utd diffraction coefficients for higher order wedge diffracted fields an asymptotic expansion of a multiple integral for knife edge diffraction is derived the expansion expresses the field in terms of single edge diffraction by considering this expression a similar expansion for higher order utd diffracted fields is proposed by means of a set of transition functions the result removes some of the shortcomings of the original set out of the utd when the incident field is not a ray optical field

an improved heuristic utd solution for multiple edge transition zone diffraction this paper describes a new heuristic approach for multiple edge diffraction modeling based on the uniform theory of diffraction utd it incorporates the slope diffraction terms which can be realized as a second order diffraction process vital for accurate predictions when more than one edge exists the predictions of this approach are compared with the vogler 1982 solution and were found to be in very good agreement for many different geometrical arrangements with low computational complexity the solution is also extended to wedge diffraction

on the variety of a highest weight module 1 2 let u g be the enveloping algebra of g detailed information on the primitive spectrum prim u g of u g was summarized by conjecture 7 4 of 181 here a main question was to relate prim u g with the springer correspondence parts i ii of this conjecture were established in i ii and a partial solution to iii was given in 21 sufficient e g to treat g of type a meanwhile borho and brylinski 5 have established iii for induced ideals while barbasch and vogan 1 2 have established a version of iii in which gr i i e prim u g is replaced by a certain wavefront set known to be contained in y gr 1 this last work involves some case by case analysis and gives little indication as to why the springer correspondence should arise in the study of prim u g

on certain zeta functions attached to two hilbert modular forms ii the case of automorphic forms on a quaternion algebra as to the meaning of the symbols the reader is referred to the introduction of part i we recall here only that f is a totally real algebraic number field of degree n and w denotes the fourier coefficients of an elliptic modular form 52 z e wq a e2q iaz in part i we investigated d assuming that ex a n q s is an l function of a cm field with an algebraic valued hecke character in the present part ii we treat the case where x a can be obtained as eigenvalues of hecke operators t a on a quaternion algebra b over f thus we begin our study by recalling the theory of automorphic forms on the product o r of r copies of the upper half plane with respect to congruence subgroups of b x here r is the number of archimedean primes of f unramified in b then we introduce in section 2 the notion of arithmeticity or q rationality of such automorphic forms and show that the space of automorphic forms can be spanned by the arithmetic ones our main theorems will be stated in section 3 and proved in section 6 sections 4 and 5 are preliminaries to the proof the central result specialized to the case r 1 asserts that if g is a q rational eigenform such that gj t a x a g for all integral ideals a of f then d s0 for certain integers so are algebraic numbers times 7tdg g where d is an integer determined by the weight of g and is the petersson inner product

primitive ideals and orbital integrals in complex exceptional groups in 2 two related problems were studied for complex classical groups determination of the primitive spectrum of the enveloping algebra as a set and fourier inversion of unipotent orbital integrals the first of these was solved completely extending work of joseph and others in sl n c and small groups although our techniques presumably solve the second as well up to determination of normalizing constants we carried out the calculations only for special unipotent classes see 181 we had of course tried to apply the same methods to the exceptional groups but they seem to be inadequate since that work was done however brylinski and kashiwara and beilinson and bernstein have established the kazhdan lusztig conjecture giving character formulas for irreducible highest weight modules 3 71 because of a conjecture of joseph proved in 24 this determines in principle the primitive ideals with a fixed regular integral infinitesimal character they are in one to one correspondence with what kazhdan and lusztig call left cells in the weyl group however the algorithm given by kazhdan and lusztig to compute these cells is enormously complicated requiring one to compute roughly 1 i polynomials of degrees on the order of half the number of positive roots this is very unsatisfactory however it

a utd solution for multiple rounded surfaces this paper describes the diffraction mechanism when a ray optical field is obstructed by a cascade of smooth convex obstructions a new formulation of the uniform theory of diffraction utd is presented that incorporates the slope diffraction term the solution follows a generic approach that can be applied to arbitrary placed convex structures it will be shown that the slope diffraction term provides the required accuracy in radio propagation predictions when more than one obstruction exists in the propagation path the proposed solution is then compared with measurements and other analytical models with excellent agreement it is also found that the presented slope utd solution seems to be superior in terms of computation time and complexity while achieving very accurate results in addition when the radius of the cylinder tends to be very small the solution approaches the utd solution for multiple knife edge predictions

