conormal varieties on the cominuscule grassmannian ii let x w be a schubert subvariety of a cominuscule grassmannian x and let mu t x rightarrow mathcal n be the springer map from the cotangent bundle of x to the nilpotent cone mathcal n in this paper we construct a resolution of singularities for the conormal variety t xx w of x w in x further for x the usual or symplectic grassmannian we compute a system of equations defining t xx w as a subvariety of the cotangent bundle t x set theoretically this also yields a system of defining equations for the corresponding orbital varieties mu t xx w inspired by the system of defining equations we conjecture a type independent equality namely t xx w pi 1 x w cap mu 1 mu t xx w the set theoretic version of this conjecture follows from this work and previous work for any cominuscule grassmannian of type a b or c

some small unipotent representations of indefinite orthogonal groups abstract we study a family of small unitary representations of indefinite orthogonal groups these representations arise as analytic continuations of the discrete series and were studied extensively by knapp in k3 we complete knapp s analysis by proving that they are irreducible in order to do so we prove that the representations are unipotent and have irreducible associated cycles in which all multiplicities are exactly one moreover we prove that the k type structure of each representation matches up to a shift the k type structure of the ring of functions on the closure a nilpotent k c orbit on p

the expression of the generalized inverse of the perturbed operator under type i perturbation in hilbert spaces let h1 h2 be two hilbert spaces over the complex field c and let t h1 h2 be a bounded linear operator with the generalized inverse t let t t t be a bounded linear operator with t t 1 suppose that dim ker t dim ker t or r t r t 0 then t has the generalized inverse t i i t t 1 i t t i t i t t 1 1 1 tt 1t x i tt tt tt i tt 1 i tt ast 1tt i tt i 1 with t t 1 t t this result gives an analogue of theorem 3 9 of m z nashed generalized inverses and applications academic press new york 1976 in hilbert spaces

on the hypersurface orbital varieties of sl n c abstract we study the structure of hypersurface orbital varieties of sl n c those that are hypersurfaces in the nilradical of some parabolic subalgebra and how information about this structure is encoded in the standard young tableau associated to it by the robinson schensted algorithm we present a conjecture for the exact form of the unique non linear defining equations of hypersurface orbital varieties and proofs of the conjecture in certain cases

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

strong and weak error estimates for elliptic partial differential equations with random coefficients we consider the problem of numerically approximating the solution of an elliptic partial differential equation with random coefficients and homogeneous dirichlet boundary conditions we focus on the case of a lognormal coefficient and deal with the lack of uniform coercivity and uniform boundedness with respect to the randomness this model is frequently used in hydrogeology we approximate this coefficient by a finite dimensional noise using a truncated karhunen loeve expansion we give estimates of the corresponding error on the solution both a strong error estimate and a weak error estimate that is an estimate of the error commited on the law of the solution we obtain a weak rate of convergence which is twice the strong one in addition we give a complete error estimate for the stochastic collocation method in this case where neither coercivity nor boundedness is stochastically uniform to conclude we apply these results of strong and weak convergence to two classical cases of covariance kernel choices the case of an exponential covariance kernel on a box and the case of an analytic covariance kernel yielding explicit weak and strong convergence rates

picard lefschetz theory for the coadjoint quotient of a semisimple lie algebra the paper develops a picard lefschetz theory for the coadjoint quotient of a semisimple lie algebra and analyzes the resulting monodromy representation of the weyl group

proof of the fundamental gap conjecture we prove the fundamental gap conjecture which states that the difference between the first two dirichlet eigenvalues the spec tral gap of a schrodinger operator with convex potential and dirichlet boundary data on a convex domain is bounded below by the spectral gap on an interval of the same diameter with zero potential more generally for an arbitrary smooth potential in higher dimensions our proof gives both a sharp lower bound for the spectral gap and a sharp modulus of concavity for the logarithm of the first eigenfunction in terms of the diameter of the domain and a modulus of convexity for the potential

