supercuspidal characters of sl l over a p adic field l a prime let f be a p adic field of characteristic 0 for odd primeswith certain tameness restric tions techniques due to assem allow us to write supercuspidal characters of sl f in terms of a family of distributions defined by kazhdan following an argument of harish chandra we derive an integral formula for these distributions results of debacker and kazhdan allow us to evaluate this integral formula this in turn gives supercuspidal character formulas for sl f explicit up to the constants appearing in the gl f local character expansion our result also describes the explicit local langlands correspondence for those positive depth supercuspidal representations of gl f which do not restrict irreducibly to sl f 1 introduction in this note using techniques due to assem see 3 we derive formulas for the supercuspidal characters of sl f whereis an odd prime here f is a p adic field of characteristic 0 i e a finite extension of qp we assume p 2 although most of the time we will only need p our results illustrate some facts about supercuspidal characters which are not immediately obvious from previous work in general there is an infinite class of supercuspidal representations of gl f which decompose intoirreducible components upon restriction to sl f the character of each component then blows up near the singular set despite the fact that supercuspidal characters of gl f are bounded on regular elliptic elements near the singular set on the other hand under a certain simple condition on the order of the residue class field of f there is another infinite class of supercuspidal representations of gl f which remain irreducible upon restriction to sl f this means that the charac ters of the restrictions have the same behaviour near the singular set on sl f as on gl f in particular the characters of these supercuspidal representations of sl f are bounded on regular elliptic elements near the singular set this has also been noted in 27 the ultimate effect of this phenomenon on harmonic analysis on sl f is not yet fully understood and will be explored in future work we briefly outline the method we will use here the construction of the supercuspidal representations of gl f is completely understood it is known that all such representations are induced from open compact modulo the centre

finite element approximations of a ladyzhenskaya model for stationary incompressible viscous flow some finite element approximation procedures are presented for a model proposed by ladyzhenskaya for stationary incompressible viscous flow the approximate problems are proved to be well posed and stable under standard assumptions on the finite element families the solutions of the approximate problems converge to the solution of the original problem under minimum regularity assumptions some error estimates are derived the optimal order of accuracy is assured with or even without using exact integration rules in the approximation procedure iterative methods for solving the discrete nonlinear problems and comments on some computational experiments are provided special attention is also paid to the common properties as well as differences between the approximation procedure presented here and the approximation for the stationary navier stokes equations

depth zero supercuspidal l packets for inner forms of gsp4 abstract we show that for any tame regular discrete series parameter of g s p 4 or its inner form g u 2 d the l packet attached by the local langlands conjecture gan and takeda 2011 14 gan and tantono submitted for publication 15 agrees with the l packet of depth zero supercuspidal representations constructed by debacker and reeder 2009 12

hecke algebras of classical groups over p adic fields and supercuspidal representations let k be a p adic field with involution 0 and let k 0 be its fixed subfield of k let v be a finite dimensional vector space defined over k equipped with e hermitian form v k 0 is large

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

endoscopic transfer of orbital integrals in large residual characteristic this article constructs shalika germs in the context of motivic integration both for ordinary orbital integrals and kappa orbital integrals based on transfer principles in motivic integration and on waldspurger s endoscopic transfer of smooth functions in characteristic zero we deduce the endoscopic transfer of smooth functions in sufficiently large residual characteristic

a nonconforming finite element method for the stationary smagorinsky model abstract in this paper we focus on a low order nonconforming finite element method fem for the stationary smagorinsky model the velocity and pressure are approximated by the constrained nonconforming rotated q1 element cn q 1 r o t and piecewise constant element respectively optimal error estimates of the velocity in the broken h1 norm and l2 norm and the pressure in the l2 norm are derived by some nonlinear analysis techniques and aubin nitsche duality argument the supercloseness and superconvergent results are also obtained under some reasonable regularity assumptions finally a numerical example is implemented to confirm our theoretical analysis