differential operators on homogeneous spaces iii characteristic varieties of harish chandra modules and of primitive ideals letg be a semi simple complex algebraic group with lie algebra and flag varietyx g b for each primitive idealj with trivial central character in the enveloping algebrau we define a characteristic variety in the cotangent bundle ofx which projects under the springer resolution mapt x onto the closure of a nilpotent orbit we prove that this characteristic variety is theg saturation of the characteristic variety of a highest weight module with annihilatorj we conjecture that it is irreducible forg sl n our conjecture would provide a geometrical explanation for the classification of primitive ideals in terms of weyl group representations as achieved by a joseph the presentation of these ideas here is simultaneously used to some extent as an opportunity to continue our more general systematic discussion of differential operators on a complete homogeneous space and to study more generally characteristic varieties of harish chandra modules

characteristic varieties and vanishing cycles on presente une etude systematique des varietes caracteristiques des systemes holonomes a singularites regulieres ce sont des sous varietes lagrangiennes coniques dans le fibre cotangent t x

a heuristic utd slope diffraction coefficient for rough lossy wedges heuristic wedge diffraction coefficients for computing propagation path loss over finitely conducting earth are extended to include slope diffraction with the assumption that propagation of energy through the wedge is negligible the slope diffraction terms for the lossy wedge are obtained in an analogous manner as for the perfectly conducting case except that special care must be taken with the factor multiplying the incident field for grazing incidence results given show that the slope diffraction term produces continuous results that behave reasonably when compared with results for perfectly conducting wedges

a slope utd solution for a cascade of multishaped canonical objects the case of multiple diffraction from a cascade of two dimensional multi shaped canonical objects such as wedges knife edges and cylinders is investigated a heuristic slope uniform theory of diffraction utd solution is presented that can give accurate and uniform field predictions at any area point and for any kind of environment the solution is then compared against measurements taken in an anechoic chamber and a very good agreement is observed furthermore an algorithm for implementing the proposed utd solution to rural area field predictions is also presented based on a novel algorithm for optimum fit of canonical shapes to the terrain irregularities comparisons between the model and measurements taken in real terrain profile show a very good agreement and uniform field predictions the importance of the choice of the used canonical object to terrain modeling is also discussed since different diffraction patterns are observed for different combination of the canonical objects

a utd triple diffraction coefficient for straight wedges in arbitrary configuration a new utd solution is presented for the description of the ray triply diffracted by straight wedges in arbitrary configuration illuminated by a spherical wavefront field the proposed utd coefficient permits to uniformly describe the field at any aspect in particular including the case of overlapping transition regions where the subsequent application of utd single wedge diffraction coefficient fails the triple diffracted ray field is expressed in an analytical form providing physical insight into the triple diffraction mechanism and a new effective engineering tool within the utd framework as required in a modern ray based code

critical values of the twisted tensor l function in the imaginary quadratic case the twisted tensor l function of f which we denote by g s f is a certain dirichlet series associated to a quadratic extension of number fields k f and a cuspidal automorphic function f over k it was introduced in 1 by asai following previous work of shimura in the case when f is a hilbert modular cusp form over a real quadratic extension k of q in the past twenty odd years this l function has been considered more generally for instance 11 and 12 deal with quadratic extensions of totally real fields 17 with imaginary quadratic extensions of q and 3 4 and 14 with general quadratic extensions of number fields all these papers have been primarily concerned with establishing analytic properties of g s f analogous to those in 1 such as meromorphic continuation to the entire complex plane location and finiteness of the number of poles and functional equation the aim of this paper is to prove a rationality result for g s f in the imaginary quadratic setting if k is an imaginary quadratic field and f a cusp form associated to k we establish that there is a period j f such that