the exotic robinson schensted correspondence abstract we study the action of the symplectic group on pairs of a vector and a flag considering the irreducible components of the conormal variety we obtain an exotic analogue of the robinson schensted correspondence conjecturally the resulting cells are related to exotic character sheaves

on orbital variety closures in sln iii geometric properties abstract this is the third paper in the series here we define a few combinatorial orders on young tableaux the first order is obtained from the induced duflo order by the extension with the help of vogan t procedure we call it the duflo vogan order the second order is obtained from the generalization of spaltenstein s construction by consideration of an orbital variety as a double chain of nilpotent orbits we call it the chain order again we use vogan s t procedure however this time to restrict the chain order we call it the vogan chain order the order on young tableaux defined by the inclusion of orbital variety closures is called the geometric order and the order on young tableaux defined by inverse inclusion of primitive ideals is called the algebraic order we get the following relations between the orders the duflo vogan order is an extension of the induced duflo order the algebraic order is an extension of the duflo vogan order the geometric order is an extension of the algebraic order the vogan chain order is an extension of the geometric order and finally the chain order is an extension of the vogan chain order the computations show that the duflo vogan and the vogan chain orders coincide on sl n for n 9 and in n 10 there is one case up to t procedure and transposition where the chain vogan order is a proper extension of the duflo vogan order in this only case the algebraic order coincides with the vogan chain order these computations permit us to conjecture that in sl n the algebraic order coincides with the geometric order as well we conjecture that the combinatorics of both the inclusions on primitive ideals and on orbital variety closures is defined by the vogan chain order on young tableaux

non integral representation theory of even multiplicity finite w algebras abstract we complete the classification of the finite dimensional irreducible representations of finite w algebras associated to even multiplicity nilpotent elements in classical lie algebras this extends earlier work in brown and goodwin 2013 bg1 where this classification is determined for such representations of integral central character

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

finite element galerkin approximation of the eigenvalues and eigenvectors of selfadjoint problems abstract this paper establishes some refined estimates for the approximation of the eigenvalues and eigenvectors of selfadjoint eigenvalue problems by finite element or more generally galerkin methods suppose lambda is an eigenvalue of multiplicity q of a selfajoint problem and let m lambda denote the space of eigenvectors corresponding to lambda keywords hilbert space

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

quantum weyl reciprocity and tilting modules quantum weyl reciprocity relates the representation theory of hecke algebras of type a with that of q schur algebras this paper establishes that weyl reciprocity holds integrally i e over the ring q q 1 of laurent polynomials and that it behaves well under base change a key ingredient in our approach involves the theory of tilting modules for q schur algebras new results obtained in that direction include an explicit determination of the ringel dual algebra of a q schur algebra in all cases in particular in the most interesting situation the ringel dual identifies with a natural quotient algebra of the hecke algebra

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

left cells in the affine weyl group of typec4 we find a representative set of left cells of the affine weyl groupwaof typec4as well as its left cell graphs by applying an algorithm this algorithm was designed in my previous paper tohoku j math 46 1994 105 124 it is reformulated and improved in a more efficient form here these representatives of left cells are presented as the vertices of so called essential graphs so that the generalized invariants of left cells ofwaare actually described explicitly the latter almost characterize the left cells ofwa some comments and conjectures are proposed on cells of affine weyl groups mostly for the case of typecl l 2