characters of odd degree we prove the mckay conjecture on characters of odd degree a major step in the proof is the verication of the inductive mckay condition for groups of lie type and primes such that a sylow subgroup or its maximal normal abelian subgroup is contained in a maximally split torus by means of a new equivariant version of harish chandra induction specics of characters of odd degree namely that most of them lie in the principal harish chandra series then allow us to deduce from it the mckay conjecture for the prime 2 hence for characters of odd degree

nonzero degree maps between closed orientable three manifolds this paper adresses the following problem given a closed orientable three manifold m are there at most finitely many closed orientable three manifolds 1 dominated by m we solve this question for the class of closed orientable graph manifolds more precisely the main result of this paper asserts that any closed orientable graph manifold 1 dominates at most finitely many orientable closed three manifolds satisfying the poincare thurston geometrization conjecture to prove this result we state a more general theorem for haken manifolds which says that any closed orientable three manifold m1 dominates at most finitely many haken manifolds whose gromov simplicial volume is sufficiently close to that of m

depth zero supercuspidal l packets and their stability in this paper we verify the local langlands correspondence for pure inner forms of unramied p adic groups and tame langlands parameters in general position for each such parameter we explicitly construct in a natural way a nite set l packet of depth zero supercuspidal representations of the appropriate p adic group and we verify some expected properties of this l packet in particular we prove with some conditions on the base eld that the appropriate sum of characters of the representations in our l packet is stable no proper subset of our l packets can form a stable combination our l packets are also consistent with the conjectures of b gross and d prasad on restriction from so2n 1 to so2n 24 these l packets are in general quite large for example sp2n has an l packet containing 2 n representations of which exactly two are generic in fact on a quasi split form eachl packet contains exactly one generic representation for every rational orbit of hyperspecial vertices in the reduced bruhattits building when the group has connected center every depth zero generic supercuspidal representation appears in one of these l packets we emphasize that there is nothing new about the representations we construct they are induced from deligne lusztig representations on subgroups of nite

a new local stabilized nonconforming finite element method for solving stationary navier stokes equations in this paper we study a new local stabilized nonconforming finite element method based on two local gauss integrals for solving the stationary navier stokes equations this nonconforming method utilizes the lowest equal order pair of mixed finite elements i e ncp 1 p 1 error estimates of optimal order are obtained and numerical results agreeing with these estimates are demonstrated numerical comparisons with other mixed finite element methods for solving the navier stokes equations are also presented to show the better performance of the present method

representations lisses modulo l de gl m d let f be a non archimedean locally compact field of residue characteristic p let d be a finite dimensional central division f algebra and let r be an algebraically closed field of characteristic different from p we classify all smooth irreducible representations of gl m d with coefficients in r in terms of multisegments generalizing works by zelevinski tadic and vigneras we prove that any irreducible r representation of gl m d has a unique supercuspidal support and thus get two classifications one by supercuspidal multisegments classifying representations with a given supercuspidal support and one by aperiodic multisegments classifying representations with a given cuspidal support these constructions are made in a purely local way with a substantial use of type theory

on the modular composition factors of the steinberg representation abstract let g be a finite group of lie type and st k be the steinberg representation of g defined over a field k we are interested in the case where k has prime characteristic l and st k is reducible tinberg has shown that the socle of st k is always simple we give a new proof of this result in terms of the hecke algebra of g with respect to a borel subgroup and show how to identify the simple socle of st k among the principal series representations of g furthermore we determine the composition length of st k when g gl n q or g is a finite classical group and l is a so called linear prime

