noncommutative noether s problem vs classic noether s problem we prove that if two complex affine irreducible varieties are birational that is their coordinate rings have isomorphic fields of fractions then their rings of differential operators are birationally equivalent it allows to address the noncommutative noether s problem on the invariants of weyl fields for linear actions of finite groups in fact we show for any field mathsf k of characteristic 0 that rationality of the quotient variety mathbb a n mathsf k g implies that the noncommutative noether s problem is positively solved for g in particular this gives affirmative answer for all pseudo reflections groups for the alternating groups n 3 4 5 and for any finite group when n 3 and mathsf k is algebraically closed covering and generalizing all previously known cases alternative proofs are given for the complex field and for all pseudo reflections groups in the later case an effective algorithm of finding the weyl generators is described

two geometric character formulas for reductive lie groups in this paper we prove two formulas for the characters of representations of reductive groups both express the character of a representation in terms of the same geometric data attached to when specialized to the case of a compact lie group one of them reduces to kirillov s character formula in the compact case and the other to an application of the atiyah bott fixed point formula to the borel weil realization of the representation to set the stage let us first recall the borel weil theorem for a connected compact lie group gr for simplicity we assume that gr is simply connected we let gr denote its lie algebra and g c r gr the complexified lie algebra via the adjoint action gr operates on igr the space of all r linear functions gr ir every gr orbit in igr coadjoint orbit for short carries a canonical grinvariant symplectic structure the orbit is said to be integral if some or equivalently any exponentiates to a character e of the isotropy subgroup gr in that case the one dimensional representation spaces of the characters e fit together into a gr equivariant real algebraic hermitian line bundle l the pair l carries a unique positive polarization a gr invariant complex structure on the manifold and the structure of a gr equivariant holomorphic line bundle on l positive in the sense of algebraic geometry the hypothesis of simple connectivity ensures that the square root k of the canonical bundle k exists as a gr equivariant holomorphic line bundle according to the borel weil theorem the natural action of gr on the space of holomorphic l valued half forms h0 o k l is irreducible moreover the association h0 o k l 1 1

a non nested galerkin multi grid method for solving linear and nonlinear solid mechanics problems abstract a non nested galerkin multi grid method is proposed in this paper for the iterative solution of general linear and nonlinear problems arising in solid mechanics as the galerkin strategy has been fully adopted in the standard multi grid method for the generation of coarse mesh equations and only the geometrical information for coarse meshes is utilised the resulting approach is relatively easy to incorporate into existing solution procedures and is particularly suitable for implementation in materially non linear cases furthermore the capability of using non nested unstructured meshes provides great flexibility to handle any problem with complex geometry key issues involved in the theoretical and practical aspects for the construction of transfer operators and efficient implementation of galerkin coarse grid approximation are extensively addressed and application to geometrically nonlinear problems is also discussed finally a selected set of numerical examples which cover major classes of problems in solid mechanics including linear elastic materially nonlinear elasto plastic in small strain geometrically nonlinear elastic and geometrically nonlinear elasto plastic problems and also include both symmetric and unsymmetric cases are presented to provide an assessment of the performance of the proposed approach

the invariant holonomic system on a semisimple lie algebra on etudie la theorie geometrique des representations du groupe de weyl du point de vue des systemes holonomes sur des algebres de lie

conservative numerical schemes for the vlasov equation abstract a new scheme for solving the vlasov equation using a phase space grid is proposed the algorithm is based on the conservation of the flux of particles and the distribution function is reconstructed using various techniques that allow control of spurious oscillations or preservation of the positivity several numerical results are presented in two and four dimensional phase space and the scheme is compared with the semilagrangian method this method is almost as accurate as the semilagrangian one and the local reconstruction technique is well suited for parallel computation

macdonald positivity via the harish chandra d module using the harish chandra d module we give a proof of haiman s theorem on the positivity of macdonald polynomials ginzburg s work on the connection between this d module and the isospectral commuting variety is fundamental to this approach