analysis of quasi monte carlo methods for elliptic eigenvalue problems with stochastic coefficients we consider the forward problem of uncertainty quantification for the generalised dirichlet eigenvalue problem for a coercive second order partial differential operator with random coefficients motivated by problems in structural mechanics photonic crystals and neutron diffusion the pde coefficients are assumed to be uniformly bounded random fields represented as infinite series parametrised by uniformly distributed i i d random variables the expectation of the fundamental eigenvalue of this problem is computed by a truncating the infinite series which define the coefficients b approximating the resulting truncated problem using lowest order conforming finite elements and a sparse matrix eigenvalue solver and c approximating the resulting finite but high dimensional integral by a randomly shifted quasi monte carlo lattice rule with specially chosen generating vector we prove error estimates for the combined error which depend on the truncation dimension s the finite element mesh diameter h and the number of quasi monte carlo samples n under suitable regularity assumptions our bounds are of the particular form mathcal o h 2 n 1 delta where delta 0 is arbitrary and the hidden constant is independent of the truncation dimension which needs to grow as h rightarrow 0 and n rightarrow infty as for the analogous pde source problem the conditions under which our error bounds hold depend on a parameter p in 0 1 representing the summability of the terms in the series expansions of the coefficients although the eigenvalue problem is nonlinear which means it is generally considered harder than the source problem in almost all cases p ne 1 we obtain error bounds that converge at the same rate as the corresponding rate for the source problem the proof involves a detailed study of the regularity of the fundamental eigenvalue as a function of the random parameters as a key intermediate result in the analysis we prove that the spectral gap between the fundamental and the second eigenvalues is uniformly positive over all realisations of the random problem

orbital varieties in slnand the smith conjecture abstract let o be a nilpotent orbit in the lie algebra s l n c that is a class of nilpotent elements for conjugation bysln c let v be an orbital variety contained in o andpbe the largest parabolic subgroup ofsln c stabilizing v the smith conjecture asserts that v contains a denseporbit this is shown to fail in general and further those nilpotent orbits for which such a dense orbit exists are determined

induction and restriction of kazhdan lusztig cells abstract barbasch and vogan gave a beautiful rule for restricting and inducing kazhdan lusztig representations of weyl groups in this paper we show that this rule implies and generalizes the littlewood richardson rule for decomposing outer products of representations of the symmetric groups a new recursive rule for computing characters of arbitrary coxeter groups follows another application is a generalization of garsia remmel s algorithm for decomposing certain tensor products of symmetric groups representations

richardson orbits for real classical groups abstract for classical real lie groups we compute the annihilators and associated varieties of the derived functor modules cohomologically induced from the trivial representation generalizing the standard terminology for complex groups the nilpotent orbits that arise as such associated varieties are called richardson orbits we show that every complex special orbit has a real form which is richardson as a consequence of the annihilator calculations we give many new infinite families of simple highest weight modules with irreducible associated varieties finally we sketch the analogous computations for singular derived functor modules in the weakly fair range and as an application outline a method to detect non normality of complex nilpotent orbit closures

on the continuity of generalized inverses of linear operators in hilbert spaces abstract let x and y be hilbert spaces and let t x y be a bounded linear operator with closed range we study the continuity problem of the generalized inverse of t and related least squares solutions to the operator equation tx y

primitive ideals and representations certain theorems of joseph and king relating primitive ideals to characters in category o are extended to the case of arbitrary g modules with finite dimensional n cohomology a derived functor construction analogous to a construction of zuckerman in the case of a nilpotent lie algebra is used to extend the results applications to harish chandra modules are also given

on the perturbation of the least squares solutions in hilbert spaces abstract let h1 and h2 be hilbert spaces and let t h1 h2 be a bounded linear operator with closed range we present some results of the perturbation analysis for the least squares solutions to the operator equation tx y

the homomorphisms between scalar generalized verma modules associated to maximal parabolic subalgebras let g be a finite dimensional simple lie algebra over c we classify the homomorphisms between g modules induced from one dimensional modules of maximal parabolic subalgebras 1

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

characters dual pairs and unipotent representations abstract we lift distribution characters of irreducible unitary representations of classical groups from the group to the lie algebra via the cayley transform then a specific class of these characters admits fourier transform supported on the closure of a single nilpotent coadjoint orbit we calculate also the wave front set of the most singular low rank representations

