isomonodromic deformations of connections with singularities of parahoric formal type in previous work the authors have developed a geometric theory of fundamental strata to study connections on the projective line with irregular singularities of parahoric formal type in this paper the moduli space of connections that contain regular fundamental strata with fixed combinatorics at each singular point is constructed as a smooth poisson reduction the authors then explicitly compute the isomonodromy equations as an integrable system this result generalizes work of jimbo miwa and ueno to connections whose singularities have parahoric formal type

symmetrization approach to concentration inequalities for empirical processes we introduce a symmetrization technique that allows us to translate a problem of controlling the deviation of some functionals on a product space from their mean into a problem of controlling the deviation between two independent copies of the functional as an application we give a new easy proof of talagrand s concentration inequality for empirical processes where besides symmetrization we use only talagrand s concentration inequality on the discrete cube 0 1 n as another application of this technique we prove new vapnik chervonenkis type inequalities for example for vc classes of functions we prove a classical inequality of vapnik and chervonenkis only with normalization by the sum of variance and sample variance

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

concentration of measure inequalities for markov chains and phi mixing processes we prove concentration inequalities for some classes of markov chains and f mixing processes with constants independent of the size of the sample that extend the inequalities for product measures of talagrand the method is based on information inequalities put forward by marton in case of contracting markov chains using a simple duality argument on entropy our results also include the family of logarithmic sobolev inequalities for convex functions applications to bounds on supremum of dependent empirical processes complete this work

displacement convexity of entropy and related inequalities on graphs we introduce the notion of an interpolating path on the set of probability measures on finite graphs using this notion we first prove a displacement convexity property of entropy along such a path and derive prekopa leindler type inequalities a talagrand transport entropy inequality certain hwi type as well as log sobolev type inequalities in discrete settings to illustrate through examples we apply our results to the complete graph and to the hypercube for which our results are optimal by passing to the limit we recover the classical log sobolev inequality for the standard gaussian measure with the optimal constant

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

depth preservation in local theta correspondence in this paper we prove that the depths of irreducible admissible representations are preserved by the local theta correspondence for any type i reductive dual pairs over a nonarchimedean local field

some inequalities for information divergence and related measures of discrimination inequalities which connect information divergence with other measures of discrimination or distance between probability distributions are used in information theory and its applications to mathematical statistics ergodic theory and other scientific fields we suggest new inequalities of this type often based on underlying identities as a consequence we obtain certain improvements of the well known pinsker inequality our study depends on two measures of discrimination called capacitory discrimination and triangular discrimination the discussion contains references to related research and comparison with other measures of discrimination e g ali silvey csiszar 1996 1966 divergences and in particular the hellinger distance

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

on existence of generic cusp forms on semisimple algebraic groups in this paper we discuss the existence of certain classes of cuspidal automorphic representations having non zero fourier coefficients for a general semisimple algebraic group g defined over a number field k such that its archimedean group g infty is not compact when g is quasi split over k we obtain a result on existence of generic cuspidal automorphic representations which generalize results of vigneras henniart and shahidi we also discuss i the existence of cuspidal automorphic forms with non zero fourier coefficients for congruence of subgroups of g infty and ii applications related to the work of bushnell and henniart on generalized whittaker models

kantorovich duality for general transport costs and applications we introduce a general notion of transport cost that encompasses many costs used in the literature including the classical one and weak transport costs introduced by talagrand and marton in the 90 s and prove a kantorovich type duality theorem as a by product we obtain various applications in different directions we give a short proof of a result by strassen on the existence of a martingale with given marginals we characterize the associated transport entropy inequalities together with the log sobolev inequality restricted to convex concave functions some explicit examples of discrete measures satisfying weak transport entropy inequalities are also given

new concentration inequalities in product spaces we introduce three new ways to measure the distance from a point to a subset of a product space and we prove corresponding concentration inequalities each of them allows to control the fluctuation of a new class of random variables

concentration for multidimensional diffusions and their boundary local times we prove that probability laws of certain multidimensional semimartingales which includes time inhomogenous diffusions under suitable assumptions satisfy quadratic transportation cost inequality under the uniform metric from this we derive concentration properties of lipschitz functions of process paths that depend on the entire history in particular we estimate concentration of boundary local time of reflected brownian motions on a polyhedral domain we work out explicit applications of consequences of measure concentration for the case of brownian motion with rank based drifts

transportation cost information inequalities and applications to random dynamical systems and diffusions we first give a characterization of the l1 transportation cost information inequality on a metric space and next find some appropriate sufficient condition to transportation cost information inequalities for dependent sequences applications to random dynamical systems and diffusions are studied

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

cocenter of p adic groups ii induction map abstract in this paper we study some relation between the cocenter h g of the hecke algebra h g of a connected reductive group g over a nonarchimedean local field and the cocenter h m of its levi subgroups m given any newton component of h g we construct the induction map i from the corresponding newton component of h m to it we show that this map is an isomorphism this leads to the bernstein lusztig type presentation of the cocenter h g which generalizes the work 11 on the affine hecke algebras we also show that the map i we constructed is adjoint to the jacquet functor

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

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

information inequalities and concentration of measure we derive inequalities of the form p q h p r h q r which hold for every choice of probability measures p q r where h p r denotes the relative entropy of p with respect to r and p q stands for a coupling type distance between p and q using the chain rule for relative entropies and then specializing to q with a given support we recover some of talagrand s concentration of measure inequalities for product spaces

counting lattices in simple lie groups the positive characteristic case in this article we prove a conjecture of a lubotzky let g g0 k where k is a local field of characteristic p 5 g0 is a simply connected absolutely almost simple k group of k rank at least 2 we give the rate of growth of x g g a lattice in g vol g x where 1 2 if and only if there is an abstract automorphism of g such that 2 1 we also study the rate of subgroup growth sx of any lattice in g as a result we show that these two functions have the same rate of growth which proves lubotzky s conjecture along the way we also study the rate of growth of the number of equivalence classes of maximal lattices in g with covolume at most x