a modified diffraction coefficient for imperfect conducting wedges and buildings with finite dimensions multiple building diffraction loss calculation is often based on the assumption that their dimensions are infinite in the direction perpendicular to the propagation path even though in reality this assumption is not valid in this paper a modified diffraction coefficient model for the calculation of a more realistic diffraction loss caused by a single building with finite dimensions is proposed the overall diffraction loss is calculated for a group of buildings in a row by using the proposed model in addition after carrying out the field measurements it is shown that for some particular cases where the transmitter and diffracting obstacles are not far away from each other i e microcells or picocells spherical wave modeling produces a better accuracy than the plane wave approximation it is also demonstrated that the theoretical results can further be improved by considering the higher order terms in the calculation of diffraction coefficients lastly excess path loss obtained for a row of buildings with spherical wave assumption is compared with our measurement data this comparison shows reasonable correspondence between our theoretical model and measurements therefore our proposed model can successfully be used for improving the theoretical methods for predicting path loss in urban environments where diffraction from a number of buildings has to be considered

on the ramanujan conjecture and finiteness of poles for certain l functions in this paper we prove a number of general results about automorphic l functions which appear in the constant terms of eisenstein series for an arbitrary quasi split connected reductive algebraic group over a number field these l functions were first introduced by langlands for chevalley groups 20 and through them he was led to make some of his deep conjectures 21 one significance of these l functions is that all the automorphic l functions studied so far are among them there are three general results first we obtain a uniform line of absolute convergence for all of these l functions theorem 5 1 a uniform estimate for the hecke eigenvalues of generic cusp forms on many absolutely simple quasi split connected reductive algebraic groups over number fields corollary 5 4 and an improvement on the best available estimate for the fourier coefficients of maass wave forms corollary 5 5 also follow next we establish a meromorphic continuation and functional equation for each of these l functions theorem 6 1 finally in theorem 6 2 we prove finiteness of poles on the whole complex plane for an important class of these l functions corollaries 6 6 through 6 10 more precisely let g be a quasi split connected reductive algebraic group over a number field f set g g a f where a f is the ring of adeles of f fix a borel subgroup b of g over f and let u be its unipotent radical let p be a maximal f parabolic subgroup of g with p d b in the context of the problems studied here nothing new will be obtained if one drops the maximality condition on p write p mn a levi decomposition n c u and let b u p m and n be the corresponding groups of adelic points for every place v of f let gv g fj similarly we have bv us pv me and nv let x xv be a generic character of u f u cf section 3 then each xv is generic let r ov 7tv be a cusp form on m we shall say 7t is

multiplicity of the adjoint representation in simple quotients of the enveloping algebra of a simple lie algebra let g be a complex simple lie algebra r a cartan subalgebra and u g the enveloping algebra of g we calculate for each maximal two sided ideal jmax a a e r of u g the number of times the adjoint representation occurs in u g jmax a this is achieved by reduction via the kazhdan lusztig polynomials to the case when a lies on a corner i e is a multiple of a fundamental weight remarkably in this case one can always present u g jmax a as a generalized principal series module and here we also calculate its goldie rank as a ring which is a question of independent interest for some of the more intransigent cases it was necessary to use recent very precise results of lusztig on left cells the results are used to show how a recent theorem of gupta established for nonspecial a can fail if a is singular finally we give a quite efficient procedure for testing if an induced ideal is maximal

degenerate whittaker functionals for real reductive groups in this paper we establish a connection between the associated variety of a representation and the existence of certain it degenerate whittaker functionals for both smooth and k finite vectors for all quasi split real reductive groups thereby generalizing results of kostant matumoto and others

multiple time domain diffraction for uwb signals the time domain td solution of the two dimensional multiple diffraction case is investigated the proposed td solution is based on the representation of the inverse fourier transform of the corresponding frequency domain fd solution in closed form as it is given by the uniform theory of diffraction utd and it incorporates the td representation of the higher order diffraction coefficients an algorithm to predict the td diffracted field after an arbitrary number of objects is also presented in the proposed algorithm different types of objects along the propagation path can be applied as well such as absorbing knife edges and metallic or nonperfectly conducting wedges the comparison between the td solution and the numerical inverse fast fourier transform of the fd solution proves the validity of the proposed solution

a general uniform double wedge diffraction coefficient the uniform theory of diffraction utd is extended to include double diffraction by an arbitrary configuration of two wedges a scalar double diffraction coefficient is defined which is uniform throughout space while reducing to the product of single diffraction coefficients outside overlapping transition regions the result is formulated in terms of double fresnel integrals which can be rapidly evaluated