adaptive finite element methods for computing band gaps in photonic crystals in this paper we propose and analyse adaptive finite element methods for computing the band structure of 2d periodic photonic crystals the problem can be reduced to the computation of the discrete spectra of each member of a family of periodic hermitian eigenvalue problems on a unit cell parametrised by a two dimensional parameter the quasimomentum these eigenvalue problems involve non coercive elliptic operators with generally discontinuous coefficients and are solved by adaptive finite elements we propose an error estimator of residual type and show it is reliable and efficient for each eigenvalue problem in the family in particular we prove that if the error estimator converges to zero then the distance of the computed eigenfunction from the true eigenspace also converges to zero and the computed eigenvalue converges to a true eigenvalue with double the rate we also prove that if the distance of a computed sequence of approximate eigenfunctions from the true eigenspace approaches zero then so must the error estimator the results hold for eigenvalues of any multiplicity we illustrate the benefits of the resulting adaptive method in practice both for fully periodic structures and also for the computation of eigenvalues in the band gap of structures with defect using the supercell method

perturbation of generalized inverses of linear operators in hilbert spaces abstract let x and y be hilbert spaces and let t x y be a bounded linear operator with closed range in this paper we present some new results of the perturbation analysis on the generalized inverse of t and the related least squares solution to the operator equation tx y

unipotent invariant matrices abstract we describe the variety of fixed points of a unipotent operator acting on the space of matrices we compute the determinant and the rank of a generic symmetric or anti symmetric matrix in the fixed variety yielding information about the generic singular locus of the corresponding bilinear form

a partial order on littlewood richardson tableaux abstract for a nilpotent linear transformation f v v of a vector space v of jordan type 1 l i e the sizes of the jordan blocks equal to the conjugate of let x d be the set of f stable subspaces of dimension d x d is partitioned indexed by littlewood richardson tableaux lr tableau t of shapes s with d each of which corresponds to the set s v t consisting of w x d such that the dimension of the vector space f r 1 v f t 1 w f r v f t 1 w f r 1 v f t w is equal to the number of cells squares in the rth row of t filled with the letter t for all r t 1 we define a partial order on the set lr d of lr tableaux of shapes s with d by t t if s v t is contained in the closure of s v t in the grassmaniann g v d we give a certain sufficient condition corollary c and a necessary condition corollary d for t t for this we introduce the notions of generic vectors the generic subspace w t the rational function field f t and the lr tableau t associated to a tableau t and then show the map s v t s v t let 1 w f w where t let 1 is the subtableau of an lr tableau t consisting of the cells filled with the letters greater than 1 is a fiber bundle with fiber isomorphic to a union of schubert cells and s v t is nonsingular theorem a the generic subspace w t of a tableau t is a generic point of s v t whose function field isomorphic to the rational function field f t associated to the lr tableau t theorem b

on the discrete series of generalized stiefel manifolds a study of the discrete series of generalized stiefel manifolds is made using the oscillator representation new infinite families of such discrete series are constructed

the unitary dual for complex classical lie groups in this paper we classify the unitary spectrum of the complex classical groups let g be any complex simply connected semisimple lie group there are two well known ways of constructing unitary representations from lower groups unitary induction from a parabolic subgroup and complementary series a reasonable way to describe the unitary dual is to find a distinguished set of representations for each group such that every other unitary representation is obtained from this set by the above constructions from strictly smaller parabolic subgroups this is entirely similar to the classification of admissible modules where the distinguished representations are the discrete series for integral infinitesimal character such a set is provided by the special unipotent representations introduced in bv1 their existence and unitarity was conjectured earlier by a they were shown to have many interesting properties such as the fact that their annihilator is a maximal primitive ideal from this it was possible to obtain character formulas for them however it is clear that this set is not large enough to describe the entire unitary spectrum for example in type c the metaplectic representation is not special unipotent or even one of the larger class of representations considered in a the definition of special unipotent representations suggests that one should consider representations with maximal primitive ideal and infinitesimal character as small as possible it is not clear which of them should be unitary imposing the condition that the annihilator be a completely prime ideal which is necessary for unitarity does not seem to be very helpful either thus one is left with the task of explicitly computing the g invariant form and ruling out the cases when it is indefinite this is an almost impossible calculation a rough estimate says that one needs to compute signatures on the k types in the principal series which have length ii 2p ql

