existence uniqueness and l q estimates for the stokes problem in an exterior domain as is well known the study of the properties of solutions to the stokes system plays an essential role in the mathematical theory of viscous fluid flows governed by the navier stokes equations it is not art overstatement to say that existence uniqueness and a priori estimates in sobolev spaces for the stokes system are fundamental in every basic question related to the navier stokes equations such as existence regularity asymptotic behavior in time etc see e g ladyzhenskaya 1969 solonni ov 1977 and te am 1977 so far however the stokes system has been fully analyzed only for bounded domains see sobolevskn 1960 ladvznenskaya 1959 solonnu ov 1960 vorovlch yot oowcr 1961 cattabrma 1961 in particular in the paper of cattabriga a complete treatment is given in the sobotev spaces w l 1 1 q c for space dimension n 3 cattabriga s results essentially state that if 2 is a bounded domain in r a of class c m m max l 2 2 l 1 then given f e wt q y2 3 gc wt l q o q e wl 2 1 q q 8o a 1 q there exists one and only one distributional solution v c w1 z q q 3 p e wt l q o to the stokes system

stable trace formula cuspidal tempered terms consider a connected reductive group g over a number field f for technical reasons we assume that the derived group of g is simply connected see l1 in l3 langlands partially stabilizes the trace formula for g after making certain assumptions he writes the elliptic regular part of the trace formula for g as a linear combination of the elliptic g regular parts of the stable trace formulas for the elliptic endoscopic groups h of g the function f used in the stable trace formula forh is obtained from the function f used in the trace formula for g by transferring orbital integrals

the local langlands conjecture for gsp 4 we prove the local langlands conjecture for gsp4 f where f is a nonarchimedean local eld of characteristic zero

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

traces for homogeneous sobolev spaces in infinite strip like domains abstract in this paper we construct a trace operator for homogeneous sobolev spaces defined on infinite strip like domains we identify an intrinsic seminorm on the resulting trace space that makes the trace operator bounded and allows us to construct a bounded right inverse the intrinsic seminorm involves two features not encountered in the trace theory of bounded lipschitz domains or half spaces first due to the strip like structure of the domain the boundary splits into two infinite disconnected components the traces onto each component are not completely independent and the intrinsic seminorm contains a term that measures the difference between the two traces second as in the usual trace theory there is a term in the seminorm measuring the fractional sobolev regularity of the trace functions with a difference quotient integral however the finite width of the strip like domain gives rise to a screening effect that bounds the range of the difference quotient perturbation the screened homogeneous fractional sobolev spaces defined by this screened seminorm on arbitrary open sets are of independent interest and we study their basic properties we conclude the paper with applications of the trace theory to partial differential equations

the dirichlet problem for nonlocal operators in this note we set up the elliptic and the parabolic dirichlet problem for linear nonlocal operators as opposed to the classical case of second order differential operators here the boundary data are prescribed on the complement of a given bounded set we formulate the problem in the classical framework of hilbert spaces and prove unique solvability using standard techniques like the fredholm alternative

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

a paradifferential approach for well posedness of the muskat problem we study the muskat problem for one fluid or two fluids with or without viscosity jump with or without rigid boundaries and in arbitrary space dimension d of the interface the muskat problem is scaling invariant in the sobolev space h s c mathbb r d where s c 1 frac d 2 employing a paradifferential approach we prove local well posedness for large data in any subcritical sobolev spaces h s mathbb r d s s c moreover the rigid boundaries are only required to be lipschitz and can have arbitrarily large variation the rayleigh taylor stability condition is assumed for the case of two fluids with viscosity jump but is proved to be automatically satisfied for the case of one fluid the starting point of this work is a reformulation solely in terms of the drichlet neumann operator the key elements of proofs are new paralinearization and contraction results for the drichlet neumann operator in rough domains

irreducible cuspidal representations with prescribed local behavior let g be a simple algebraic group defined over the global field k in this paper we use the simple trace formula to determine the sum of the multiplicities of the irreducible representations in the cuspidal spectrum of g with specified local behavior at a finite set of places of k and unramified elsewhere this sum is expressed as the product of the values of modified artin l functions at negative integers

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

formal degrees of unipotent discrete series representations and the exotic fourier transform we introduce a notion of elliptic fake degrees for unipotent elliptic representations of a semisimple p adic group we conjecture and verify in some cases that the relation between the formal degrees of unipotent discrete series representations of a semisimple p adic group and the elliptic fake degrees is given by the exotic fourier transform matrix introduced by lusztig in the study of representations of finite groups of lie type