p adic l functions of bianchi modular forms the theory of overconvergent modular symbols developed by rob pollack and glenn stevens gives a beautiful and effective construction of the p adic l function of a modular form in this paper we give an analogue of their results for bianchi modular forms that is modular forms over imaginary quadratic fields in particular we prove control theorems that say that the canonical specialisation map from overconvergent to classical bianchi modular symbols is an isomorphism on small slope eigenspaces of suitable hecke operators we also give an explicit link between the classical modular symbol attached to a bianchi modular form and critical values of its l function which then allows us to construct p adic l functions of bianchi modular forms

dimensions of irreducible modules over w algebras and goldie ranks the main goal of this paper is to compute two related numerical invariants of a primitive ideal in the universal enveloping algebra of a semisimple lie algebra the first one very classical is the goldie rank of an ideal the second one is the dimension of an irreducible module corresponding to this ideal over an appropriate finite w algebra we concentrate on the integral central character case we prove modulo a conjecture that in this case the two are equal our conjecture asserts that there is a one dimensional module over the w algebra with certain additional properties the conjecture is proved for the classical types also modulo the same conjecture we compute certain scale factors introduced by joseph this allows to compute the goldie ranks of the algebras of locally finite endomorphisms of simples in the bgg category mathcal o this completes a program of computing goldie ranks proposed by joseph in the 80 s for integral central characters and modulo our conjecture we also provide an essentially kazhdan lusztig type formula for computing the characters of the irreducibles in the brundan goodwin kleshchev category mathcal o for a w algebra again under the assumption that the central character is integral in particular this allows to compute the dimensions of the finite dimensional irreducible modules the formula is based on a certain functor from an appropriate parabolic category mathcal o to the w algebra category mathcal o this functor can be regarded as a generalization of functors previously constructed by soergel and by brundan kleshchev we prove a number of properties of this functor including the quotient property and the double centralizer property we develop several side topics related to our generalized soergel functor for example we discuss its analog for the category of harish chandra modules we also discuss generalizations to the case of categories mathcal o over dixmier algebras the most interesting example of this situation comes from the theory of quantum groups we prove that an algebra that is a mild quotient of luszitg s form of a quantum group at a root of unity is a dixmier algebra for this we check that the quantum frobenius epimorphism splits

the character and the wave front set correspondence in the stable range abstract we relate the distribution characters and the wave front sets of unitary representation for real reductive dual pairs of type i in the stable range

a uniform gtd formulation for the diffraction by a wedge with impedance faces a uniform high frequency solution is presented for the diffraction by a wedge with impedance faces illuminated by a plane wave perpendicularly incident on its edge arbitrary uniform isotropic impedance boundary conditions may be imposed on the faces of the wedge and both the transverse electric te and transverse magnetic tm cases are considered this solution is formulated in terms of a diffraction coefficient which has the same structure as that of the uniform geometrical theory of diffraction utd for a perfectly conducting wedge its extension to the present case is achieved by introducing suitable multiplying factors which have been derived from an asymptotic evaluation of the exact solution given by maliuzhinets when the field point is located on the surface near the edge a more accurate asymptotic evaluation is employed to obtain a high frequency expression for the diffracted field which is suitable for several specific applications the formulation described in this paper may provide a useful rigorous basis to search for a more numerically efficient but yet accurate approximation

finite dimensional representations of w algebras w algebras of finite type are certain finitely generated associative algebras closely related to the universal enveloping algebras of semisimple lie algebras in this paper we prove a conjecture of premet that gives an almost complete classification of finite dimensional irreducible modules for w algebras also we study a relation between harish chandra bimodules and bimodules over w algebras

a hybrid asymptotic solution for the scattering by a pair of parallel perfectly conducting wedges the overlapping transition regions of the double diffraction by a pair of parallel wedge edges are considered for the hybrid case where the gap between the edges is small compared to the distances from the source and the observation point plane wave far field limit and the scatterer as a whole is large or infinite a closed form asymptotic solution for the scattered field continuous at all angles of incidence and scattering is constructed for this case the peculiar feature of this solution is a hybrid representation of the field singly diffracted by the first wedge a part of it is described by a nonuniform geometrical theory of diffraction gtd expression while the other part is described in terms of the uniform theory of diffraction utd the rest of the diffracted ray fields are described by nonuniform expressions with singularities mutually canceling on summation this solution is applied to the scattering by a perfectly conducting rectangular cylinder with appropriate geometrical parameters and agreement with moment method calculation is demonstrated