acyclicity of schneider and stuhler s coefficient systems another approach in the level 0 case let f be a non archimedean local field and g be the locally profinite group gl n f n 1 we denote by x the bruhat tits building of g for any smooth complex representation v of g and for any level n 1 schneider and stuhler have constructed a coefficient system c c v n on the simplicial complex x they proved that if v is generated by its fixed vectors under the principal congruence subgroup of level n then the augmented complex c or x c v of oriented chains of x with coefficients in c is a resolution of v in the category of smooth complex representations of g in this paper we give another proof of this result in the level 0 case and assuming moreover that v is generated by its fixed vectors under an iwahori subgroup i of g here level 0 refers to bushnell and kutzko s terminology that is to the case n 1 0 our approach is different we strongly use the fact that the trivial character of i is a type in the sense of bushnell and kutzko

distinguished regular supercuspidal representations based on recent work of kaletha we apply hakim murnaghan s result to study distinguished regular supercuspidal representations of tamely ramified reductive p adic groups assuming p is sufficiently large we obtain a necessary and sufficient condition for regular supercuspidal representations to be distinguished we also investigate the relation between the distinction problem and the langlands functoriality and confirm a conjecture of lapid for regular depth zero or epipelagic supercuspidal representations

finite order automorphisms and a parametrization of nilpotent orbits in p adic lie algebras let k be a field with a nontrivial discrete valuation which is complete and has perfect residue field let g be the group of k rational points of a reductive linear algebraic group bf g equipped with a finite cyclic group l of order m acting on bf g by algebraic automorphisms defined over k we assume that the lie algebra of g decomposes into a direct sum of eigenspaces under the action of l with generator theta we let frak g i denote the eigenspace corresponding to eigenvalue xi i where xi is a primitive m th root of unity that is x in frak g i if and only if theta x xi ix if bf h is a k subgroup of bf g l the group of l fixed points which contains the neutral component of bf g l then h bf h k acts on each eigenspace of frak g let r in bbb r under mild restrictions on the residual characteristic of k the set of nilpotent h orbits in the xi eigenspace frak g 1 is parametrized by equivalence classes of noticed moy prasad cosets of depth r which lie in frak g 1

formulas for tamely ramified supercuspidal characters of let f denote a p adic local field of residual characteristic p 3 this article gives formulas valid on the regular elliptic set for the irreducible supercuspidal characters of gl 3 f which correspond to characters of a ramified cartan subgroup in the case in which f does not contain cube roots of unity i e the case in which ramified cubic extensions of degree 3 over f cannot be galois base change results concerning simple types due to bushnell and henniart 1996 are used in the proofs

supercuspidal l packets of positive depth and twisted coxeter elements the local langlands correspondence is a conjectural connection between representations of groups g k for connected reductive groups g over a padic field k and certain homomorphisms langlands parameters from the galois or weil deligne group of k into a complex lie group g which is dual in a certain sense to g and which encodes the splitting structure of g over k more introductory remarks on the local langlands correspondence can be found in 21 wheng gl1 this correspondence should reduce to local abelian class field theory for g gln the langlands correspondence is uniquely determined by local factors 24 and was shown to exist in 23 and 25 so far this correspondence is not completely explicit but much progress has been made in this direction see 9 10 for example for groups other than gln or pgln the theory is much less advanced new phenomena appear arising on the arithmetic side from the difference between conjugacy and stable conjugacy and on the dual side from nontrivial monodromy of langlands parameters this means that a single langlands parameter should determine not just one but a finite set of representations these are the l packets of the title however since local factors have not been defined in general there is no precise characterization of an l packet for general groups one can at present only hope to define finite sets of representations attached to langlands parameters and show that they have properties expected or perhaps unexpected of l packets see 14 chap 3 for some of these properties one is thereby proposing a definition of local factors for the representations in the sets cf 4 chap 3 this paper is a sequel to 14 the aim of both papers is to verify in an explicit and natural way the local langlands correspondence for the simplest kinds of non abelian extensions of k and the simplest kinds of

