duality for ext groups and extensions of discrete series for graded hecke algebras in this paper we study extensions of graded affine hecke algebra modules in particular based on an explicit projective resolution on graded affine hecke algebra modules we prove a duality result for ext groups this duality result with an ind res resolution gives an algebraic proof of the fact that all higher ext groups between a tempered module and a discrete series vanish

inverse maximal eigenvalues problems for leslie and doubly leslie matrices abstract in this paper we deal with leslie and doubly leslie matrices of order n in particular with the companion and doubly companion matrices we study three inverse eigenvalues problems which consist of constructing these matrices from the maximal eigenvalues of its all leading principal submatrices for leslie and doubly companion matrices an eigenvector associated with the maximal eigenvalue of the matrix is additionally considered and for the doubly leslie matrix also an eigenvector associated with the maximal eigenvalue of leading principal submatrix of order n 1 is required we give necessary and sufficient conditions for the existence of a leslie matrix and a companion matrix and sufficient conditions for the existence of a doubly leslie matrix and a doubly companion matrix our results are constructive and generate an algorithmic procedure to construct these special kinds of matrices

homological algebra for affine hecke algebras abstract in this paper we study homological properties of modules over an affine hecke algebra h in particular we prove a comparison result for higher extensions of tempered modules when passing to the schwartz algebra s a certain topological completion of the affine hecke algebra the proof is self contained and based on a direct construction of a bounded contraction of certain standard resolutions of h modules this construction applies for all positive parameters of the affine hecke algebra this is an important feature since it is an ingredient to analyse how the irreducible discrete series representations of h arise in generic families over the parameter space of h for irreducible non simply laced affine hecke algebras this will enable us to give a complete classification of the discrete series characters for all positive parameters we will report on this application in a separate article

on a class of degenerate and singular elliptic systems in bounded domains abstract this paper deals with the nonexistence and multiplicity of nonnegative nontrivial solutions to a class of degenerate and singular elliptic systems of the form div h 1 x u f u x u v in div h 2 x v f v x u v in where is a bounded domain with smooth boundary in r n n 2 and h i 0 h i l loc 1 h i i 1 2 are allowed to have essential zeroes at some points in f u f v f and is a positive parameter our proofs rely essentially on the critical point theory tools combined with a variant of the caffarelli kohn nirenberg inequality in p caldiroli r musina on a variational degenerate elliptic problem nodea nonlinear differential equations appl 7 2000 189 199

critical point theory for nondifferentiable functionals and applications abstract we consider minimization results for locally lipschitzian functionals on a banach space by introducing a compactness condition of the palais smale type which is suggested by the variational principle of ekeland applications are given to some semilinear elliptic problems with discontinuous nonlinearities

dirac operator and its cohomology for the quantum group uq x1d598 x1d591 x1d7d0 we introduce a dirac operator d for the quantum group uq x1d530 x1d529 2 as an element of the tensor product of uq x1d530 x1d529 2 with the clifford algebra on two generators we study the properties of d including an analogue of vogan s conjecture we compute the cohomology of d acting on various uq x1d530 x1d529 2 modules

dirac cohomology for graded affine hecke algebras we define an analogue of the casimir element for a graded affine hecke algebra mathbb h and then introduce an approximate square root called the dirac element using it we define the dirac cohomology hd x of an mathbb h module x and show that hd x carries a representation of a canonical double cover of the weyl group widetilde w our main result shows that the widetilde w structure on the dirac cohomology of an irreducible mathbb h module x determines the central character of x in a precise way this can be interpreted as p adic analogue of a conjecture of vogan for harish chandra modules we also apply our results to the study of unitary representations of mathbb h

multiplicity result for a class of elliptic problems with non differentiable terms in rn abstract using non smooth critical point theory and a multiplicity result based on local linking the original smooth version is due to brezis and nirenberg we prove the existence of multiple solutions for a class of schrodinger type systems with non differentiable terms in r n in addition we give two examples to illustrate our multiplicity result