on the discrete dynamic nature of the conjugate gradient method the primary objective of this technical note is to establish an equivalence between the preconditioned conjugate gradient pcg algorithm and a special central difference based dr procedure thereby revealing a discrete dynamic nature of the cg iterative procedure this may therefore provide an alterative viewpoint to gain a further understanding of the cg method and its variants

a numerical method for solving the vlasov poisson equation based on the conservative ido scheme we have applied the conservative form of the interpolated differential operator ido cf scheme in order to solve the vlasov poisson equation which is one of the multi moment schemes through numerical tests of the nonlinear landau damping and two stream instability we compared the present scheme with other schemes such as the spline and cip ones we mainly investigated the conservation property of the l1 norm energy entropy and phase space area for each scheme and demonstrated that the ido cf scheme is capable of performing stable long time scale simulation while maintaining high accuracy the scheme is based on an eulerian approach and it can thus be directly used for fokker planck high dimensional vlasov poisson and also guiding center drift simulations aiming at particular problems of plasma physics the benchmark tests for such simulations have shown that the ido cf scheme is superior in keeping the conservation properties without causing serious phase error

on polarizations in invariant theory abstract given a reductive algebraic group g and a finite dimensional algebraic g module v we study how close is the algebra of g invariant polynomials on v n to the subalgebra generated by polarizations of g invariant polynomials on v we address this problem in a more general setting of g actions on arbitrary affine varieties

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

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

vlasov simulations using velocity scaled hermite representations the efficiency accuracy and stability of two different pseudo spectral methods using scaled hermite basis and weight functions applied to the nonlinear vlasov poisson equations in one dimension 1d 1v are explored and compared a variable velocity scaleuis introduced into the hermite basis and is shown to yield orders of magnitude reduction in errors as compared to linear kinetic theory with no increase in workload a set of fourier hermite coefficients representing a periodic gaussian distribution function are advanced through time with ano t2 accurate splitting method within this splitting scheme the advection and acceleration terms of the vlasov equation are solved separately using ano t4 accurate runge kutta method the asymmetrically weighted aw hermite basis which has been used previously by many authors conserves particles and momentum exactly and total energy too t3 however the aw hermite method doesnotconserve the square integral of the distribution and is in fact numerically unstable the symmetrically weighted sw hermite algorithm applied here to the vlasov system for the first time can either conserve particles and energy fornueven or momentum fornuodd as t 0 wherenuis the largest hermite mode number the sw hermite method conserves the square integral of the distribution and therefore remains numerically stable in addition careful velocity scaling improves the conservation properties of the sw hermite method damping and growth rates oscillation frequencies e field saturation levels and phase space evolution are seen to be qualitatively correct during simulations relative errors with respect to linear landau damping and linear bump on tail instability are shown to be less than 1 using only 64 velocity scaled hermite functions comparisons to particle in cell pic simulations show that as the number of particles increases to more than 106 the pic solutions converge to scaled sw hermite solutions that were found in only 1 10 of the run time the sw hermite method with velocity scaling is well suited to kinetic simulations of warm plasmas

finite dimensional representations of invariant differential operators ii abstract in part i of this paper g w schwarz finite dimensional representations of invariant differential operators j algebra 258 2002 160 204 we considered the representation theory of the algebra b d g g where g sl 3 c and d g g denotes the algebra of g invariant polynomial differential operators on the lie algebra g of g we also considered the representation theory of the subalgebra a of b where a is generated by the invariant functions o g g b and the invariant constant coefficient differential operators s g g b among other things we found that the finite dimensional representations of a and b are completely reducible and we could reduce the study of the finite dimensional irreducible representations of b to those of a irreducible finite dimensional representations of a are quotients of verma modules we found sufficient conditions for the irreducible quotients of verma modules to be finite dimensional and we conjectured that these sufficient conditions are also necessary in this paper we establish the conjecture giving a complete classification of the finite dimensional representations of a and b

microlocalization of rational cherednik algebras we construct a microlocalization of the rational cherednik algebras h of type sn this is achieved by a quantization of the hilbert scheme hilb c2 of n points in c2 we then prove the equivalence of the category of h modules and that of modules over its microlocalization under certain conditions on the parameter

