conformal blocks and generalized theta functions let yp be the moduli space of rank r vector bundles with trivial determinant on a riemann surface x this space carries a natural line bundle the determinant line bundle s we describe a canonical isomorphism of the space of global sections of sk with the space of conformal blocks defined in terms of representations of the lie algebra 5 c z it follows in particular that the dimension of h yk is given by the verlinde formula

symplectic surgery and the spinc dirac operator abstract letgbe a compact connected lie group and m a compact hamiltoniang space with moment mapj m g under the assumption that these data are pre quantizable one can construct an associated spinc dirac operator formula whose equivariant index yields a virtual representation ofg we prove a conjecture of guillemin and sternberg that if 0 is a regular value ofj the multiplicityn 0 of the trivial representation in the index space formula is equal to the index of the spinc dirac operator for the symplectic quotientm0 j 1 0 g this generalizes previous results for the case thatg tis abelian i e a torus

infinite grassmannians and moduli spaces of g bundles let c be a smooth projective irreducible algebraic curve over c of any genus and g a connected simply connected simple affine algebraic group over c in this paper we elucidate the relationship between 1 the space of vacua conformal blocks defined in conformal field theory using an integrable highest weight representation of the affine kac moody algebra associated to g and 2 the space of regular sections generalised theta functions of a line bundle on the moduli space ta of semistable principal g bundles on c fix a point p in c and let p resp kp be the completion of the local ring p of c at p resp the quotient field of c p let g kp the kp rational points of the algebraic group g be the loop group of g and let g p be the standard maximal parahoric subgroup of then the generalised flag variety x n is an inductive limit of projective varieties in fact of generalised schubert varieties one has a natural r line bundle s on x cf sect 2 2 and the picard group pic x is isomorphic to z which is generated by s proposition 2 3 where f is the universal central extension of by the multiplicative group c cf sect 2 2 by an analogue of the borel weil theorem proved in the kac moody setting by kumar and also by mathieu the space h 162 s of the regular sections of the line bundle y dxo y xo for any d 0 is canonically isomorphic with the full vector space dual v dxo of the integrable highest weight irreducible module v dxo of the affine kacmoody group or of the associated affine kac moody algebra which is a certain one dimensional central extension of the loop algebra lie g with highest weight dxo cf sect 6 1

equivariant dirac operators and differentiable geometric invariant theory in this paper we give a geometric expression for the multiplicities of the equivariant index of a spin c dirac operator

geometric quantization and multiplicities of group representations the heisenberg uncertainty principle says that it is impossible to determine simultaneously the position and momentum of a quantum mechanical particle this can be rephrased as follows the smallest subsets of classical phase space in which the presence of a quantum mechanical particle can be detected are its lagrangian submanifolds for this reason it makes sense to regard the lagrangian submanifolds of phase space as being its true points see weinstein 17 now let g be a compact lie group and g x x x a hamiltonian action of g on x see w for definitions it is well known that the fixed points of this action form a symplectic submanifold of x see for instance guillemin and sternberg 5 however what can one say about the fixed points of g we will show that they are also the points of a symplectic manifold xc this manifold is the marsden weinstein reduction of x with respect to the zero orbit in g and will be described in sect 2 it was introduced in a completely different context fl om ours by marsden and weinstein 12 problems in classical mechanics can often be reduced to the study of hamiltonian systems on symplectic manifolds and problems in quantum mechanics to the study of linear operators on hilbert space this fact has inspired a number of efforts to quantize symplectic geometry by devising schemes for associating hilbert space to symplectic manifolds the no go theorems of groenwald and van hove impose some embarrassing limitations on all such schemes however it seems to be a useful idea heuristically to think of every symplectic manifold x a ic 1 as being symbiotically associated with a hilbert space xq a t in such a way that the classical observables on the first space correspond to quantum observables on the second space the heuristics further suggests that if g is a group of symmetries of x la i j it should also be a group of symmetries of xq a tum in this heuristic spirit we will state the main conjecture of this paper

the quantization conjecture revisited a strong version of the quantization conjecture of guillemin and sternberg is proved for a reductive group action on a smooth compact polarized variety x l the cohomologies of l over the git quotient x g equal the invariant part of the cohomologies over x this generalizes the theorem of gs on global sections and strengthens its subsequent extensions jk li to riemannroch numbers remarkable by products are the invariance of cohomology of vector bundles over x g under a small change in the defining polarization or under shift desingularization as well as a new proof of boutot s theorem also studied are equivariant holomorphic forms and the equivariant hodgeto de rham spectral sequences for x and its strata whose collapse is shown one application is a new proof of the borel weil bott theorem of ti for the moduli stack of c bundles over a curve and of analogous statements for the moduli stacks and spaces of bundles with parabolic structures collapse of the hodge to de rham sequences for these stacks is also shown

