nonsymmetric macdonald polynomials and demazure characters we establish a connection between a specialization of the nonsymmetric macdonald polynomials and the demazure characters of the corresponding affine kac moody algebra this allows us to obtain a representation theoretical interpretation of the coefficients of the expansion of the specialized symmetric macdonald polynomials in the basis formed by the irreducible characters of the associated finite lie algebra

lie groups and twistor spaces on considere un espace symetrique de riemann simplement connexe m la connexion de levi civita et le fibre torseur metrique 7b f m j m dont les fibres 7b f x m pour x m est constitue de structures complexes orthogonales

the gauss kronecker curvature of minimal hypersurfaces in four dimensional space forms let mathbb q 4 c be a four dimensional space form of constant curvature c in this paper we show that the infimum of the absolute value of the gauss kronecker curvature of a complete minimal hypersurface in mathbb q 4 c c leq 0 whose ricci curvature is bounded from below is equal to zero further we study the connected minimal hypersurfaces m 3 of a space form mathbb q 4 c with constant gauss kronecker curvature k for the case c 0 we prove by a local argument that if k is constant then k must be equal to zero we also present a classification of complete minimal hypersurfaces of mathbb q 4 c with k constant

a scattering approach to the quantization of hamiltonians in two dimensions application to the wedge billiard we extend the scattering approach to quantize general hamiltonian systems in two dimensions and demonstrate it for the wedge billiard the resulting energy levels are given by the zeros of a secular equation the corresponding wavefunctions may also be constructed we verify analytically that this quantization condition is exact in the case of the integrable wedges this method has been applied successfully to computing numerically a few thousand levels for various values of the angle using these data we can check the gutzwiller trace formula and other semiclassical relations involving the scattering matrix and periodic orbits for which we find excellent agreement

crystal energy functions via the charge in types a and c the ram yip formula for macdonald polynomials at t 0 provides a statistic which we call charge in types a and c it can be defined on tensor products of kashiwara nakashima single column crystals in this paper we prove that the charge is equal to the negative of the energy function on affine crystals the algorithm for computing charge is much simpler and can be more efficiently computed than the recursive definition of energy in terms of the combinatorial r matrix

affine structures and a tableau model for e_6 crystals abstract we provide the unique affine crystal structure for type e 6 1 kirillov reshetikhin crystals corresponding to the multiples of fundamental weights s 1 s 2 and s 6 for all s 1 in bourbaki s labeling of the dynkin nodes where 2 is the adjoint node our methods introduce a generalized tableaux model for classical highest weight crystals of type e and use the order three automorphism of the affine e 6 1 dynkin diagram in addition we provide a conjecture for the affine crystal structure of type e 7 1 kirillov reshetikhin crystals corresponding to the adjoint node

bogomolny s semiclassical transfer operator for rotationally invariant integrable systems the transfer operator due to bogomolny provides a convenient method for obtaining a semiclassical approximation to the energy eigenvalues of a quantum system no matter what the nature of the analogous classical system in this paper the method is applied to integrable systems which are rotationally invariant in two and three dimensions in two dimensions the transfer operator is expanded in a fourier series in the angle variable while in three dimensions it is expanded in spherical harmonics in both cases when the fourier coefficients are evaluated using the stationary phase approximation we arrive at the einstein brillouin keller quantization conditions the associated maslov indices are shown to agree with the results calculated by well known simple rules the theory is applied to several rotationally invariant systems including the hydrogen atom and the isotropic harmonic oscillator in two and three dimensions the circle billiard a billiard inside a spherical cavity and a harmonic potential with a singular magnetic flux line

on the nullity distribution of a submanifold of a space form if m is a submanifold of a space form the nullity distribution mathcal n of its second fundamental form is when defined the common kernel of its shape operators in this paper we will give a local description of any submanifold of the euclidean space by means of its nullity distribution we will also show the following global result if m is a complete irreducible submanifold of the euclidean space or the sphere then its complementary distribution mathcal n bot is completely non integrable this means that any two points in m can be joined by a curve everywhere perpendicular to mathcal n we will finally show that this statement is false for a submanifold of the hyperbolic space