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

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

test vectors for finite periods and base change abstract let e f be a quadratic extension of finite fields by a result of gow an irreducible representation of g gl n e has at most one non zero h invariant vector up to multiplication by scalars when h is gl n f or u n e f if does have an h invariant vector it is said to be h distinguished it is known from the work of gow that h distinction is characterized by base change from u n e f due to kawanaka when h is gl n f resp from gl n f due to shintani when h is u n e f assuming is generic and h distinguished we give an explicit description of the h invariant vector in terms of the bessel function of let be a non degenerate character of n g n h and let b be the normalized bessel function of on the whittaker model for the h average w 1 h h h h b of the bessel function we prove that w i n dim dim gl n e gl n f u n e f where is the irreducible representation of u n e f resp gl n f that base changes to when h is gl n f resp u n e f as an application we classify the members of a generic l packet of sl n e that admit invariant vectors for sl n f finally we prove a p adic analogue of our result for square integrable representations in terms of formal degrees by employing the formal degree conjecture of hiraga ichino ikeda 19

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

traces of weighted sobolev spaces old and new abstract we give short simple proofs of uspenski s results characterizing besov spaces as trace spaces of weighted sobolev spaces we generalize uspenski s results and prove the optimality of these generalizations we next show how classical results on the functional calculus in the besov spaces can be obtained as straightforward consequences of the theory of weighted sobolev spaces

on a uniqueness property of supercuspidal unipotent representations abstract the formal degree of a unipotent discrete series character of a simple linear algebraic group over a non archimedean local field in the sense of lusztig 17 is a rational function of q evaluated at q q the cardinality of the residue field the irreducible factors of this rational function are q and cyclotomic polynomials we prove that the formal degree of a supercuspidal unipotent representation determines its lusztig langlands parameter up to twisting by weakly unramified characters for split exceptional groups this result follows from the work of m reeder 28 and for the remaining exceptional cases this is verified in 7 in the present paper we treat the classical families the main result of this article characterizes unramified lusztig langlands parameters which support a cuspidal local system in terms of formal degrees the result implies the uniqueness of so called cuspidal spectral transfer morphisms as introduced in 22 between unipotent affine hecke algebras up to twisting by unramified characters in 23 the essential uniqueness of arbitrary unipotent spectral transfer morphisms was reduced to the cuspidal case

rigid inner forms of real and p adic groups we dene a new cohomology set h 1 u w z g for an ane algebraic group g and a nite central subgroup both dened over a local eld of characteristic zero which is an enlargement of the usual rst galois cohomology set of g we show how this set can be used to give a precise conjectural description of the internal structure and endoscopic transfer of tempered l packets for arbitrary connected reductive groups that extends the well known conjectural description for quasi split groups in the case of real groups we show that this description is correct using shelstad s work

on characters and formal degrees of discrete series of affine hecke algebras of classical types we address two fundamental questions in the representation theory of affine hecke algebras of classical types one is an inductive algorithm to compute characters of tempered modules and the other is the determination of the constants in the formal degrees of discrete series in the form conjectured by reeder j reine angew math 520 37 93 2000 the former is completely different from the lusztig shoji algorithm shoji in invent math 74 239 267 1983 lusztig in ann math 131 355 408 1990 and it is more effective in a number of cases the main idea in our proof is to introduce a new family of representations which behave like tempered modules but for which it is easier to analyze the effect of parameter specializations our proof also requires a comparison of the c theoretic results of opdam delorme slooten solleveld j inst math jussieu 3 531 648 2004 arxiv 0909 1227 int math res not 2008 adv math 220 1549 1601 2009 acta math 205 105 187 2010 and the geometric construction from kato duke math j 148 305 371 2009 am j math 133 518 553 2011 ciubotaru and kato adv math 226 1538 1590 2011

geometric representations of graded and rational cherednik algebras abstract we provide geometric constructions of modules over the graded cherednik algebra h gr and the rational cherednik algebra h rat attached to a simple algebraic group g together with a pinned automorphism these modules are realized on the cohomology of affine springer fibers of finite type that admit c actions in the rational cherednik algebra case the standard grading on these modules is derived from the perverse filtration on the cohomology of affine springer fibers coming from its global analog hitchin fibers when is trivial we show that our construction gives the irreducible finite dimensional spherical modules l triv of h gr and of h rat we give a formula for the dimension of l triv and give a geometric interpretation of its frobenius algebra structure the rank two cases are studied in further details