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

zero cycles of degree one on principal homogeneous spaces abstract let k be a field of characteristic different from 2 let g be an absolutely simple simply connected or adjoint semisimple algebraic k group of classical type we show that if a principal homogeneous space under g over k admits a zero cycle of degree 1 then it has a k rational point

concentration inequalities for dependent random variables via the martingale method the martingale method is used to establish concentration inequalities for a class of dependent random sequences on a countable state space with the constants in the inequalities expressed in terms of certain mixing coefficients along the way bounds are obtained on martingale differences associated with the random sequences which may be of independent interest as applications of the main result concentration inequalities are also derived for inhomogeneous markov chains and hidden markov chains and an extremal property associated with their martingale difference bounds is established this work complements and generalizes certain concentration inequalities obtained by marton and samson while also providing different proofs of some known results

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

totaro s question on zero cycles on torsors let g be a smooth connected linear algebraic group and let x be a g torsor totaro asked if x admits a zero cycle of degree d 1 then does x have a closed etale point of degree dividing d while the literature contains affirmative answers in some special cases we give examples to show that the answer is negative in general

on endoscopic transfer of deligne lusztig functions in this paper we prove the fundamental lemma for deligne lusztig functions namely for every deligne lusztig functionon a p adic group g we write down an explicit linear combinationh of deligne lusztig functions on an endoscopic group h such thatandh have matching orbital integrals in particular we prove a conjecture of kottwitz ko4 more precisely we do it under some mild restriction on p

the local langlands correspondence for gsp 4 over local function fields we prove the local langlands correspondence for rm gsp 4 f where f is a non archimedean local field of positive characteristic with residue characteristic 2

sur l homologie des fibres de springer affines tronquees en suivant les travaux de goresky kottwitz et macpherson 12 on calcule l homologie des fibres de springer affines tronquees dans le cas non ramifie mais sous une hypothese de purete dans le cas equivalue on demontre cette hypothese de purete la troncature depend d un parametre qui est un diviseur sur une variete torique l adique dans les cas non ramifies et equivalues pour chaque fibre de springer affine tronquee on definit un faisceau coherent gradue sur cette variete torique dont l espace des sections globales s identifie a l homologie l adique de la fibre consideree pour certaines familles de groupes endoscopiques ces faisceaux apparaissent dans des suites exactes on en deduit alors le lemme fondamental pondere d arthur 5 conjecture 5 1 dans les cas equivalues et non ramifies

a general duality theorem for marginal problems given probability spaces xi ai pi i 1 2 letm p1 p2 denote the set of all probabilities on the product space with marginalsp1 andp2 and leth be a measurable function on x1 x2 a1 a2 in order to determine supfh dp where the supremum is taken overp inm p1 p2 a general duality theorem is proved only the perfectness of one of the coordinate spaces is imposed without any further topological or tightness assumptions an example without any further topological or tightness assumptions an example is given to show that the assumption of perfectness is essential applications to probabilities with given marginals and given supports stochastic order and probability metrics are included

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

a simple proof of the blowing up lemma corresp the blowing up lemma says that if the probability with respect to a product measure of a set a subseteq cal x n cal x finite n large is not exponentially small then its l n neighborhood has probability almost one for some l n o n here an information theoretic proof of the blowing up lemma generalizing it to continuous alphabets is given

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

concentration inequalities using the entropy method we investigate a new methodology worked out by ledoux and massart to prove concentration of measure inequalities the method is based on certain modified logarithmic sobolev inequalities we provide some very simple and general ready to use inequalities one of these inequalities may be considered as an exponential version of the efron stein inequality the main purpose of this paper is to point out the simplicity and the generality of the approach we show how the new method can recover many of talagrand s revolutionary inequalities and provide new applications in a variety of problems including rademacher averages rademacher chaos the number of certain small subgraphs in a random graph and the minimum of the empirical risk in some statistical estimation problems

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

affine springer fibers of type a and combinatorics of diagonal coinvariants abstract we calculate the borel moore homology of affine springer fibers of type a associated with some regular semisimple nil elliptic elements as a result we obtain bigraded s n modules whose bigraded frobenius series are a generalization of the symmetric functions introduced by haglund haiman loehr remmel and ulyanov

on branching rules of depth zero representations abstract using bruhat tits theory we analyze the restriction of depth zero representations of a semisimple simply connected p adic group g to a maximal compact subgroup k we prove the coincidence of branching rules within classes of deligne lusztig supercuspidal representations furthermore we show that under obvious compatibility conditions the restriction to k of a deligne lusztig supercuspidal representation of g intertwines with the restriction of a depth zero principal series representation in infinitely many distinct components of arbitrarily large depth several qualitative and quantitative results are obtained and their use is illustrated in an example

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 interactions between ergodic theory and information theory information theorists frequently use the ergodic theorem likewise entropy concepts are often used in information theory recently the two subjects have become partially intertwined as deeper results from each discipline find use in the other a brief history of this interaction is presented in this paper together with a more detailed look at three areas of connection namely recurrence theory blowing up bounds and direct sample path methods

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 a mixture of brenier and strassen theorems we give a characterization of optimal transport plans for a variant of the usual quadratic transport cost introduced in 33 optimal plans are composition of a deterministic transport given by the gradient of a continuously differentiable convex function followed by a martingale coupling we also establish some connections with caffarelli s contraction theorem 14

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