conformal hypersurfaces with the same gauss map in this paper we provide a complete classification of all hypersurfaces of euclidean space which admit conformal deformations other than the ones obtained through conformal diffeomorphisms of the ambient space preserving the gauss map elie cartan in one of his earlier papers in differential geometry classified all euclidean hypersurfaces of dimension at least five which admit conformal deformations other than the trivial ones obtained through compositions with conformal diffeomorphisms of the ambient space when all possible conformal deformations are trivial a hypersurface is called conformally rigid first cartan proved that a hypersurface is conformally rigid if at no point does there exist a principal curvature of multiplicity at least n 2 see ca cd2 or da using this he concluded that a conformally deformable hypersurface is either conformally flat or a 2 parameter envelope of spheres or planes of some very special type moreover the set of all conformal deformations is either a 1parameter family or there is just one other deformation in this paper we classify all euclidean hypersurfaces which admit nontrivial conformal deformations preserving the gauss map in the isometric case a complete classification for any codimension has been obtained by dajczer and gromoll dg2 while the conformal case but only for surfaces is due to vergasta ve2 theorem let f g mn rn 1 n 3 be conformal immersions of an ndimensional connected riemannian manifold with the same gauss map assume that on no open subset do they differ by a conformal diffeomorphism of rn l then f mn is part of one of the following examples while g mn is of the same type i a minimal real kaehler hypersurface ii a rotation hypersurface over a plane curve iii a rotation hypersurface over a minimal surface in r3 minimal real kaehler hypersurfaces have been completely classified by dajczer and gromoll in dg2 the assumption that f and g do not differ locally by a conformal deformation of the ambient space has been introduced in order to produce a global result without that assumption our proof shows that the only other possibility is to have an open subset where up to homothety received by the editors december 10 1993 1991 mathematics subject classification primary 53c42 1995 american mathematical society

schensted type correspondence plactic monoid and jeu de taquin for type cn we use kashiwara s theory of crystal bases to study the plactic monoid for uq sp2n then we describe the corresponding insertion and sliding algorithms the sliding algorithm is essentially the symplectic jeu de taquin defined by sheats and our construction gives the proof of its compatibility with plactic relations

affine crystals one dimensional sums and parabolic lusztig q analogues this paper is concerned with one dimensional sums in classical affine types we prove a conjecture of shimozono and zabrocki j algebra 299 33 61 2006 by showing they all decompose in terms of one dimensional sums related to affine type a provided the rank of the root system considered is sufficiently large as a consequence any one dimensional sum associated to a classical affine root system with sufficiently large rank can be regarded as a parabolic lusztig q analogue

quantum section method for the soft stadium abstract the soft stadium is defined by a monomial potential with exponent as a parameter such that corresponds to the billiard the practical use of the quantum section method depends only on the partial separability of the system on both sides of the section which holds for all s in particular for 1 0 the system becomes globally separable allowing for a general test of the method for various values of the parameter we also tested the use of the asymptotic wkb type approximation in the construction of green s functions and asymptotic overlap integrals to obtain higher energy eigenvalues we show these approximations to be reliable

exact bogomolny sections for separable systems a good poincare section of a classical dynamical system defines a mapping that provides essential qualitative information about its orbits bogomolny s sections in quantum mechanics lead to an eigenvalue condition that depends on the poincare map in the semi classical limit the conditions for the construction of purely quantum mechanical bogomolny green functions are discussed here with reference to separable systems for appropriately chosen green functions the ensuing eigenvalue condition is shown to be exact this result is valid even if not a single classical orbit crosses the surface of section

on minimal surfaces in a kahler manifold of constant holomorphic sectional curvature this paper studies minimal surfaces in kshler manifolds of constant holomorphic sectional curvature using the technique of the moving frame in particular we provide a classification of the minimal two spheres in cpst complex projective sf space equipped with the fubini study metric this classification can be described as follows to each holomorphic curve in c pt1 classically there is associated a particular framing of c l l called the frenet frame each element of the frenet frame induces a minimal surface in cp the classification theorem states that all minimal surfaces of topological type of the two sphere occur in this manner the theorem is proved using holomorphic differentials that occur naturally on minimal surfaces in kshler manifolds of constant holomorphic sectional curvature together with the riemann roch theorem