on the de rham cohomology of differential and algebraic stacks abstract we introduce the notion of cofoliation on a stack a cofoliation is a change of the differentiable structure which amounts to giving a full representable smooth epimorphism cofoliations are uniquely determined by their associated lie algebroids cofoliations on stacks arise from flat connections on groupoids connections on groupoids generalize connections on gerbes and bundles in a natural way a flat connection on a groupoid is an integrable distribution of the morphism space compatible with the groupoid structure and complementary to both source and target fibres a cofoliation of a stack determines the flat groupoid up to etale equivalence we show how a cofoliation on a stack gives rise to a refinement of the hodge to de rham spectral sequence where the e 1 term consists entirely of vector bundle valued cohomology groups our theory works for differentiable holomorphic and algebraic stacks

on differential operators on complete symmetric varieties of type a1 and a2 abstract in this paper we consider the algebra of global differential operators d x on wonderful compactifications x of symmetric spaces g h of type a 1 and a 2 we will first construct a global differential operator on these varieties that does not come from the infinitesimal action of the lie algebra of g we will then focus on type a 2 where we will show that d x is an algebra of finite type and that for any invertible sheaf l on x h 0 x l is either 0 or a simple left d x l module finally we will show with the help of local cohomology that this still holds for higher cohomology groups h i x l

the yang mills heat flow on the moduli space of framed bundles on a surface we study the analog of the yang mills heat flow on the moduli space of framed bundles on a cut surface existence and convergence of the heat flow give a stratification of morse type invariant under the action of the loop group we use the stratification to prove versions of kahler quantization commutes with reduction and kirwan surjectivity

the strong macdonald conjecture and hodge theory on the loop grassmannian we prove the strong macdonald conjecture of hanlon and feigin for reductive groups g in a geometric reformulation we show that the dolbeault cohomology hq x op of the loop grassmannian x is freely generated by de rham s forms on the disk coupled to the indecomposables of h bg equating the two euler characteristics gives an identity independently known to macdonald m which generalises ramanujan s 1 1 sum for simply laced root systems at level 1 we also find a strong form of bailey s 4 4 sum failure of hodge decomposition implies the singularity of x and of the algebraic loop groups some of our results were announced in t2

formality in an equivariant setting we define and discuss g formality for certain spaces endowed with an action by a compact lie group this concept is essentially formality of the borel construction of the space in a category of commutative differential graded algebras over r h bg these results may be applied in computing the equivariant cohomology of their loop spaces

the index formula for the moduli of g bundles on a curve we prove the formulae conjectured by the first author for the index of k theory classes over the moduli stack of algebraic g bundles on a smooth projective curve the formulae generalize e verlinde s for line bundles and have witten s integrals over the moduli space of stable bundles as their large level limits as an application we prove the newstead ramanan conjecture on the vanishing of high chern classes of certain moduli spaces of semi stable g bundles

borel weil bott theory on the moduli stack of g bundles over a curve let g be a semi simple group and m the moduli stack of g bundles over a smooth complex projective curve using representation theoretic methods i prove the pure dimensionality of sheaf cohomology for certain evaluation vector bundles over m twisted by powers of the fundamental line bundle this result is used to prove a borel weil bott theorem conjectured by g segal for certain generalized flag varieties of loop groups along the way the homotopy type of the group of algebraic maps from an affine curve to g and the homotopy type the hodge theory and the picard group of m are described one auxiliary result in appendix a is the alexander cohomology theorem conjectured in gro2 a self contained account of the uniformization theorem of ls for the stack m is given including a proof of a key result of drinfeld and simpson in characteristic 0 a basic survey of the simplicial theory of stacks is outlined in appendix b

dimension of zero weight space an algebro geometric approach abstract let g be a connected adjoint simple algebraic group over the complex numbers with a maximal torus t and a borel subgroup b containing t the study of zero weight spaces in irreducible representations of g has been a topic of considerable interest there are many works which study the zero weight space as a representation space for the weyl group in this paper we study the variation on the dimension of the zero weight space as the highest weight of the irreducible representation varies over the set of dominant integral weights of t which are lattice points in a certain polyhedral cone the theorem proved here asserts that the zero weight spaces have dimensions which are piecewise quasi polynomial functions on the polyhedral cone of dominant integral weights the main tool we use are the geometric invariant theory and the riemann roch theorem

der kanonische modul eines invariantenrings abstract we calculate the canonical module of a ring of invariants r of a reductive group acting on an affine variety from that we derive a criterion for the gorenstein property to hold for r in case r is the ring of invariants of a graded ring we obtain lower and upper bounds for the degree of the generating function of r

trace functionals on noncommutative deformations of moduli spaces of flat connections abstract we describe an efficient construction of a canonical noncommutative deformation of the algebraic functions on the moduli spaces of flat connections on a riemann surface the resulting algebra is a variant of the quantum moduli algebra introduced by alekseev grosse and schomerus and buffenoir and roche we construct a natural trace functional on this algebra and show that it is related to the canonical trace in the formal index theory of fedosov and nest and tsygan via verlinde s formula

gauged gromov witten theory for small spheres we relate the genus zero gauged gromov witten invariants of a smooth projective variety for sufficiently small area with equivariant gromov witten invariants as an application we deduce a gauged version of abelianization for gromov witten invariants in the small area chamber in the symplectic setting we prove that any sequence of genus zero symplectic vortices with vanishing area has a subsequence that converges after gauge transformation to a holomorphic map with zero average moment map