discrete thermal element modelling of heat conduction in particle systems basic formulations this paper proposes a novel numerical methodology termed the discrete thermal element method dtem for the effective modelling of heat conduction in systems comprising a large number of circular particles in 2d cases based on an existing analytical integral solution for the temperature distribution over a circular domain subjected to the neumann boundary condition a linear algebraic system of thermal conductivity equations for each particle is derived in terms of the average temperatures and the resultant fluxes at the contact zones with its neighboring particles thus each particle is treated as an individual element with the number of temperature unknowns equal to the number of particles that it is in contact with the element thermal conductivity matrix can be very effectively evaluated and is entirely dependent on the characteristics of the contact zones including the contact positions and contact angles this new element shares the same form and properties with its conventional thermal finite element counterpart in particular the whole solution procedure can follow exactly the same steps as those involved in the finite element analysis unlike finite elements or other modern numerical techniques however no discretization errors are involved in the discrete thermal elements the modelling error mainly stems from the assumption made about the heat flux distribution within the contact zones based on some theoretical work an enhanced version is suggested to improve the approximation the numerical assessment against finite element results indicates that the basic version of dtem can achieve a very reasonable solution accuracy while the enhanced version further improves the accuracy to a high level in addition thermal resistance phenomena between the contact zones can be readily incorporated into the current modelling framework

a high order time splitting method based on integral deferred correction for semi lagrangian vlasov simulations semi lagrangian schemes with various splitting methods and with different reconstruction interpolation strategies have been applied to kinetic simulations for example the order of spatial accuracy of the algorithms proposed in qiu and christlieb 2010 29 is very high as high as ninth order however the temporal error is dominated by the operator splitting error which is second order for strang splitting it is therefore important to overcome such low order splitting error in order to have numerical algorithms that achieve higher orders of accuracy in both space and time in this paper we propose to use the integral deferred correction idc method to reduce the splitting error specifically the temporal order accuracy is increased by r with each correction loop in the idc framework where r 1 2r 1 2 for coupling the first order splitting and the strang splitting respectively the proposed algorithm is applied to the vlasov poisson system the guiding center model and two dimensional incompressible flow simulations in the vorticity stream function formulation we show numerically that the idc procedure can automatically increase the order of accuracy in time we also investigate numerical stability of the proposed algorithm via performing fourier analysis to a linear model problem

a method for the automatic evaluation of the dynamic relaxation parameters abstract dynamic relaxation a vector iteration method which belongs to the family of methods under the title of three term recursive formulae is described with viscous and kinetic damping an automatic procedure is developed for the evaluation of the iteration parameters thus avoiding any trial run or any eigenvalue analysis of the modified stiffness matrix starting values for the maximum eigenvalue could be obtained from the gershgorin bound and for the minimum eigenvalue any positive number less than the estimated maximum eigenvalue the method is applied to geometrically and material nonlinear problems and comparisons are made with an improved conjugate gradient method and a direct stiffness method

extending tensors on polar manifolds given a polar action on a riemannian manifold we prove surjectivity of restriction to the section for general invariant tensors and a sharper surjectivity result in the special case of metrics these are related to the chevalley restriction theorem and michor s basic forms theorem the proofs rely on results in the invariant theory of finite reflection groups and symmetric pairs some of which may be of independent interest

hamiltonian reduction and nearby cycles for mirabolic d modules abstract we study holonomic d modules on sl n c c n called mirabolic modules analogous to lusztig s character sheaves we describe the supports of simple mirabolic modules we show that a mirabolic module is killed by the functor of hamiltonian reduction from the category of mirabolic modules to the category of representations of the trigonometric cherednik algebra if and only if the characteristic variety of the module is contained in the unstable locus we introduce an analogue of verdier s specialization functor for representations of cherednik algebras which agrees on category o with the restriction functor of bezrukavnikov and etingof in type a we also consider a verdier specialization functor on mirabolic d modules we show that hamiltonian reduction intertwines specialization functors on mirabolic d modules with the corresponding functors on representations of the cherednik algebra this allows us to apply known hodge theoretic purity results for nearby cycles in the setting considered by bezrukavnikov and etingof