on a class of semilinear schrodinger equations involving critical growth and discontinuous nonlinearities abstract in this paper we deal with a class of schrodinger equation in r n involving critical sobolev exponent and jump discontinuities the basic tool employed here is an approximation technique with periodic functions and variational arguments based on a linking theorem for locally lipschitz functionals

variational methods for non differentiable functionals and their applications to partial differential equations the interest in extending the theory of nonlinear partial differential equations pde to pde with discontinuous nonlinearities dnde has been increasing daily the reasons are twofold 1 many free boundary problems may be reduced to boundary value problems of dnde cf chang 16 71 chang and jiang 9 and fleishman and mahar 121 2 sometimes in dealing with a pde problem it is beneficial to put the original pde into a large category for instance dnde cf chang 7 simply speaking a dnde is as follows

dirac cohomology unitary representations and a proof of a conjecture of vogan u radu je dokazana slutnja d vogana koja odredjuje infinitesimalni karakter ireducibilnog g k modula koji ima diracovu kohomologiju razlicitu od nula infinitesimalni karakter je dan kao rho translat najvece tezine k tipa u diracovoj kohomologiji diracova kohomologija se definira kao ker d im d cap ker d gdje d oznacava diracov operator na zadanom g k modulu tenzoriranom sa spin modulom

characters of springer representations on elliptic conjugacy classes for a weyl group w we investigate simple closed formulas valid on elliptic conjugacy classes for the character of the representation of w in the homology of a springer fiber we also give a formula valid again on elliptic conjugacy classes of the w character of an irreducible discrete series representation with real central character of a graded affine hecke algebra with arbitrary parameters in both cases the pin double cover of w and the dirac operator for graded affine hecke algebras play key roles

an inverse eigenvalue problem for symmetrical tridiagonal matrices we consider the following inverse eigenvalue problem to construct a symmetrical tridiagonal matrix from the minimal and maximal eigenvalues of all its leading principal submatrices we give a necessary and sufficient condition for the existence of such a matrix and for the existence of a nonnegative symmetrical tridiagonal matrix our results are constructive in the sense that they generate an algorithmic procedure to construct the matrix

inverse spectral problems for collections of leading principal submatrices of tridiagonal matrices abstract which assignments from 2 n 1 arbitrary distinct real numbers as eigenvalues of designated leading principal submatrices permit a real symmetric tridiagonal matrix we raise this question motivated both by known results and recent work on multiplicities and interlacing equalities in symmetric matrices whose graph is a given tree known results are reviewed a general conjecture is given and several new partial results are proved

reduction to real infinitesimal character in affine hecke algebras the main result of this paper is to show that the problem of the determination of the unramified unitary dual of a split p adic group is equivalent to the problem of determining the unitary dual of the corresponding graded hecke algebra in bm the authors established this equivalence in the case of iwahori spherical representations under a certain restriction namely it was essential for the infinitesimal character to be real in the terminology of bmj in terms of the langlands deligne lusztig parameters s u p kul the restriction is that s e lg be a purely hyperbolic element the technique used in bm was to combine the notion of the signature of a k character in v with some facts which follow from kl namely that the 2w characters of tempered representations are linearly independent this is essentially true precisely when the infinitesimal character is real for if not s has an elliptic part se such that

green polynomials of weyl groups elliptic pairings and the extended dirac index abstract in this paper we give a uniform construction of irreducible genuine characters of the pin cover w of a weyl group w and put them into the context of theory of springer representations in the process we provide a direct connection between springer theory via green polynomials the irreducible representations of w and an extended dirac operator for graded hecke algebras we also introduce a q elliptic pairing for w with respect to the reflection representation v these constructions are of independent interest the q elliptic pairing is a generalization of the elliptic pairing of w introduced by reeder and it is also related to s kato s notion of graded kostka systems for the semidirect product a w c w s v