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

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

estimates for eigenvalues of the paneitz operator abstract for an n dimensional compact submanifold m n in the euclidean space r n we study estimates for eigenvalues of the paneitz operator on m n our estimates for eigenvalues are sharp

mathematical and numerical analysis of linear peridynamic models with nonlocal boundary conditions in this paper we study the linear bond based nonlocal peridynamic models with a particular focus on problems associated with nonstandard nonlocal displacement loading conditions both stationary and time dependent problems are considered for a one dimensional scalar equation defined on a finite bar and for a two dimensional system defined on a square the related peridynamic operators and associated functional spaces are analyzed applications to the numerical analysis of the finite dimensional approximations to peridynamic models are also illustrated

langlands parameters for epipelagic representations of hbox gl _ n let f be a non archimedean local field an irreducible cuspidal representation of mathrm gl n f is epipelagic if its swan conductor equals 1 we give a full and explicit description of the langlands parameters of such representations in terms of elementary representation theory and local class field theory

on formulae decoupling the total variation of bv functions abstract in this paper we prove several formulae that enable one to capture the singular portion of the measure derivative of a function of bounded variation as a limit of non local functionals one special case shows that rescalings of the fractional laplacian of a function u s b v converge strictly to the singular portion of d u

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

stability of the reeb vector field of h contact manifolds it is well known that a hopf vector field on the unit sphere s 2n 1 is the reeb vector field of a natural sasakian structure on s 2n 1 a contact metric manifold whose reeb vector field is a harmonic vector field is called an h contact manifold sasakian and k contact manifolds generalized k spaces and contact metric three manifolds with strongly normal are h contact manifolds in this paper we study in dimension three the stability with respect to the energy of the reeb vector field for such special classes of h contact manifolds and with respect to the volume when is also minimal in terms of webster scalar curvature finally we extend for the reeb vector field of a compact k contact 2n 1 manifold the obtained results for the hopf vector fields to minimize the energy functional with mean curvature correction

haar measure and the artin conductor let g be a connected reductive group defined over a local non archimedean field k the group g k is locally compact and unimodular in gr a haar measure g was defined on g k using the theory of bruhat and tits in this note we give another construction of the measure g using the artin conductor of the motive m of g over k the equivalence of the two constructions is deduced from a result of prasad p 1 the root datum and motive of g in this section k is an arbitrary field and g is a connected reductive group over k we let k be an algebraic closure of k ks the separable closure of k in k and gal ks k let t b g be a maximal torus contained in a borel subgroup defined over ks let g b t be the based root datum defined by this choice we recall cf sp pg3 12 that x t t b x t t b 1 1 with x t and x t the character and cocharacter groups of t respectively and and the simple roots and coroots determined by b respectively let w ng t t be the weyl group of the finite group w acts as automorphisms of x t and is generated by the reflections s x x x 1 2 for the galois group acts as automorphisms of ie as automorphisms of the group x t preserving the finite set as follows if then we can find g g ks such that int g t g t g t and int g b g b g b with g well defined up to left multiplication by t ks hence it induces a well defined automorphism x t x t date 14th january 1997 1 2 benedict h gross and wee teck gan preserving hence we get a group homomorphism aut via acts on aut by inner automorphisms similarly if f g g is any automorphism of g over ks it induces an automorphism f of which depends only on the image of f in the quotient group outks g of outer automorphisms the resulting map outks g aut is an isomorphism which respects the respective galois actions on the two groups cf sp pg10 the galois group also acts on w via the formula s s 1 3 and the semi direct product w acts on the rational vector space e x t q 1 4 let r sym e w which is a graded q module let r be the ideal of elements of positive degree in r and define v r r 2 d 1vd 1 5 this is a graded q module and chevalley proved that dim v dim e cf ch steinberg extended the proof to show that e and v are isomorphic modules cf st pg22 we sketch a proof of this result that does not involve the classification of irreducible root systems proposition 1 6 the q modules e and v are isomorphic proof by the criterion in se pg 104 it suffices to show that for all the fixed spaces e and v have the same dimension for any graded module a am we define the poincare series of by p a t tr am t then p a b p a p b steinberg showed that there is an isomorphism of graded modules s e s vd a here a is finite dimensional with basis bw w w and action given by bw b w the degree of bw is the length l w of w with respect to the generators s furnished by this isomorphism yields the following identity of poincare series det 1 t e d 1 det 1 t vd

preservation of depth in the local geometric langlands correspondence it is expected that under mild conditions the local langlands correspondence preserves depths of representations in this article we formulate a conjectural geometrisation of this expectation we prove half of this conjecture by showing that the depth of a categorical representation of the loop group is greater than or equal to the depth of its underlying geometric langlands parameter a key ingredient of our proof is a new definition of the slope of a meromorphic connection a definition which uses opers