on the orbit method and the homomorphism of harish chandra abstract let g be a complex semisimple lie algebra with adjoint group g let u g be the enveloping algebra of g following an idea of j dixmier sur la methode des orbites in proceedings de la conference non commutative harmonic analysis marseille luminy 1978 lecture notes in mathematics vol 728 springer verlag new york berlin we construct a bijection from the set of regular coadjoint orbits onto the set of minimal primitive ideals in u g without using the notion of polarization see j dixmier algebres enveloppantes gauthier villars paris 1974 this is a partial answer to the problem posed by j dixmier in the former reference

arbitrary order time accurate semi lagrangian spectral approximations of the vlasov poisson system abstract the vlasov poisson system modeling the evolution of non collisional plasmas in the electrostatic limit is approximated by a semi lagrangian technique spectral methods of periodic type are implemented through a collocation approach groups of particles are represented by the fourier lagrangian basis and evolve for a single timestep along a high order accurate representation of the local characteristic lines the time advancing technique is based on truncated taylor series that can be in principle of any order of accuracy a variant is obtained by coupling the phase space discretization with high order accurate backward differentiation formulas bdf at each timestep particle displacements are reinterpolated and expressed in the original basis to guarantee the order of accuracy in all the variables at relatively low costs thus these techniques combine the excellent features of spectral approximations with high order time integration the resulting method has excellent conservation properties indeed it can be proven that the total number of particles proportional to the total mass and charge is conserved up to the machine precision series of numerical experiments are performed in order to assess the real performance in particular comparisons with standard benchmarks are examined

preconditioning techniques for large linear systems a survey this article surveys preconditioning techniques for the iterative solution of large linear systems with a focus on algebraic methods suitable for general sparse matrices covered topics include progress in incomplete factorization methods sparse approximate inverses reorderings parallelization issues and block and multilevel extensions some of the challenges ahead are also discussed an extensive bibliography completes the paper

on categories of equivariant d modules abstract let x be a variety with an action by an algebraic group g in this paper we discuss various properties of g equivariant d modules on x such as the decompositions of their global sections as representations of g when g is reductive and descriptions of the categories that they form when g acts on x with finitely many orbits the category of equivariant d modules is equivalent to the category of finite dimensional representations of a finite quiver with relations we describe explicitly these categories for irreducible g modules x that are spherical varieties and show that in such cases the quivers are almost always representation finite i e with finitely many indecomposable representations

bicontinuity of the dixmier map let g be a solvable lie algebra over c and let v be its algebraic adjoint group the dixmier map i is a bijective and continuous correspondence between the set g of coadjoint orbits and the set prim g of all the primitive ideals of u g actually the map i which is the infinitesimal analog of the kirillov orbits method k ak has been defined by j dixmier dil di2j the continuity of i has been proved by n berline conze m vergne and m duflo cv cd the surjectivity by m duflo du du2 and the injectivity by r rentschler r bgr dixmier s book di3 contains proofs of all these results we prove that i is bicontinuous i e that the inverse of i is continuous

a legendre fourier spectral method with exact conservation laws for the vlasov poisson system we present the design and implementation of an l 2 stable spectral method for the discretization of the vlasov poisson model of a collisionless plasma in one space and velocity dimension the velocity and space dependence of the vlasov equation are resolved through a truncated spectral expansion based on legendre and fourier basis functions respectively the poisson equation which is coupled to the vlasov equation is also resolved through a fourier expansion the resulting system of ordinary differential equation is discretized by the implicit second order accurate crank nicolson time discretization the non linear dependence between the vlasov and poisson equations is iteratively solved at any time cycle by a jacobian free newton krylov method in this work we analyze the structure of the main conservation laws of the resulting legendre fourier model e g mass momentum and energy and prove that they are exactly satisfied in the semi discrete and discrete setting the l 2 stability of the method is ensured by discretizing the boundary conditions of the distribution function at the boundaries of the velocity domain by a suitable penalty term the impact of the penalty term on the conservation properties is investigated theoretically and numerically an implementation of the penalty term that does not affect the conservation of mass momentum and energy is also proposed and studied a collisional term is introduced in the discrete model to control the filamentation effect but does not affect the conservation properties of the system numerical results on a set of standard test problems illustrate the performance of the method