lie superalgebras clifford algebras induced modules and nilpotent orbits let g be a classical simple lie superalgebra to every nilpotent orbit o in g0 we associate a clifford algebra over the field of rational functions on o we find the rank k o of the bilinear form defining this clifford algebra and deduce a lower bound on the multiplicity of a u g module with o or an orbital subvariety of o as associated variety in some cases we obtain modules where the lower bound on multiplicity is attained using parabolic induction the invariant k o is in many cases equal to the odd dimension of the orbit g o where g is a lie supergroup with lie superalgebra g

finite dimensional irreducible representations of finite w algebras associated to even multiplicity nilpotent orbits in classical lie algebras we consider finite w algebras u mathfrak g e associated to even multiplicity nilpotent elements in classical lie algebras we give a classification of finite dimensional irreducible u mathfrak g e modules with integral central character in terms of the highest weight theory from brundan et al int math res notices 15 art id rnn051 2008 as a corollary we obtain a parametrization of primitive ideals of u mathfrak g with associated variety the closure of the adjoint orbit of e and integral central character

character d modules via drinfeld center of harish chandra bimodules the category of character d modules is realized as drinfeld center of the abelian monoidal category of harish chandra bimodules tensor product of harish chandra bimodules is related to convolution of d modules via the long intertwining functor radon transform by a result of beilinson and ginzburg represent theory 3 1 31 1999 exactness property of the long intertwining functor on a cell subquotient of the harish chandra bimodules category shows that the truncated convolution category of lusztig adv math 129 85 98 1997 can be realized as a subquotient of the category of harish chandra bimodules together with the description of the truncated convolution category bezrukavnikov et al in isr j math 170 207 234 2009 this allows us to derive under a mild technical assumption a classification of irreducible character sheaves over obtained by lusztig by a different method

sheets and topology of primitive spectra for semisimple lie algebras abstract let g be a semisimple lie algebra consider the set x of primitive ideals of the enveloping algebra u g given the jacobson topology a basic open problem is to describe x as a countable union of algebraic varieties v with strata v as large as possible towards this aim the notion of a sheet in x is introduced here in analogy with the notion of sheet in the adjoint orbit space g g the sheets in x are classified and given a purely topological description moreover goldie rank is constant and maximal on a dense open subset of a sheet for every positive integer n it is shown that there are only finitely many sheets in x with maximal goldie rank n for n 1 which corresponds to completely prime ideals an extension of the dixmier map from some adjoint orbits to primitive ideals is introduced with the aim of showing that the sheets with n 1 are homeomorphic to sheets in g g and hence explicitly described as algebraic varieties this goal is partly though not fully achieved the proofs of the above results require a detailed knowledge of goldie rank polynomials and even a new difficult result concerning the support of their restrictions to the walls an extension of a theorem of soergel on the topology of x is also required and for this a new approach is given which is both simpler and more general

a uniform geometrical theory of diffraction for an imperfectly conducting half plane diffraction tensors are presented in the context of the uniform geometrical theory of diffraction utd for the high frequency scattering by an impedance half plane at normal and oblique skew incidence these are based on the exact wiener hopf solution and were derived according to the utd ansatz in addition unlike previous uniform diffraction coefficients the ones given here reduce to the known utd diffraction coefficients for the perfectly conducting case the coefficients are explicit and therefore appropriate for practical applications several scattering patterns are also presented and compared to a previous heuristic solution

calculation of higher order diffracted fields for multiple edge transition zone diffraction a well known problem in conventional uniform theory of diffraction utd is multiple edge transition zone diffraction here higher order diffracted fields have to be used in order to improve the result and these fields are added by means of a series unfortunately this series will converge slowly for many edges which means that a good enough result might require a large number of terms however by summing up the series properly the computation time can be kept reasonable which will be shown for knife edges the summation technique is though valid for wedges as well