Duke Mathematical Journal

For a function $\varphi$ in $L^2(0,1)$, extended to the whole real line as an odd periodic function of period 2, we ask when the collection of dilates $\varphi(nx)$, $n=1,2,3,\ldots$, constitutes a Riesz basis or a complete sequence in $L^2(0,1)$. The problem translates into a question concerning multipliers and cyclic vectors in the Hilbert space $\cal H$ of Dirichlet series $f(s)=\sum_n a_nn^{-s}$, where the coefficients $a_n$ are square summable. It proves useful to model $\cal H$ as the $H^2$ space of the infinite-dimensional polydisk, or, which is the same, the $H^2$ space of the character space, where a character is a multiplicative homomorphism from the positive integers to the unit circle. For given $f$ in $\cal H$ and characters $\chi$, $f_\chi(s)=\sum_na_n\chi(n)n^{-s}$ is a vertical limit function of $f$. We study certain probabilistic properties of these vertical limit functions.

We establish a twisted analog of our recent work on vertex representations and the McKay correspondence. For each finite group $\Gamma$ and a virtual character of $\Gamma$ we construct twisted vertex operators on the Fock space spanned by the super spin characters of the spin wreath products $\Gamma\wr\widetilde{S}_n$ of $\Gamma$ and a double cover of the symmetric group $S_n$ for all $n$. When $\Gamma$ is a subgroup of $SL_2(\mathbb C)$ with the McKay virtual character, our construction gives a group theoretic realization of the basic representations of the twisted affine and twisted toroidal algebras. When $\Gamma$ is an arbitrary finite group and the virtual character is trivial, our vertex operator construction yields the spin character tables for $\Gamma\wr\widetilde{S}_n$.

We prove an analogue of the Lefschetz (1,1) Theorem characterizing cohomology classes of Cartier divisors (or equivalently first Chern classes of line bundles) in the second integral cohomology. Let $X$ be a normal complex projective variety. We show that the classes of Cartier divisors in $H^2(X,Z)$ are precisely the classes $x$ such that (i) the image of $x$ in $H^2(X,C)$ (cohomology with complex coefficients) lies in $F^1 H^2(X,C)$ (first level of the Hodge filtration for Deligne's mixed Hodge structure), and (ii) $x$ is Zariski-locally trivial, i.e., there is a covering of $X$ by Zariski open sets $U$ such that $x$ has zero image in $H^2(U,Z)$. For normal quasi-projective varieties, this positively answers a question of Barbieri-Viale and Srinivas (J. Reine Ang. Math. 450 (1994)), where examples are given to show that divisor classes are not characterized by either one of the above conditions (i), (ii), taken by itself, unlike in the case of non-singular varieties. The present paper also contains an example of a non-normal projective variety for which (i) and (ii) do not suffice to characterize divisor classes.

We prove the Tate conjecture for divisor classes and the Mumford-Tate conjecture for the cohomology in degree 2 for varieties with $h^{2,0}=1$ over a finitely generated field of characteristic 0, under a mild assumption on their moduli. As an application of this general result, we prove the Tate and Mumford-Tate conjectures for some classes of algebraic surfaces with $p_g=1$.

We derive two multivariate generating functions for three-dimensional Young diagrams (also called plane partitions). The variables correspond to a colouring of the boxes according to a finite Abelian subgroup G of SO(3). We use the vertex operator methods of Okounkov--Reshetikhin--Vafa for the easy case G = Z/n; to handle the considerably more difficult case G=Z/2 x Z/2, we will also use a refinement of the author's recent q--enumeration of pyramid partitions. In the appendix, we relate the diagram generating functions to the Donaldson-Thomas partition functions of the orbifold C^3/G. We find a relationship between the Donaldson-Thomas partition functions of the orbifold and its G-Hilbert scheme resolution. We formulate a crepant resolution conjecture for the Donaldson-Thomas theory of local orbifolds satisfying the Hard Lefschetz condition.