on finite group actions on reductive groups and buildings let h be a connected reductive group over a non archimedean local field k and let f autk h be a finite group of order not divisible by p the residual characteristic of k let g h f be the identity component of the subgroup of h consisting of points fixed by f the main theorem of this paper asserts that the bruhat tits building b g of g can be identified with the set of f fixed points of b h several special cases of this theorem have been known previously when e k is a finite totally ramified galois extension h rese k g and f gal e k the condition p f is simply that e k is tamely ramified in this case our main theorem is a well known but unpublished theorem of g rousseau recently one of us g p found a simple proof of this theorem p when g is a classical group realized in the standard way as the identity component of the group of fixed points of an involution of a general linear group h bruhat and tits bt4 have given a description of b g as a subset of b h in particular it follows from their description that b g b h when p 2 recently this fact has been rediscovered by j kim and a moy km and independently a simple proof has been given by gy let h be the split form of spin 8 then one can choose an outer automorphism aut h of order 3 such that g h is of type g2 from a joint work gy by w t gan and one of us j y it is known that b g b h is true for arbitrary p

derived categories and deligne lusztig varieties ii this paper is a continuation and a completion of boro1 we extend the jordan decomposition of blocks we show that blocks of finite groups of lie type in non describing characteristic are morita equivalent to blocks of subgroups associated to isolated elements of the dual group the key new result is the invariance of the part of the cohomology in a given modular series of deligne lusztig varieties associated to a given levi subgroup under certain variations of parabolic subgroups we also show that the equivalence arises from a splendid rickard equivalence even in the setting of boro1 the finer homotopy equivalence was unknown as a consequence the equivalence preserves defect groups and categories of subpairs we finally determine when deligne lusztig induced representations of tori generate the derived category of representations

depth zero base change for ramified u 2 1 we give an explicit description of l packets and quadratic base change for depth zero representations of ramified unitary groups in two and three variables we show that this base change lifting is compatible with a certain lifting of families of representations of finite groups we conjecture that such a compatibility is valid in much greater generality

a new second order nonconforming mixed finite element scheme for the stationary stokes and navier stokes equations abstract in this paper a new stable nonconforming mixed finite element scheme with second order accuracy is proposed for the stationary stokes and navier stokes equations in which a new nonconforming rectangular element is taken for approximating space the velocity and the bilinear element for the pressure the optimal error estimates for the approximation of both the velocity and the pressure in l 2 norm are established as well as one in a broken h 1 norm for the velocity numerical experiments are given which are consistent with our theoretical analysis

construction of tame supercuspidal representations the notion of depth is defined by moy prasad mp2 the notion of a generic character will be defined in 9 when g gln or g is the multiplicative group of a central division algebra of dimension n with n p 1 our generic characters are just the generic characters in my where the definition is due to kutzko moreover in these cases our construction literally specializes to howe s construction as formulated in my and it is known that the construction yields all supercuspidal representations my hm notice that the initial datum in our construction consists of very simple objects linear characters and supercuspidal representations of depth zero the latter are well understood by the work of moy prasad and morris see 3 by the work of howe irreducible supercuspidal representations of gln f with n p 1 are parametrized by certain characters of e as e varies through

character expansions and unrefined minimal k types let k be a p adic field let g be the group of k rational points of a connected reductive group g defined over k and let inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i be its lie algebra under certain hypotheses on g and k we get a new character expansion called a asymptotic expansion for representations of positive depth inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i this character expansion is valid on the g domain inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 03i which is bigger than that where the harish chandra howe character expansion is known to be valid