on the quotient ring by diagonal invariants for a finite coxeter group w and its reflection representation we find the character and hilbert series for a quotient ring of by an ideal containing the w invariant polynomials without constant term this confirms conjectures of haiman

semisimple orbital integrals on the symplectic space for a real reductive dual pair we prove a weyl harish chandra integration formula for the action of a reductive dual pair on the corresponding symplectic space w as an intermediate step we introduce a notion of a cartan subspace and a notion of an almost semisimple element in the symplectic space w we prove that the almost semisimple elements are dense in w finally we provide estimates for the orbital integrals associated with the different cartan subspaces in w

irregularity of hypergeometric systems via slopes along coordinate subspaces we study the irregularity sheaves attached to the a hypergeometric d module ma introduced by gel fand et al where a z d n is pointed of full rank and c more precisely we investigate the slopes of this module along coordinate subspaces in the process we describe the associated graded ring to a positive semigroup ring for a filtration defined by an arbitrary weight vector l on torus equivariant generators to this end we introduce the a l umbrella a simplicial complex determined by a and l and identify its facets with the components of the associated graded ring we then establish a correspondence between the full a l umbrella and the components of the l characteristic variety of ma we compute in combinatorial terms the multiplicities of these components in the l characteristic cycle of the associated euler koszul complex identifying them with certain intersection multiplicities we deduce from this that slopes of ma are combinatorial independent of and in one to one correspondence with jumps of the a l umbrella this confirms a conjecture of sturmfels and gives a converse of a theorem of hotta ma is regular if and only if a defines a projective variety 1991 mathematics subject classification 13n10 14m25 16w70

invariant differential operators and an homomorphism of harish chandra let g be a reductive complex lie algebra with adjoint group g cartan subalgebra h and weyl group w then g acts naturally on the algebra of polynomial functions o g and hence on the ring of differential operators with polynomial coefficients d g similarly w acts on h and hence on d h in hc2 harish chandra defined an algebra homomorphism d g d h recently wallach proved that if g has no factors of type e6 e7 or e8 then this map is surjective wa theorem 3 1 the significance of wallach s result is that it enables him to give an easy proof of an important theorem of harish chandra about invariant distributions and to give an elegant new approach to the springer correspondence the main aim of this paper is to give an elementary proof of wa theorem 3 1 that also works for all reductive lie algebras set

a unipotent support for irreducible representations 1 1 let g f be the group of fq rational points of a connected reductive algebraic group g defined over a finite field f throughout this paper we shall assume that the characteristic of the ground field is large the following is part of problem ii in l9 let p be an irreducible complex character of g f show that there is a unique umpotent class c in g which has the property that cga c cfqi p g 0 and has maximal dimension among classes with this property one of the results of this paper is a solution of this problem we also prove the following refinement if g e gf is such that tr g p 0 then the unipotent part of g lies in c or in a conjugacy class of dimension dim c we call c the unipotent support of p another map from the set of irreducible representations of gf to unipotent classes in g has been defined in l8 13 43 this was the composition of two maps one from irreducible representations of gf to a set of parameters in the langlands dual of g and one from this set of parameters to unipotent classes in g another conjectural definition of this map which did not pass through the langlands dual has been proposed by kawanaka kl k2 k3 in terms of generalized gelfand graev representations in analogy with real lie groups he uses the name wave front set for the unipotent class attached to a representation one of the results of this paper is a proof see section 11 of kawanaka s conjecture that his wave front set is well defined and that it coincides with the one defined in l8 this has been known earlier from the work of kawanaka in certain special cases we also show that the unipotent support of an irreducible representation coincides with the wave front set of the dual representation