fast algorithms for component by component construction of rank 1 lattice rules in shift invariant reproducing kernel hilbert spaces we reformulate the original component by component algorithm for rank 1 lattices in a matrix vector notation so as to highlight its structural properties for function spaces similar to a weighted korobov space we derive a technique which has construction cost o snlog n in contrast with the original algorithm which has construction cost o sn 2 herein s is the number of dimensions and n the number of points taken prime in contrast to other approaches to speed up construction our fast algorithm computes exactly the same quantity as the original algorithm the presented algorithm can also be used to construct randomly shifted lattice rules in weighted sobolev spaces

perturbation analysis of the least squares solution in hilbert spaces abstract let h 1 h 2 be two hilbert spaces over the same field and let t h 1 h 2 be a bounded linear operator with closed range we give a complete description of the perturbation analysis for the least squares solution to the operator equation t x y where x h 1 y h 2

on intersections of orbital varieties and components of springer fiber abstract we consider springer fibers and orbital varieties for gl n we show that the irreducible components of an intersection of components of springer fiber are in bijection with the irreducible components of intersection of orbital varieties moreover the corresponding irreducible components in this correspondence have the same codimension finally we give a sufficient condition to have an intersection in codimension one

a method for solving stochastic eigenvalue problems ii we present a novel approach for calculating the stochastic eigenvalues of a linear operator this is an extension of our earlier work on the subject and introduces a significant simplification that reduces the amount of numerical work required when evaluating the polynomial chaos expansions improves the accuracy and enables higher harmonics to be studied examples based on several types of problem involving differential equations and matrices have been analysed and detailed numerical results obtained

the leading coefficient of certain kazhdan lusztig polynomials of the permutation group sn abstract in this paper we show that the leading coefficient y w of certain kazhdan lusztig polynomials p y w of the permutation group s n of 1 2 n are not greater than 1 more precisely we show that the leading coefficients y w are not greater than 1 whenever a y a w where a s n n is the function defined in g lusztig cells in affine weyl groups in algebraic groups and related topics in adv stud pure math vol 6 kinokunia north holland 1985 pp 255 287 see 1 5 for a simple interpretation of the function a

flag varieties and interpretations of young tableau algorithms abstract the conjugacy class of nilpotent n n matrices can be parameterized by partitions of n and for a nilpotent in the class parameterized by the variety f of stable flags has its irreducible components parameterized by the standard young tableaux of shape we indicate how several algorithmic constructions defined for young tableaux have significance in this context thus extending steinberg s result that the relative position of flags generically chosen in the irreducible components of f parameterized by tableaux p and q is the permutation associated to p q under the robinson schensted correspondence other constructions for which we give interpretations are schutzenberger s involution of the set of young tableaux jeu de taquin leading also to an interpretation of littlewood richardson coefficients and the transpose robinson schensted correspondence defined using column insertion in each case we use a doubly indexed family of partitions defined in terms of the flag or pair of flags determined by a point chosen in the variety under consideration and we show that for generic choices the family satisfies combinatorial relations that make it correspond to an instance of the algorithmic operation being interpreted as described in m a a van leeuwen electron j combin 3 no 2 1996 r15

perturbation analysis for the operator equationtx bin banach spaces letx1 x2be two banach spaces over the complex fieldcand lett x1 x2be a bounded linear operator with the generalized inverset lett t tbe a bounded linear operator with t t 1 suppose thatdimkert dimkert orr t kert 0 thenthas the generalized inverset i t t 1t with t t 1 t t this result generalizes theorem 3 9 ofna using this result we give some results of the perturbation analysis for the operator equationtx b