finite group actions on 3 manifolds if g is a finite group acting smoothly on a closed surface f it is well known that g leaves invariant some riemannian metric of constant curvature on f thus any action of g on the 2 sphere s 2 is conjugate in diff s 2 to an orthogonal action if g acts on the torus sx s 1 there is a g invariant flat metric on s a s 1 and if g acts on a surface f with negative euler number then f admits a g invariant hyperbolic metric recently thurston th 1 th 2 th 3 has described the eight 3 dimensional geometries which provide geometric structures for closed 3 manifolds in the same way that the 2 sphere s 2 the euclidean plane e 2 and the hyperbolic plane h 2 provide geometric structures for surfaces see also the survey article by scott sc4 thurston also conjectured that if m is a closed 3 manifold with a geometric structure modelled on one of these eight geometries say x then any smooth action of a finite group g on m should leave invariant some metric on m inducing the geometry x we will say that g preserves the geometric structure on m in this case it should be noted that the restriction to smooth actions of g on m is essential for bing bi showed that there are involutions of s 3 whose fixed set is a wild 2 sphere however in dimension two it was proved by eilenberg ei that any action of a finite group on a surface is conjugate to a smooth action in this paper our main result asserts that thurston s conjecture holds for five of the eight geometries the result is the following

theta lifts and distinction for regular supercuspidal representations this article has a twofold purpose first by recent works of kaletha and loke ma we give an explicit description of the local theta correspondence between regular supercuspidal representations in the equal rank symplectic orthogonal case second based on this description we show that the local theta correspondence preserves distinction with respect to unramified galois involutions

supercuspidal characters of reductive p adic groups we compute the characters of many supercuspidal representations of reductive p adic groups specifically we deal with representations that arise via yu s construction from data satisfying a certain compactness condition each character is expressed in terms of a depth zero character of a smaller group the linear characters appearing in yu s construction fourier transforms of orbital integrals and certain signs and cardinalities that are described explicitly in terms of the datum associated to the representation and of the element at which the character is evaluated

error estimates of two level finite element method for smagorinsky model two level finite element method for simulating smagorinsky model in large eddy simulation is investigated in this two level algorithm a linearized discrete problem is solved by using the variational multiscale method on the coarse mesh corresponding to newton linearization method the linearized smagorinsky model is solved on the fine mesh the error estimates derived in this paper implies that by choosing appropriate mesh sizes and the radius of the spatial filter used in smagorinsky model the two level method provides the optimal convergence rates for the velocity in h1 norm and the pressure in l2 norm meanwhile the two different numerical experiments are given to support the optimal convergence rates and the high efficiency of two level algorithm

supercuspidal l packets via isocrystals in a recent paper debacker and reeder construct and parameterize l packets on pure inner forms of unramified p adic groups that consist of depth zero supercuspidal representations we generalize their work to non pure inner forms by providing an alternative construction based on the theory of isocrystals with additional structure due to kottwitz furthermore we show the stability and endoscopic transfer for these l packets

sato tate theorem for families and low lying zeros of automorphic l functions we consider certain families of automorphic representations over number fields arising from the principle of functoriality of langlands let g be a reductive group over a number field f which admits discrete series representations at infinity let l g widehat g rtimes mathrm gal bar f f be the associated l group and r l g rightarrow mathrm gl d mathbb c a continuous homomorphism which is irreducible and does not factor through mathrm gal bar f f the families under consideration consist of discrete automorphic representations of g mathbb a f of given weight and level and we let either the weight or the level grow to infinity we establish a quantitative plancherel and a quantitative sato tate equidistribution theorem for the satake parameters of these families this generalizes earlier results in the subject notably of sarnak prog math 70 321 331 1987 and serre j am math soc 10 1 75 102 1997 as an application we study the distribution of the low lying zeros of the associated family of l functions l s pi r assuming from the principle of functoriality that these l functions are automorphic we find that the distribution of the 1 level densities coincides with the distribution of the 1 level densities of eigenvalues of one of the unitary symplectic and orthogonal ensembles in accordance with the katz sarnak heuristics we provide a criterion based on the frobenius schur indicator to determine this symmetry type if r is not isomorphic to its dual r vee then the symmetry type is unitary otherwise there is a bilinear form on mathbb c d which realizes the isomorphism between r and r vee if the bilinear form is symmetric resp alternating then r is real resp quaternionic and the symmetry type is symplectic resp orthogonal