a berger type normal holonomy theorem for complex submanifolds we prove a berger type theorem for the normal holonomy phi perp i e the holonomy group of the normal connection of a full complete complex submanifold m of the complex projective space mathbb c p n namely if phi perp does not act transitively then m is the complex orbit in the complex projective space of the isotropy representation of an irreducible hermitian symmetric space of rank greater or equal to 3 moreover we show that for complete irreducible complex submanifolds of mathbb c n the normal holonomy is generic i e it acts transitively on the unit sphere of the normal space the methods in the proofs rely heavily on the singular data of appropriate holonomy tubes after lifting the submanifold to the complex euclidean space in the mathbb c p n case and basic facts of complex submanifolds

numerical accuracy of bogomolny s semiclassical quantization scheme in quantum billiards we use the semiclassical quantization scheme of bogomolny to calculate eigenvalues of the limacon quantum billiard corresponding to a conformal map of the circle billiard we use the entire billiard boundary as the chosen surface of section and use a finite approximation for the transfer operator in coordinate space computation of the eigenvalues of this matrix combined with a quantization condition determines a set of semiclassical eigenvalues which are compared with those obtained by solving the schrodinger equation the classical dynamics of this billiard system undergoes a smooth transition from integrable circle to completely chaotic motion thus providing a test of bogomolny s semiclassical method in coordinate space in terms of the morphology of the wavefunction we analyse the results for billiards which exhibit both soft and hard chaos

a combinatorial formula for macdonald polynomials abstract in this paper we use the combinatorics of alcove walks to give uniform combinatorial formulas for macdonald polynomials for all lie types these formulas resemble the formulas of haglund haiman and loehr for macdonald polynomials of type gl n at q 0 these formulas specialize to the formula of schwer for the macdonald spherical function in terms of positively folded alcove walks and at q t 0 these formulas specialize to the formula for the weyl character in terms of the littelmann path model in the positively folded gallery form of gaussent and littelmann

the twistor equation in lorentzian spin geometry in this paper we discuss the twistor equation in lorentzian spin geometry in particular we explain the local conformal structure of lorentzian manifolds which admit twistor spinors inducing lightlike dirac currents furthermore we derive all local geometries with singularity free twistor spinors that occur up to dimension 7

on the fundamental group of the space of harmonic 2 spheres in the n sphere harmonic maps s s are of interest from the point of view of differential geometry as branched minimal immersions and from the point of view of gauge theory as classical solutions of the s non linear sigma model from the work of chern calabi barbosa and others there is now a complete description of such harmonic maps in terms of holomorphic data however very little is known about the global properties of the space harm s s of all harmonic maps since the analogous object in yang mills theory the space of solutions to the yang mills equations has proved to be of great significance the space of harmonic maps is also of some interest in this paper we shall give an approach to the study of the topology of harm s s

complete minimal hypersurfaces in the hyperbolic space mathbb h 4 with vanishing gauss kronecker curvature we investigate 3 dimensional complete minimal hypersurfaces in the hyperbolic space h 4 with gauss kronecker curvature identically zero more precisely we give a classification of complete minimal hypersurfaces with gauss kronecker curvature identically zero a nowhere vanishing second fundamental form and a scalar curvature bounded from below

complete minimal submanifolds with nullity in euclidean space in this paper we investigate minimal submanifolds in euclidean space with positive index of relative nullity let m m be a complete riemannian manifold and let f m m rightarrow mathbb r n be a minimal isometric immersion with index of relative nullity at least m 2 at any point we show that if the omori yau maximum principle for the laplacian holds on m m for instance if the scalar curvature of m m does not decrease to infty too fast or if the immersion f is proper then the submanifold must be a cylinder over a minimal surface