We study the collapsing behaviour of Ricci-flat Kahler metrics on a projective Calabi-Yau manifold which admits an abelian fibration, when the volume of the fibers approaches zero. We show that away from the critical locus of the fibration the metrics collapse with locally bounded curvature, and along the fibers the rescaled metrics become flat in the limit. The limit metric on the base minus the critical locus is locally isometric to an open dense subset of any Gromov-Hausdorff limit space of the Ricci-flat metrics. We then apply these results to study metric degenerations of families of polarized hyperkahler manifolds in the large complex structure limit. In this setting we prove an analog of a result of Gross-Wilson for K3 surfaces, which is motivated by the Strominger-Yau-Zaslow picture of mirror symmetry.

The Fourier-Mukai transform is lifted to the derived category of sheaves with connection on abelian varieties. The case of flat connections (D-modules) is discussed in detail.

The Abelian sandpile growth model is a diffusion process for configurations of chips placed on vertices of the integer lattice $\mathbb{Z}^d$, in which sites with at least 2d chips {\em topple}, distributing 1 chip to each of their neighbors in the lattice, until no more topplings are possible. From an initial configuration consisting of $n$ chips placed at a single vertex, the rescaled stable configuration seems to converge to a particular fractal pattern as $n\to \infty$. However, little has been proved about the appearance of the stable configurations. We use PDE techniques to prove that the rescaled stable configurations do indeed converge to a unique limit as $n \to \infty$. We characterize the limit as the Laplacian of the solution to an elliptic obstacle problem.

We give a generalization to higher genera of the famous formula $12 \lambda=\delta$ for genus 1. We also compute the classes of certain strata in the Satake compactification as elements of the push down of the tautological ring.

We consider some classical maps from the theory of abelian varieties and their moduli spaces and prove their definability, on restricted domains, in the o-minimal structure $\Rae$. In particular, we prove that the embedding of moduli space of principally polarized ableian varierty, $Sp(2g,\Z)\backslash \CH_g$, is definable in $\Rae$, when restricted to Siegel's fundamental set $\fF_g$. We also prove the definability, on appropriate domains, of embeddings of families of abelian varieties into projective space.

Let $E$ be an elliptic curve defined over the rationals without complex multiplication. The field $F$ generated by all torsion points of $E$ is an infinite, non-abelian Galois extension of the rationals which has unbounded, wild ramification above all primes. We prove that the absolute logarithmic Weil height of an element of $F$ is either zero or bounded from below by a positive constant depending only on $E$. We also show that the N\'eron-Tate height has a similar gap on $E(F)$ and use this to determine the structure of the group $E(F)$.

In this paper, we give explicit descriptions of Hyodo and Kato's Frobenius and Monodromy operators on the first $p$-adic de Rham cohomology groups of curves and Abelian varieties with semi-stable reduction over local fields of mixed characteristic. This paper was motivated by the first author's paper "A $p$-adic Shimura isomorphism and periods of modular forms," where conjectural definitions of these operators for curves with semi-stable reduction were given.

We determine the abelianizations of the following three kinds of graded Lie algebras in certain stable ranges: derivations of the free associative algebra, derivations of the free Lie algebra and symplectic derivations of the free associative algebra. In each case, we consider both the whole derivation Lie algebra and its ideal consisting of derivations with positive degrees. As an application of the last case, and by making use of a theorem of Kontsevich, we obtain a new proof of the vanishing theorem of Harer concerning the top rational cohomology group of the mapping class group with respect to its virtual cohomological dimension.

For any smooth complex projective variety X and smooth very ample hypersurface Y in X, we develop the technique of genus zero relative Gromov-Witten invariants of Y in X in algebro-geometric terms. We prove an equality of cycles in the Chow groups of the moduli spaces of relative stable maps that relates these relative invariants to the Gromov-Witten invariants of X and Y. Given the Gromov-Witten invariants of X, we show that these relations are sufficient to compute all relative invariants, as well as all genus zero Gromov-Witten invariants of Y whose homology and cohomology classes are induced by X.

In this work we establish a connection between two classical notions, unrelated so far: Harmonic functions on the one hand and absolutely monotonic functions on the other hand. We use this to prove convexity type and propagation of smallness results for harmonic functions on the lattice. We also find an apparently new algorithm to transform harmonic functions in $\mathbb{R}^d$ to discrete harmonic functions.