asymptotic behavior of supercuspidal representations and sato tate equidistribution for families abstract we establish properties of families of automorphic representations as we vary prescribed supercuspidal representations at a given finite set of primes for the tame supercuspidals constructed by j k yu we prove the limit multiplicity property with error terms thereby we obtain a sato tate equidistribution for the hecke eigenvalues of these families the main new ingredient is to show that the orbital integrals of matrix coefficients of tame supercuspidal representations with increasing formal degree on a connected reductive p adic group tend to zero uniformly for every noncentral semisimple element

local theta correspondences between epipelagic supercuspidal representations in this paper we study the local theta correspondences between epipelagic supercupsidal representations of a type i classical dual pair g g over p adic fields we show that besides an exceptional case an epipelagic supercupsidal representation pi of widetilde g lifts to an epipelagic supercupsidal representation pi of widetilde g if and only if the epipelagic data of pi and pi are related by the moment maps

the supercuspidal representations of p adic classical groups let g be a unitary symplectic or special orthogonal group over a locally compact non archimedean local field of odd residual characteristic we construct many new supercuspidal representations of g and bushnell kutzko types for these representations moreover we prove that every irreducible supercuspidal representation of g arises from our constructions

supercuspidal representations an exhaustion theorem let be a reductive adic group we prove that all supercuspidal representations of arise through yu s construction subject to certain hypotheses on depending on as a corollary under the same hypotheses we see that any supercuspidal representation is compactly induced from a representation of an open subgroup which is compact modulo the center

epipelagic l packets and rectifying characters we provide an explicit construction of the local langlands correspondence for general tamely ramified reductive p adic groups and a class of wildly ramified langlands parameters furthermore we verify that our construction satisfies many expected properties of such a correspondence more precisely we show that each l packet we construct admits a parameterization in terms of the langlands dual group contains a unique generic element for a fixed whittaker datum satisfies the formal degree conjecture is compatible with central and cocentral characters provides a stable virtual character and satisfies the expected endoscopic character identities moreover we show that in the case of mathrm gl n our construction coincides with the established local langlands correspondence our techniques provide a general approach to the construction of the local langlands correspondence for tamely ramified groups and regular supercuspidal parameters

epipelagic representations and invariant theory letg be a reductive p adic group we give a new construction of small depth epipelagic supercuspidal representations ofg k using stable orbits in geometric invariant theory git in contrast to previously known methods this construction works without any restrictions on the residue characteristic p we construct appropriate langlands parameters for these representations with some restrictions on p the git arises from graded moy prasad filtrations which we show are isomorphic to the graded lie algebras whose git was analyzed by vinberg and levy this leads to a classification of stable orbits in moy prasad filtrations as well as epipelagic supercuspidal representations in terms of z regular elliptic automorphisms of the absolute root system of g transferred to the l group of g generates the image of tame inertia under the corresponding wild langlands parameter for unramified groups and sufficiently large p we also classify the moy prasad filtrations which have semi stable orbits which solves the long outstanding problem of classifying non degenerate k types for g k

semisimple characters for p adic classical groups let g be a unitary symplectic or orthogonal group over a non archimedean local field of residual characteristic different from 2 considered as the fixed point subgroup in a general linear group of an involution following previous work of bushnell and kutzko and of the author we generalize the notion of a semisimple character for and for g in particular following the formalism of bushnell and henniart we show that these semisimple characters have certain functorial properties finally we show that any positive level supercuspidal representation of g contains a semisimple character

a cheapest nonconforming rectangular finite element for the stationary stokes problem abstract we introduce a least degrees of freedom nonconforming finite element pair to approximate the stokes equations based on quadrilateral meshes the finite element space for the velocity field is composed of the p 1 nonconforming quadrilateral element plus only one additional global dssy douglas santos sheen ye or rt rannacher turek type of bubble function the pressure field is approximated by the piecewise constant function we show that the discrete problem is nonsingular and the element pair satisfies a weak discrete inf sup condition several numerical examples are shown to confirm the efficiency and reliability of the proposed method