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

on well posedness of the muskat problem with surface tension abstract we consider the muskat problem with surface tension for one fluid or two fluids with or without viscosity jump with infinite depth or lipschitz rigid boundaries and in arbitrary dimension d of the interface the problem is nonlocal quasilinear and to leading order is scaling invariant in the sobolev space h s c r d with s c 1 d 2 we prove local well posedness for large data in all subcritical sobolev spaces h s r d s s c allowing for initial interfaces whose curvatures are unbounded and furthermore when d 1 not locally square integrable to the best of our knowledge this is the first large data well posedness result that covers all subcritical sobolev spaces for the muskat problem with surface tension we reformulate the problem in terms of the dirichlet neumann operator and use a paradifferential approach to reduce the problem to an explicit parabolic equation which is of independent interest

conformal energy in four dimension the conformal energy for 4 manifolds using the paneitz operator is introduced in this article the conformal invariance of the energy functional allows us to find a sharp lower bound in terms of the conformal volume we also demonstrate certain obstruction to existence of minimal immersions to spheres using the fourth order curvature invariance associated to the operator

kloosterman sheaves for reductive groups deligne constructed a remarkable local system on p 1 f 0 1g attached to a family of kloosterman sums katz calculated its monodromy and asked whether there are kloosterman sheaves for general reductive groups and which automorphic forms should be attached to these local systems under the langlands correspondence motivated by work of gross and frenkel gross we nd an explicit family of such automorphic forms and even a simple family of automorphic sheaves in the framework of the geometric langlands program we use these automorphic sheaves to construct adic kloosterman sheaves for any reductive group in a uniform way and describe the local and global monodromy of these kloosterman sheaves in particular they give motivic galois representations with exceptional monodromy groups g2 f4 e7 and e8 this also gives an example of the geometric langlands correspondence with wild ramication for any reductive group

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

second eigenvalue of paneitz operators and mean curvature for n 7 we give the optimal estimate for the second eigenvalue of paneitz operators for compact n dimensional submanifolds in an n p dimensional space form in terms of the mean curvature and the q curvature

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

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

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

analysis and approximation of nonlocal diffusion problems with volume constraints a recently developed nonlocal vector calculus is exploited to provide a variational analysis for a general class of nonlocal diffusion problems described by a linear integral equation on bounded domains in mbrn the nonlocal vector calculus also enables striking analogies to be drawn between the nonlocal model and classical models for diffusion including a notion of nonlocal flux the ubiquity of the nonlocal operator in applications is illustrated by a number of examples ranging from continuum mechanics to graph theory in particular it is shown that fractional laplacian and fractional derivative models for anomalous diffusion are special cases of the nonlocal model for diffusion that we consider the numerous applications elucidate different interpretations of the operator and the associated governing equations for example a probabilistic perspective explains that the nonlocal spatial operator appearing in our model corresponds to the infinitesimal generator for a symmetric jump process sufficie

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

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

on depth zero l packets for classical groups by computing reducibility points of parabolically induced representations we construct to within at most two unramified quadratic characters the langlands parameter of an arbitrary depth zero irreducible cuspidal representation formula presented of a classical group which may be not quasi split over a non archimedean local field of odd residual characteristic from this we can explicitly describe all the irreducible cuspidal representations in the union of one two or four formula presented packets containing formula presented these results generalize the work of debacker reeder in the case of classical groups from regular to arbitrary tame langlands parameters

minimal immersions of surfaces by the first eigenfunctions and conformal area let 0 m s be a minimal immersion of a compact surface into a unit sphere then the linear functions of 0 are eigenfunctions for the laplacian of m corresponding to the eigenvalue 2 2 the main purpose of this paper is to study those minimal surfaces for which 2 is exactly the first non zero eigenvalue of its laplacian this kind of immersions have a peculiar behaviour among all compact minimal surfaces of the sphere and they appear naturally when one considers different geometric problems as li and yau have shown in 6 the methods that we use in this paper are based for the most part on 6 it is known that the only metric on a 2 dimensional sphere admitting a minimal immersion into s by the first eigenfunctions is the standard one this follows for example from the fact that the multiplicity of the first eigenvalue for such a metric is at most three see the cheng work 3 our first result shows that it is possible to extend this property for an arbitrary compact surface in the following way

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

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

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

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