multi dimensional fully implicit spectral method for the vlasov maxwell equations with exact conservation laws in discrete form a spectral method for the numerical solution of the multi dimensional vlasov maxwell equations is presented the plasma distribution function is expanded in fourier for the spatial part and hermite for the velocity part basis functions leading to a truncated system of ordinary differential equations for the expansion coefficients moments that is discretized with an implicit second order accurate crank nicolson time discretization the discrete non linear system is solved with a preconditioned jacobian free newton krylov method it is shown analytically that the fourier hermite method features exact conservation laws for total mass momentum and energy in discrete form standard tests involving plasma waves and the whistler instability confirm the validity of the conservation laws numerically the whistler instability test also shows that we can step over the fastest time scale in the system without incurring in numerical instabilities some preconditioning strategies are presented showing that the number of linear iterations of the krylov solver can be drastically reduced and a significant gain in performance can be obtained

rational cherednik algebras and hilbert schemes ii representations and sheaves let h c be the rational cherednik algebra of type a n 1 with spherical subalgebra u c e h c e then u c is filtered by order of differential operators with associated graded ring gr u c mathbb c mathfrac h mathfrac h w where is the n th symmetric group using the z algebra construction from gs it is also possible to associate to a filtered h c or u c module hat m on the hilbert scheme hilb n using this technique we study the representation theory of u c and h c and we relate it to hilb n and to the resolution of singularities hilb n mathfrac h mathfrac h w for example we prove the following if c 1 n so that l c triv is the unique one dimensional simple h c module then hat e l c triv mathcal o z n where z n 1 0 is the punctual hilbert scheme if c 1 n k for k in mathbb n then under a canonical filtration on the finite dimensional module l c triv gr e l c triv has a natural bigraded structure that coincides with that on h 0 z n mathscr l k where mathscr l mathcal o hilb n 1 this confirms conjectures of berest etingof and ginzburg beg2 conjectures 7 2 7 3 under mild restrictions on c the characteristic cycle of hat e c equals sum k z where k are kostka numbers and the z are known irreducible components of 1 mathfrak h w

a capelli harish chandra homomorphism for a real reductive dual pair the capelli identities define a homomorphism c from the center of the universal enveloping algebra of the larger group to the center of the universal enveloping algebra of the smaller group in terms of the harish chandra isomorphism this map involves a shift we view a dual pair as a lie supergroup and offer a construction of the homomorphism c based solely on the harish chandra s radial component maps thus we provide a geometric interpretation of the shift

invariant eigendistributions on a semisimple lie algebra and homology classes on the conormal variety ii representations of weyl groups abstract we construct representations of w in the homology of certain subvarieties of the cotangent bundle of the flag manifold of g in particular in the homology of the conormal variety and of springer varieties using the integral formula we prove that in top degree the former is isomorphic with the natural representation of w on coherent families of invariant eigendistributions and decomposes into a sum of the latter according to the decomposition of the nilpotent variety into g0 orbits as consequences we obtain springer s theorem on irreducible representations of weyl groups a formula for irreducible characters of g0 as integrals over characteristic varieties of d modules an identification of the harmonic polynomials occuring in the asymptotic expansion at zero of invariant eigendistributions as homology classes on the flag manifold a formula for the fourier transforms of nilpotent g0 orbits and a proof of a conjecture of joseph on the characteristic polynomials of intersections of nilpotent g0 orbits with n0

symplectic reflection algebras calogero moser space and deformed harish chandra homomorphism to any finite group sp v of automorphisms of a symplectic vector space v we associate a new multi parameter deformation h of the algebra v smash product of with the polynomial algebra on v the parameter runs over points of r where r number of conjugacy classes of symplectic reflections in the algebra h called a symplectic reflection algebra is related to the coordinate ring of a poisson deformation of the quotient singularity v this leads to a symplectic analogue of mckay correspondence which is most complete in case of wreath products if is the weyl group of a root system in a vector space and v then the algebras h are certain rational degenerations of the double affine hecke algebra introduced earlier by cherednik