a construction of the supercuspidal representations of _ adic let f be a nondiscrete locally compact non archimedean field in this paper we construct all irreducible supercuspidal representations of g gl n f for each such representation which we may as well assume is unitary we give a subgroup j of g that is compact mod the center z of g and a finite dimensional representation of j such that inducing to g gives the proof that all supercuspidals have been constructed appeals to a theorem the matching theorem that has been proved by global methods

parametrizing nilpotent orbits via bruhat tits theory let k denote a field with nontrivial discrete valuation we assume that k is complete with perfect residue field let g be the group of k rational points of a reductive linear algebraic group defined over k let g denote the lie algebra of g fix r r subject to some restrictions we show that the set of distinguished degenerate moy prasad cosets of depth r up to an equivalence relation parametrizes the nilpotent orbits in g

branching rules for supercuspidal representations of sl2 k for k a p adic field abstract the restriction of a supercuspidal representation of sl 2 k to a maximal compact subgroup decomposes as a multiplicity free direct sum of irreducible representations we explicitly describe this decomposition and determine how the spectrum of this decomposition varies as a function of the parameters describing the supercuspidal representation

the unitary dual for the multiplicative group of arbitrary division algebras over local fields let f be a locally compact nondiscrete non archimedean field i e a finite extension of qp or a field of formal laurent series over a finite field and let d 2 be a division algebra of degree n over f so that d f n in this paper we give a construction of all irreducible unitary representations of dx there does not seem to be an easy way to parametrize these representations theorem 5 5 is a start in that direction the problem of determining dx is interesting not only in itself but also for the light it sheds on the problem of determining all irreducible supercuspidal representations of gl f the problems are connected in two ways the construction of supercuspidals for gl f has used procedures quite similar to those used to construct dx in previous papers dx has been determined when p2 n or p2 t n see 11 10 3 4 5 the supercuspidal representations of gl f are known when p t n and when n is the product of at most two distinct primes see 12 17 2 6 and there is usually a close similarity in methods of construction to the corresponding case for division algebras furthermore one important method of proving that the construction of supercuspidals is complete uses the matching theorem of deligne and kazhdan see 8 which sets up a natural correspondence between dx and the set of discrete series representations of gl f see e g 2 or 17 for this kind of proof it seems likely that the construction in this paper will also apply to give a construction of all supercuspidal representations of gl f this matter will be dealt with in a future paper in this connection it is worth noting the results in 15 about supercuspidals in gl f n the product of two primes describing the general method of the construction requires some notation let k be the residue class field of f and suppose that k has q elements recall from e g chapter i of 18 that d contains an unramified extension

p1 nonconforming quadrilateral finite element methods for second order elliptic problems a p1 nonconforming quadrilateral finite element is introduced for second order elliptic problems in two dimensions unlike the usual quadrilateral nonconforming finite elements which contain quadratic polynomials or polynomials of degree greater than 2 our element consists of only piecewise linear polynomials that are continuous at the midpoints of edges one of the benefits of using our element is convenience in using rectangular or quadrilateral meshes with the least degrees of freedom among the nonconforming quadrilateral elements an optimal rate of convergence is obtained also a nonparametric reference scheme is introduced in order to systematically compute stiffness and mass matrices on each quadrilateral an extension of the p1 nonconforming element to three dimensions is also given finally several numerical results are reported to confirm the effective nature of the proposed new element

the local character expansion near a tame semisimple element consider the character of an irreducible admissible representation of a p adic reductive group the harish chandra howe local expansion expresses this character near a semisimple element as a linear combination of fourier transforms of nilpotent orbital integrals under mild hypotheses we describe an explicit region on which the local character expansion is valid we assume neither that the group is connected nor that the underlying field has characteristic zero

on an exact mass formula of shimura in a series of recent papers g shimura obtained an exact formula for the mass of a maximal lattice in a quadratic or hermitian space over a totally real number field using bruhat tits theory we obtain a quick and more conceptual proof of his formula when the form is totally definite

