin factors for real groups introduction the purpose of this paper is to prove the equality of certain local coefficients of arithmetic significance which were attached to representations of quasi split real reductive algebraic groups in 27 with their corresponding artin factors attached by local class field theory 21 as a consequence we establish an identity satisfied by certain normalized intertwining operators it seems to be useful in applications of the trace formula 1 29 more precisely let g be the group of real points of a quasi split reductive algebraic group over r let a be the set of simple roots defined by a fixed minimal parabolic subgroup po m0a0u of g fix 6 c a and let p p9 be the corresponding standard parabolic subgroup of g and write p man for its langlands decomposition fix a nondegenerate character x f u i a h o be an irreducible admissible x gic banach in particular x generic unitary representation of m cf section 1 given v g a the complex dual of the lie algebra of a let i v o 0 be the continuously quasi unitarily if o is unitary induced representation indp ga e v 9 and let v v o 0 be its space section 0 then f a vfoo o now let w be the weyl group of ao in g choose w g w such that w 0 c a let n be the unipotent group opposite to n define n u c wn w where w is a representative of w in g for g f a 00 define

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

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

local langlands correspondence for mathrm gl _ n and the exterior and symmetric square varepsilon factors let f be a p adic field that is a finite extension of qp for some prime p the local langlands correspondence llc attaches to each continuous n dimensional semisimple representation of w f the weil deligne group for f f an irreducible admissible representation of gln f such that among other things the local l and e factors of pairs are preserved this correspondence should be robust and preserve various parallel operations on the arithmetic and analytic sides such as taking the exterior square or symmetric square in this article we show that this is the case for the local arithmetic and analytic symmetric square and exterior square e factors that is that e s 2 e s 2 and e s sym2 e s sym2 the agreement of the l functions also follows by our methods but this was already known by henniart the proof is a robust deformation argument combined with local global techniques which reduces the problem to the stability of the analytic factor s 2 under highly ramified twists when is supercuspidal this last step is achieved by relating the factor to a mellin transform of a partial bessel function attached to the representation and then analyzing the asymptotics of the partial bessel function inspired in part by the theory of shalika germs for bessel integrals the stability for every irreducible admissible representation then follows from those of the corresponding arithmetic factors as a corollary

generic transfer for general spin groups we prove langlands functoriality for the generic spectrum of general spin groups both odd and even contrary to other recent instances of functoriality our resulting automorphic representations on the general linear group are not self dual together with cases of classical groups this completes the list of cases of split reductive groups whose l groups have classical derived groups the important transfer from gsp4 to gl4 follows from our result as a special case

on optimization problems with prescribed rearrangements on considere certains resultats concernant des fonctions avec des distributions imposees et des prroblemes d optimisation qui y sont relies

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

extremal energies of laplacian operator different configurations for steady vortices abstract in this paper we study a maximization and a minimization problem associated with a poisson boundary value problem optimal solutions in a set of rearrangements of a given function define stationary and stable flows of an ideal fluid in two dimensions the main contribution of this paper is to determine the optimal solutions at first we determine a nearly optimal solution which is an approximation of the optimal solution when the problems are in low contrast regime secondly for the high contrast regime two optimization algorithms are developed for the minimization problem we prove that our algorithm converges to the global minimizer regardless of the initializer the maximization algorithm is capable of deriving all local maximizers including the global one numerical experiments lead us to a conjecture about the location of the maximizers in the set of rearrangements of a function

twisted endoscopy and reducibility of induced representations for p adic groups 1 introduction this is the first in a series of papers in which we study the reducibility of representations induced from discrete series representations of the levi factors ofmaximal parabolic subgroups of p adic groups on the unitary axis the problem is equivalent to determining certain local langlands l functions which are of arithmetic significance by themselves one of the main aims of this paper is to interpret our results in the direction of the parametrization problem by means of the theory of twisted endoscopy for which our method seems to be very suitable when the representation is supercuspidal we also study the reducibility of the induced representations off the unitary axis let f be a p adic field of characteristic zero fix a positive integer n 1 and let g be either of the groups sp2n s02n or s02n l in all three cases there is a conjugacy class of maximal parabolic subgroups which has gl as its levi factor let p mn be the standard parabolic subgroup ofg in this conjugacy class then m gl let tr be a discrete series representation of m gln f and given s c let i s tr be the representation of g g f induced from tr r idet ls let i tr i 0 tr

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

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