geometric proof of a conjecture of fulton we give a geometric proof of a conjecture of fulton on the multiplicities of irreducible representations in a tensor product of irreducible representations for gl r

on parametric state feedback design an explicit parametric expression for the controller gain matrix in linear state variable feedback is derived it involves both the case where the set of closed loop eigenvalues does not include any open loop eigenvalue and the case where the two eigenvalue sets have elements in common some advantages of this new explicit parameterization of the controller gain matrix are illustrated by two numerical examples

descent of coherent sheaves and complexes to geometric invariant theory quotients abstract fix a scheme x over a field of characteristic zero that is equipped with an action of a reductive algebraic group g we give necessary and sufficient conditions for a g equivariant coherent sheaf on x or a bounded above complex of g equivariant coherent sheaves on x to descend to a good quotient x g this gives a description of the coherent derived category of x g as an admissible subcategory of the equivariant derived category of x

backlund transformation and the painleve property when a differential equation possesses the painleve property it is possible for specific equations to define a backlund transformation by truncating an expansion about the singular manifold at the constant level term from the backlund transformation it is then possible to derive the lax pair modified equations and miura transformations associated with the completely integrable system under consideration in this paper completely integrable systems are considered for which backlund transformations as defined above may not be directly defined these systems are of two classes the first class consists of equations of toda lattice type e g sine gordon bullough dodd equations we find that these equations can be realized as the minus one equation of sequences of integrable systems although the backlund transformation may or may not exist for the minus one equation it is shown for specific sequences that the backlund transformation does exist for the positive equations of the sequence this in turn allows the derivation of lax pairs and the recursion operation for the entire sequence the second class of equations consists of sequences of harry dym type these equations have branch point singularities and thus do not directly possess the painleve property yet by a process similar to the uniformization of algebraic curves their solutions may be parametrically represented by meromorphic functions for specific systems this is shown to provide a natural extension of the painleve property

gromov witten gauge theory we introduce a modular completion of the stack of maps from stable marked curves to the quotient stack pt c x and use this stack to construct some gauge theoretic analogues of the gromov witten invariants we also indicate the generalization of these invariants to the quotient stacks x c x where x is a smooth proper complex algebraic variety

the witten reshetikhin turaev invariant for links in finite order mapping tori i abstract we state asymptotic expansion and growth rate conjectures for the witten reshetikhin turaev invariants of arbitrary framed links in 3 manifolds and we prove these conjectures for the natural links in mapping tori of finite order automorphisms of marked surfaces our approach is based upon geometric quantisation of the moduli space of parabolic bundles on the surface which we show coincides with the construction of the witten reshetikhin turaev invariants using conformal field theory as was recently completed by andersen and ueno

properness for scaled gauged maps we prove properness of moduli stacks of gauged maps satisfying a stability conditition introduced by mundet 42 schmitt 48 and ziltener 59 the proof combines a git construction of schmitt 48 properness for twisted stable maps by abramovich vistoli 1 a variation of a boundedness argument due to ciocan fontanine kim maulik 13 and a removal of singularities for bundles on surfaces in colliot thelene sansuc 14

an analytic proof of the geometric quantization conjecture of guillemin sternberg we present a direct analytic approach to the guillemin sternberg conjecture gs that geometric quantization commutes with symplectic reduction which was proved recently by meinrenken m1 m2 and vergne v1 v2 et al besides providing a new proof of this conjecture our methods also lead immediately to further extensions in various contexts

moduli of vector bundles frobenius splitting and invariant theory let x be an irreducible projective curve of genus g over an algebraically closed field of positive characteristic 2 3 in part i we prove adapting the degeneration arguments of n tr that moduli spaces of rank two parabolic bundles on x are frobenius split m r for generic smooth x a similar result holds for x nodal a consequence is the verlinde formula in positive characteristic in part ii we give a direct proof of the fact that the local rings of the moduli spaces are f split and further that they are cohen macaulay this involves showing that the ring of invariants of g copies of 2 x 2 matrices under the adjoint action of sl 2 is f split and cohen macaulay

multiplicities formula for geometric quantization part ii 1 introduction let p be a compact manifold let h be a compact lie group acting on the right on p we assume that the stabilizer of each element y e p is a finite subgroup of h the space m p h is an orbifold and every orbifold can be presented this way if h acts freely then m is a manifold if ae graded space of h invariant solutions of the h horizontal dirac operator on p twisted by the line bundle let m o 9 be the moment map for the g action assume 0 is a regular value of let mred be the reduced orbifold of m that is mred o g consider the reduced orbifold line bundle rd lu lto g on m oa in the ease where both g andh are torus we prove here the formula

on the equivariant formality of kahler manifolds with torus group actions we consider simply connected compact kahler manifolds which have a holomorphic action of a torus group we use the existing equivariant models for rational homotopy to show that these spaces satisfy an equivariant formality condition over the complex numbers