deformed universal characters for classical and affine algebras abstract creation operators are given for the three distinguished bases of the type bcd universal character ring of koike and terada yielding an elegant way of treating computations for all three types in a unified manner deformed versions of these operators create symmetric function bases whose expansion in the universal character basis has polynomial coefficients in q with nonnegative integer coefficients we conjecture that these polynomials are one dimensional sums associated with crystal bases of finite dimensional modules over quantized affine algebras for all nonexceptional affine types these polynomials satisfy a macdonald type duality

rigidity for complete weingarten hypersurfaces we classify locally and globally the ruled weingarten hypersurfaces of the euclidean space as a consequence of the local classification and a rigidity theorem of dajczer and gromoll it follows that a complete weingarten hypersurface which does not contain an open subset of the form l3 x rn 3 where l3 is unbounded and n 3 is rigid

the space of minimal embeddings of a surface into a three dimensional manifold of positive ricci curvature in this paper we obtain a curvature estimate for embedded minimal surfaces in a three dimensional manifold of positive ricci curvature in terms of the geometry of the ambient manifold and the genus of the minimal surface it should be mentioned that there are two main points in our result one is the absence of a stability assumption and the other is the requirement of being embedded most known curvature estimates require the stability assumption and once the stability assumption is dropped many of these known results cease to be valid see ss and an for another example of a curvature estimate without the assumption of stability the embeddedness condition is rather subtle because of the way it enters in our proof our proof depends on the eigenvalue estimate and the area bound due to the first author and wang cw which require embeddedness in an essential way present knowledge indicates that closed embedded minimal surfaces in s 3 are rare while immersed surfaces are more plentiful for example only a finite number of minimal embeddings of a given genus are known on the other hand otsuki o constructed infinitely many immersed minimal tori with arbitrarily large area we obtain the curvature estimate indirectly by proving the smooth compactness theorem theorem 1 theorem 1 has many interesting consequences it shows that the set of conformal structures that can be realized on a minimal embedding in s 3 is a compact subset of the moduli space this is in contrast with the result of bryant b who showed that every riemann surface is conformally and minimally immersed in s 4 in view of our compactness result and the scarcity of examples it is very tempting to conjecture that there are only finitely many embedded minimal surfaces up to rigid motion in s 3 for each fixed genus throughout this paper manifold means manifold without boundary unless explictly stated otherwise when we say a sequence m i of surfaces converges to a

ls galleries the path model and mv cycles we give an interpretation of the path model of a representation 18 of a complex semisimple algebraic group g in terms of the geometry of its affine grassmannian in this setting the paths are replaced by ls galleries in the affine coxeter complex associated to the weyl group of g to explain the connection with geometry consider a demazure hansen bott samelson desingularization of the closure of an orbit g c t in the affine grassmannian the homology of has a basis given by bia lynicki birula cell s which are indexed by the t fixed points in now the points of can be identified with galleries of a fixed type in the affine tits building associated to g and the t fixed points correspond in this language to combinatorial galleries of a fixed type in the affine coxeter complex we determine those galleries such that the associated cell has a non empty intersection with g c t identified with an open subset of and we show that the closures of the strata associated to ls galleries are exactly the mv cycles 24 which form a basis of the representation v for the langland s dual group g

application of bogomolny s transfer operator to a circular harmonic oscillator plus potential bogomolny s transfer operator has been used to find an analytical solution for the semiclassical energy eigenvalues of a simple two dimensional integrable system the system studied consists of a particle moving in an isotropic harmonic oscillator potential plus a potential the classical trajectories are used to construct the transfer matrix and an expression is derived for the eigenvalues of this matrix as a function of the energy these eigenvalue curves yield the semiclassical energy eigenvalues for the quantum system which turn out to be exactly the same as the results obtained by solving the schrodinger equation some insight into this unexpected agreement is provided by considering an exact transfer operator we show that when this operator is expanded in powers of planck s constant the leading term in the expansion is bogomolny s transfer operator

crystal graphs for representations of the q analogue of classical lie algebras the explicit form of the crystal graphs for the finite dimensional representations of the q analogue of the universal enveloping algebras of type a b c and d is given in terms of semi standard tableaux and its analogues