three dimensional manifolds kleinian groups and hyperbolic geometry 1 a conjectural picture of 3 manifolds a major thrust of mathematics in the late 19th century in which poincare had a large role was the uniformization theory for riemann surfaces that every conformai structure on a closed oriented surface is represented by a riemannian metric of constant curvature for the typical case of negative euler characteristic genus greater than 1 such a metric gives a hyperbolic structure any small neighborhood in the surface is isometric to a neighborhood in the hyperbolic plane and the surface itself is the quotient of the hyperbolic plane by a discrete group of motions the exceptional cases the sphere and the torus have spherical and euclidean structures three manifolds are greatly more complicated than surfaces and i think it is fair to say that until recently there was little reason to expect any analogous theory for manifolds of dimension 3 or more except perhaps for the fact that so many 3 manifolds are beautiful the situation has changed so that i feel fairly confident in proposing the

bounds on the dimension of ext for finite groups of lie type abstract let g be a finite group of lie type defined in characteristic p and let k be an algebraically closed field of characteristic r 0 we will assume that r p so we are in the non defining characteristic case let v be a finite dimensional irreducible left kg module in 2011 guralnick and tiep found bounds on the dimension of h 1 g v in non defining characteristic which are independent of v the aim of this paper is to generalize the work of gurlanick and tiep we assume that g is split and use methods of modular harish chandra theory to find bounds on the dimension of ext k g 1 y v where y and v are irreducible kg modules we then use dipper and du s algorithms to illustrate our bounds in a series of examples

arithmetic invariants of discrete langlands parameters the local langlands correspondence can be used as a tool for making verifiable predictions about irreducible complex representations of p adic groups and their langlands parameters which are homomorphisms from the local weil deligne group to the l group in this article we refine a conjecture of hiraga ichino and ikeda which relates the formal degree of a discrete series representation to the value of the local gamma factor of its parameter we attach a rational function in x with rational coefficients to each discrete parameter which specializes at x q the cardinality of the residue field to the quotient of this local gamma factor by the gamma factor of the steinberg parameter the order of this rational function at x 0 is also an important invariant of the parameter it leads to a conjectural inequality for the swan conductor of a discrete parameter acting on the adjoint representation of the l group we verify this conjecture in many cases when we impose equality we obtain a prediction for the existence of simple wild parameters and simple supercuspidal representations both of which are found and described in this article

sp2n fq2 invariants in irreducible unipotent representations of sp4n fq abstract we show that for any irreducible representation of sp 4 n f q the subspace of all its sp 2 n f q 2 invariants is at most one dimensional in terms of lusztig symbols we give a complete list of irreducible unipotent representations of sp 4 n f q which have a non zero sp 2 n f q 2 invariant and in particular we prove that every irreducible unipotent cuspidal representation has a one dimensional subspace of sp 2 n f q 2 invariants as an application we give an elementary proof of the fact that the unipotent cuspidal representation is defined over q which was proved by lusztig in 12

nonconforming mixed finite element approximation to the stationary navier stokes equations on anisotropic meshes abstract the main aim of this paper is to study the error estimates of a rectangular nonconforming finite element for the stationary navier stokes equations under anisotropic meshes that is the nonconforming rectangular element is taken as approximation space for the velocity and the piecewise constant element for the pressure the convergence analysis is presented and the optimal error estimates both in a broken h 1 norm for the velocity and in an l 2 norm for the pressure are derived on anisotropic meshes

unrefined minimal k types for p adic groups via their restriction to compact open subgroups was begun by mautner shalika and tanaka for groups of type ai in contrast to real reductive groups where the representation theory of a maximal compact subgroup k is given in terms of slight modifications to cartan s theory of the highest weight the representation theory of a compact parahoric subgroup is quite complicated there is still no comprehensive theory for classifying the irreducible representa tions in the case of