We prove for the square Fibonacci Hamiltonian that the density of states measure is absolutely continuous for almost all pairs of small coupling constants. This is obtained from a new result we establish about the absolute continuity of convolutions of measures arising in hyperbolic dynamics with exact-dimensional measures.

Let A be a unital separable simple C*-algebra with a unique tracial state. We prove that if A is nuclear and quasidiagonal, then A tensored with the universal UHF-algebra has decomposition rank at most one. Then it is proved that A is nuclear, quasidiagonal and has strict comparison if and only if A has finite decomposition rank. For such A, we also give a direct proof that A tensored with a UHF-algebra has tracial rank zero. Applying this characterization, we obtain a counter-example to the Powers-Sakai conjecture.

Shokurov conjectured that the set of all log canonical thresholds on varieties of bounded dimension satisfies the ascending chain condition. In this paper we prove that the conjecture holds for log canonical thresholds on smooth varieties and, more generally, on locally complete intersection varieties and on varieties with quotient singularities. Comment: 20 pages. This supersedes arXiv:0811.4642 and arXiv:0811.4644. As opposed to these older versions, we now give a proof of Koll\'ar's m-adic semicontinuity result using only the Connectedness Theorem; v.3: section 4 has been rewritten; to appear in Duke Math. J

We define a renormalized characteristic class for Einstein asymptotically complex hyperbolic (ACHE) manifolds of dimension 4: for any such manifold, the polynomial in the curvature associated to the characteristic class euler-3signature is shown to converge. This extends a work of Burns and Epstein in the Kahler-Einstein case. This extends a work of Burns and Epstein in the Kahler-Einstein case. We also define a new global invariant for any 3-dimensional pseudoconvex CR manifold, by a renormalization procedure of the eta invariant of a sequence of metrics which approximate the CR structure. Finally, we get a formula relating the renormalized characteristic class to the topological number euler-3signature and the invariant of the CR structure arising at infinity.

Reflection laws of high order elliptic differential equations in two independent variables with constant coefficients and unequal characteristics across analytic boundary conditions

We analyse volume-preserving actions of product groups on Riemannian manifolds. To this end, we establish a new superrigidity theorem for ergodic cocycles of product groups ranging in linear groups. There are no a priori assumptions on the acting groups, except a spectral gap assumption on their action. Our main application to manifolds concerns irreducible actions of Kazhdan product groups. We prove the following dichotomy: Either the action is infinitesimally linear, which means that the derivative cocycle arises from unbounded linear representations of all factors. Otherwise, the action is measurably isometric, in which case there are at most two factors in the product group. As a first application, this provides lower bounds on the dimension of the manifold in terms of the number of factors in the acting group. Another application is a strong restriction for actions of non-linear groups. Comment: To appear in the Duke Mathematical Journal; 32 pages. Minor revisions, including the addition of a variation on Theorem A

We strengthen the local-global compatibility of Langlands correspondences for $GL_{n}$ in the case when $n$ is even and $l\not=p$. Let $L$ be a CM field and $\Pi$ be a cuspidal automorphic representation of $GL_{n}(\mathbb{A}_{L})$ which is conjugate self-dual. Assume that $\Pi_{\infty}$ is cohomological and not "slightly regular", as defined by Shin. In this case, Chenevier and Harris constructed an $l$-adic Galois representation $R_{l}(\Pi)$ and proved the local-global compatibility up to semisimplification at primes $v$ not dividing $l$. We extend this compatibility by showing that the Frobenius semisimplification of the restriction of $R_{l}(\Pi)$ to the decomposition group at $v$ corresponds to the image of $\Pi_{v}$ via the local Langlands correspondence. We follow the strategy of Taylor-Yoshida, where it was assumed that $\Pi$ is square-integrable at a finite place. To make the argument work, we study the action of the monodromy operator $N$ on the complex of nearby cycles on a scheme which is locally etale over a product of semistable schemes and derive a generalization of the weight-spectral sequence in this case. We also prove the Ramanujan-Petersson conjecture for $\Pi$ as above. Comment: 88 pages