semi simplicity of invariant holonomic systems on a reductive lie algebra let inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i be a reductive complex lie algebra with adjoint group g let g act on the ring of differential operators inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i via the adjoint action and write inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 03i inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 04i for the differential of this action fix inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 05i generalizing work of hotta and kashiwara we prove that the invariant holonomic system inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 06i is semisimple the simple summands of inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 07i are parametrized by the irreducible representations of w the stabilizer of in the weyl group consequently the subcategory generated by inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 08i is equivalent to the category of finite dimensional representations of w

characteristic cycles for the loop grassmannian and nilpotent orbits x which is a linear combination of closures of conormal bundles to submanifolds of x intuitively the microlocal multiplicities c me asurethesingularity of at in settings related to representation theory a group g acts on x is g equivariant and the microlocal multiplicities play a significant but only partially understood role in representation theory see ro sv abv and kasa e g in this paper we compute microlocal multiplicities for certain cases of interest in representation theory let g be a connected reductive group with loop group lg and let p bethesubgroup of lg consisting of loops with positive fourier coefficients then p orbits on theloop grassmannian lg p correspond to the irreducible representations l of thedual re ductivegroup g for dominant weights and of a torus of

d modules bernstein sato polynomials and f invariants of direct summands abstract we study the structure of d modules over a ring r which is a direct summand of a polynomial or a power series ring s with coefficients over a field we relate properties of d modules over r to d modules over s we show that the localization r f and the local cohomology module h i i r have finite length as d modules over r furthermore we show the existence of the bernstein sato polynomial for elements in r in positive characteristic we use this relation between d modules over r and s to show that the set of f jumping numbers of an ideal i r is contained in the set of f jumping numbers of its extension in s as a consequence the f jumping numbers of i in r form a discrete set of rational numbers we also relate the bernstein sato polynomial in r with the f thresholds and the f jumping numbers in r

invariant distributions supported on the nilpotent cone of a semisimple lie algebra let g be a semisimple complex lie algebra with adjoint group g and 1 g be the algebra of differential operators with polynomial coefficients on g if go is a real form of g we give the decomposition of the semisimple 1 g g module of invariant distributions on go supported on the nilpotent cone

closed formulae for a b function associated with a weighted homogeneous isolated complete intersection singularity abstract following ideas of b malgrange we give explicit formulae for a b function associated with a weighted homogeneous polynomial morphism which defines an isolated complete intersection singularity

outflow boundary conditions for the fourier transformed three dimensional vlasov maxwell system a problem with the solution of the vlasov equation is its tendency to become filamented oscillatory in velocity space which in numerical simulations can give rise to unphysical oscillations and recurrence effects we present a three dimensional vlasov maxwell solver three spatial and velocity dimensions plus time in which the vlasov equation is fourier transformed in velocity space and the resulting equations solved numerically by designing absorbing outflow boundary conditions in the fourier transformed velocity space the highest fourier modes in velocity space are removed from the numerical solution this introduces a dissipative effect in velocity space and the numerical recurrence effect is strongly reduced the well posedness of the boundary conditions is proved analytically while the stability of the numerical implementation is assessed by long time numerical simulations well known wave modes in magnetized plasmas are shown to be reproduced by the numerical scheme

invariant differential operators on a reductive lie algebra and weyl group representations let v be a finite dimensional vector space and let g be a subgroup of gl v set d v equal to the algebra of differential operators on v with polynomial coefficients and d v g equal to the g invariants in d v if g is a reductive lie algebra over c then t c g is a cartan subgroup of g and if g is the adjoint group of g then w is the weyl group of g harish chandra introduced an algebra homomorphism 5 of d g g to d t w h3 a is given by the obvious restriction mapping on the subalgebra of invariant polynomials and on the invariant constant coefficient differential operators and ker is

some combinatorial identities related to commuting varieties and hilbert schemes in this article we explore some of the combinatorial consequences of recent results relating the isospectral commuting variety and the hilbert scheme of points in the plane