the implicit construction of multiplicity lists for classes of trees and verification of some conjectures abstract for the problem of understanding what multiplicities are possible for eigenvalues among real symmetric matrices with a given graph constructing matrices with conjectured multiplicities is generally more difficult than finding constraining conditions here the implicit function theorem method for constructing matrices with a given graph and given multiplicity list is refined and extended in particular the breadth of known circumstances in which the jacobian is nonsingular is increased this allows characterization of all multiplicity lists for binary diametric depth one trees in addition the degree conjecture and a conjecture about the minimum number of multiplicities equal to 1 is proven for diametric trees finally an intriguing conjecture about the eigenvalues of a matrix whose graph is a path and its submatrices is given along with a discussion of some ides that would support a proof of the degree conjecture and the minimum number of 1 s conjecture in general

functors for unitary representations of classical real groups and affine hecke algebras we define exact functors from categories of harish chandra modules for certain real classical groups to finite dimensional modules over an associated graded affine hecke algebra with parameters we then study some of the basic properties of these functors in particular we show that they map irreducible spherical representations to irreducible spherical representations and moreover that they preserve unitarity in the case of split classical groups we thus obtain a functorial inclusion of the real spherical unitary dual with real infinitesimal character into the corresponding p adic spherical unitary dual

spin representations of real reflection groups of noncrystallographic root systems abstract a uniform parametrization for the irreducible spin representations of weyl groups in terms of nilpotent orbits is recently achieved by ciubotaru 2011 this paper is a generalization of this result to other real reflection groups let v 0 r v 0 r be a root system with the real reflection group w we define a special subset of points in v 0 which will be called solvable points those solvable points in the case r crystallographic correspond to the nilpotent orbits whose elements have a solvable centralizer in the corresponding lie algebra then a connection between the irreducible spin representations of w and those solvable points in v 0 is established

nonlinear elliptic eigenvalue problems with discontinuities abstract in this paper we study the existence of solution for two different eigenvalue problems the first is nonlinear and the second is semilinear our approach is based on results from the nonsmooth critical point theory in the first theorem we prove the existence of at least two nontrivial solutions when is in a half axis in the second theorem based on a nonsmooth variant of the generalized mountain pass theorem we prove the existence of at least one nontrivial solution for every r

a variational approach to discontinuous problems with critical sobolev exponents abstract we employ variational techniques to study the existence and multiplicity of positive solutions of semilinear equations of the form u h x h u a uq u2 1 in rn where a 0 are parameters h x is both nonnegative and integrable on rn h is the heaviside function 2 is the critical sobolev exponent and 0 q

on a class of schrodinger systems with discontinuous nonlinearities using a non smooth critical point theory for locally lipschitz functionals we investigate a class of stationary schrodinger systems with subcritical discontinuous nonlinearities and lower bounded potentials that blow up at infinity the existence of nontrivial solution is obtained

on the classification of unitary representations of reductive lie groups each subset is identified conjecturally conjecture 0 6 with a collection of unitary representations of a certain subgroup g u of g we will give strong evidence and partial results for this conjecture in this way the problem of classifying u g would be reduced by induction on the dimension of g to the case g u g before considering the general program in more detail we describe it in the familiar case g sl 2 r this example will be treated more com

existence and multiplicity results for some elliptic systems with discontinuous nonlinearities abstract in this paper we complete two tasks we first extend the critical point theorem obtained by li and willem s j li m willem applications of local linking to critical point theory j math anal appl 189 1995 6 32 to the nonsmooth case in which the energy functional is locally lipschitz and satisfies the weaker nonsmooth cerami condition then we study some semilinear elliptic systems with discontinuous nonlinearities and obtain some existence and multiplicity results

a vanishing theorem for dirac cohomology of standard modules abstract this paper studies the dirac cohomology of standard modules in the setting of graded hecke algebras with geometric parameters we prove that the dirac cohomology of a standard module vanishes if and only if the module is not twisted elliptic tempered the proof makes use of two deep results one is some structural information from the generalized springer correspondence obtained by s kato and lusztig another one is a computation of the dirac cohomology of tempered modules by barbasch ciubotaru trapa and ciubotaru we apply our result to compute the dirac cohomology of ladder representations for type a n for each of such representations with non zero dirac cohomology we associate to a canonical weyl group representation we use the dirac cohomology to conclude that such representations appear with multiplicity one