perturbation and expression for inner inverses in banach spaces and its applications abstract it is well known that if t is invertible and t 1 is its inverse then t 1 i t t t 1 1 is the inverse of t for t t t 1 1 the main purpose of this paper is to study the following problem if t is an inner inverse of t and t t t 1 is t i t t t 1 an inner inverse of t if the answer is no when is t i t t t 1 an inner inverse of t in this paper we give a complete answer to these problems and provide some characterizations for t i t t t 1 to be an inner inverse of t in banach spaces utilizing these results we investigate similar problems for the generalized inverse 1 3 inverse 1 4 inverse 1 5 inverse 1 2 3 inverse 1 2 4 inverse moore penrose inverse group inverse and drazin inverse the results obtained in this paper extend and improve many recent results in this area

higher order qmc petrov galerkin discretization for affine parametric operator equations with random field inputs we construct quasi monte carlo methods to approximate the expected values of linear functionals of petrov galerkin discretizations of parametric operator equations which depend on a possibly infinite sequence of parameters such problems arise in the numerical solution of differential and integral equations with random field inputs we analyze the regularity of the solutions with respect to the parameters in terms of the rate of decay of the fluctuations of the input field if p in 0 1 denotes the summability exponent corresponding to the fluctuations in affine parametric families of operators then we prove that deterministic interlaced polynomial lattice rules of order alpha lfloor 1 p rfloor 1 in s dimensions with n points can be constructed using a fast component by component algorithm in mathcal o alpha s n log n alpha 2 s 2 n operations to achieve a convergence rate of mathcal o n 1 p with the implied constant independent of s this dimension independen

ribbon tableaux hall littlewood functions quantum affine algebras and unipotent varieties we introduce a new family of symmetric functions which are q analogs of products of schur functions defined in terms of ribbon tableaux these functions can be interpreted in terms of the fock space representation fq of uq sln and are related to hall littlewood functions via the geometry of flag varieties we present a series of conjectures and prove them in special cases the essential step in proving that these functions are actually symmetric consists in the calculation of a basis of highest weight vectors of fq using ribbon tableaux

analysis of eigenvalues and modal interaction of stochastic systems an eigenvalue spectral analysis of stochastic engineering systems is presented a comparative numerical study between approximations based on monte carlo sampling a taylor series based perturbation approach and the polynomial chaos representation is conducted it is observed that the polynomial chaos representation gives more accurate estimates of the statistical moments than the perturbation method especially for the higher modes the differences of accuracy in the two methods are more pronounced as the system variability increases moreover the chaos expansion gives a more detailed probabilistic description of the eigenvalues and the eigenvectors in addition a method for representing the statistical modal overlapping is presented that characterizes the statistical interaction between the various modes

perturbation analysis for the reduced minimum modulus of bounded linear operator in banach spaces let x and y be two banach spaces over the complex field c and let t x y be a bounded linear operator with the generalized inverse t let t t dt be a bounded linear operator with t dt 1 2 let c t denote the reduced minimum modulus of t we first establish the upper semi continuity theorem for c t furthermore if dimkert dimkert or r t kert 0 then c t is continuous under the operator norm also we have a corollary in hilbert spaces which improves the result in linear algebra appl 262 1997 229 242 theorem 4 1

on orbital variety closures in sln i induced duflo order abstract for a semisimple lie algebra g the orbit method attempts to assign representations of g to coadjoint orbits in g orbital varieties are particular lagrangian subvarieties of such orbits leading to highest weight representations of g in sl n orbital varieties are described by young tableaux inclusion relation on orbital variety closures defines a partial order on young tableaux our aim is to describe this order the paper is devoted to the combinatorial description of induced duflo order on young tableaux the order generated by inclusion of generating subspaces of orbital varieties this is a very interesting and complex combinatorial question this is the first paper in the series in parts ii and iii we use repeatedly the results of this paper as a basis for further study of orbital variety closures

constructing randomly shifted lattice rules in weighted sobolev spaces shifted rank 1 lattice rules a special class of quasi monte carlo methods have recently been proposed by the present authors for the integration of functions belonging to certain weighted sobolev spaces the shifts in these rules were generated in a deterministic manner in contrast in this paper we generate these shifts randomly this allows probabilistic estimates for the error in a given integral it also reduces the number of operations required to find the generating vectors for the underlying lattice rules component by component the rules thus constructed achieve a worst case strong tractability error bound in an average or probabilistic sense

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

kazhdan lusztig representations and whittaker space of some genuine representations we prove a formula for the dimension of whittaker functionals of irreducible constituents of a regular unramified genuine principal series for covering groups the formula explicitly relates such dimension to the kazhdan lusztig representations associated with certain right cells of the weyl group we also state a refined version of the formula which is proved under some natural assumption the refined formula is also verified unconditionally in several important cases

a recursive rule for kazhdan lusztig characters abstract a formula for computing kazhdan lusztig characters in hecke algebras of coxeter groups is given this formula holds for products of coxeter elements of parabolic subgroups in an arbitrary coxeter group where the parabolic subgroups are of type a and the kazhdan lusztig representations are of finite dimension the formula yields new combinatorial interpretations of murnaghan nakayama type to these characters and to other related characters

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

left cells in the weyl group of type e8 abstract the aim of the present paper is to give an explicit description for all the left cells of the weyl group w of type e 8 we first find a representative set for the left cells of w and then use it to describe these cells our results show that most of the left cells in w can be characterized by their generalized invariants alone then the remaining ones which are called exceptional left cells are characterized by the generalized invariants together with the a values

representation theory of rectangular finite w algebras abstract we classify the finite dimensional irreducible representations of rectangular finite w algebras i e the finite w algebras u g e where g is a symplectic or orthogonal lie algebra and e g is a nilpotent element with jordan blocks all the same size

fast non overlapping schwarz domain decomposition methods for solving the neutron diffusion equation studying numerically the steady state of a nuclear core reactor is expensive in terms of memory storage and computational time in order to address both requirements one can use a domain decomposition method implemented on a parallel computer we present here such a method for the mixed neutron diffusion equations discretized with raviart thomas nedelec finite elements this method is based on the schwarz iterative algorithm with robin interface conditions to handle communications we analyse this method from the continuous point of view to the discrete point of view and we give some numerical results in a realistic highly heterogeneous 3d configuration computations are carried out with the minos solver of the apollo3 r neutronics code

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

reduced basis method for band structure calculations we describe the application of the reduced basis method in rapid and accurate determination of band energies in band structure calculations the method is well suited for problems requiring repetitive evaluations of the band energies especially in the many query limit we demonstrate the efficacy of the method in the determination of the spectral properties of crystalline silicon

joint relations on elements of the symmetric group abstract we introduce two methods which reduce the checking of the joint relations on element pairs x y i e checking if x y 0 see 1 2 inside a two sided cell of the symmetric group s n to some special cases up to star operations one involves the involutions in s n and the other involves the set for n see 3 1 and 5 1 we give a detailed investigation for the properties of the set in terms of standard tableaux also we study the sets u s and u s and give an affirmative answer to a question of n h xi see proposition 2 9

estimates for the errors in eigenvalue and eigenvector approximation by galerkinmethods with particular attention to the case of multiple eigenvalues refined estimates for the errors in eigenvalue and eigenvector approximation by finite element or more generally galerkin methods as they apply to selfadjoint problems are presented particular attention is given to the case of multiple eigenvalues the results are new in this case the proof is based on a novel approach which yields the known results for simple eigenvalues in a simple way numerical computations are presented and analyzed in light of the theoretical results

quasi monte carlo finite element methods for a class of elliptic partial differential equations with random coefficients in this paper quasi monte carlo qmc methods are applied to a class of elliptic partial differential equations pdes with random coefficients where the random coefficient is parametrized by a co

the steinberg variety and representations of reductive groups abstract we give an overview of some of the main results in geometric representation theory that have been proved by means of the steinberg variety steinberg s insight was to use such a variety of triples in order to prove a conjectured formula by grothendieck the steinberg variety was later used to give an alternative approach to springer s representations and played a central role in the proof of the deligne langlands conjecture for hecke algebras by kazhdan and lusztig