We study conformal actions of connected nilpotent Lie groups on compact pseudo-Riemannian manifolds. We prove that if a type-(p,q) compact manifold M supports a conformal action of a connected nilpotent group H, then the degree of nilpotence of H is at most 2p+1, assuming p <= q; further, if this maximal degree is attained, then M is conformally equivalent to the universal type-(p,q), compact, conformally flat space, up to finite covers. The proofs make use of the canonical Cartan geometry associated to a pseudo-Riemannian conformal structure. Comment: 41 pages, 3 figures. Article has been shortened from previous version, and several corrections have been made according to referees' suggestions

We show that if r ≥ 3 and α is a faithful ℤ( r-Cartan action on a torus T{double-struck} d by automorphisms, then any closed subset of (T{double-struck} d) 2 which is invariant and topologically transitive under the diagonal ℤ r -action by α is homogeneous, in the sense that it is either the full torus (T{double-struck} d) 2, or a finite set of rational points, or a finite disjoint union of parallel translates of some d-dimensional invariant subtorus. A counterexample is constructed for the rank 2 case.

We construct holomorphic families of proper holomorphic embeddings of $\C^k$ into $\C^n$ ($0<k<n-1$), so that for any two different parameters in the family no holomorphic automorphism of $\C^n$ can map the image of the corresponding two embeddings onto each other. As an application to the study of the group of holomorphic automorphisms of $\C^n$ we derive the existence of families of holomorphic $\C^*$-actions on $\C^n$ ($n\ge 5$) so that different actions in the family are not conjugate. This result is surprising in view of the long standing Holomorphic Linearization Problem, which in particular asked whether there would be more than one conjugacy class of $\C^*$ actions on $\C^n$ (with prescribed linear part at a fixed point).

We prove a decomposition theorem for the equivariant K-theory of actions of affine group schemes G of finite type over a field on regular separated noetherian algebraic spaces, under the hypothesis that the actions have finite geometric stabilizers and satisfy a rationality condition together with a technical condition which holds e.g. for G abelian or smooth. We describe in an Appendix various complicial bi-Waldhausen categories (in Thomason's terminology) modelling the equivariant K-theory of regular noetherian separated algebraic spaces.

Consider a free ergodic measure preserving profinite action $\Gamma\curvearrowright X$ (i.e. an inverse limit of actions $\Gamma\curvearrowright X_n$, with $X_n$ finite) of a countable property (T) group $\Gamma$ (more generally of a group $\Gamma$ which admits an infinite normal subgroup $\Gamma_0$ such that the inclusion $\Gamma_0\subset\Gamma$ has relative property (T) and $\Gamma/\Gamma_0$ is finitely generated) on a standard probability space $X$. We prove that if $w:\Gamma\times X\to \Lambda$ is a measurable cocycle with values in a countable group $\Lambda$, then $w$ is cohomologous to a cocycle $w'$ which factors through the map $\Gamma\times X\to \Gamma\times X_n$, for some $n$. As a corollary, we show that any orbit equivalence of $\Gamma\curvearrowright X$ with any free ergodic measure preserving action $\Lambda\curvearrowright Y$ comes from a (virtual) conjugacy of actions.

Consider homogeneous G/H and G/F, for an S-algebraic group G. A lattice {\Gamma} acts on the left strictly conservatively. The following rigidity results are obtained: morphisms, factors and joinings defined apriori only in the measurable category are in fact algebraically constrained. Arguing in an elementary fashion we manage to classify all the measurable {\Phi} commuting with the {\Gamma}-action: assuming ergodicity, we find they are algebraically defined.

We introduce new methods from p-adic integration into the study of representation zeta functions associated to compact p-adic analytic groups and arithmetic groups. They allow us to establish that the representation zeta functions of generic members of families of p-adic analytic pro-p groups obtained from a global, `perfect' Lie lattice satisfy functional equations. In the case of `semisimple' compact p-adic analytic groups, we exhibit a link between the relevant p-adic integrals and a natural filtration of the locus of irregular elements in the associated semisimple Lie algebra, defined by centraliser dimension. Based on this algebro-geometric description, we compute explicit formulae for the representation zeta functions of principal congruence subgroups of the groups SL_3(O), where O is a compact discrete valuation ring of characteristic 0, and of the corresponding unitary groups. These formulae, combined with approximative Clifford theory, allow us to determine the abscissae of convergence of representation zeta functions associated to arithmetic subgroups of algebraic groups of type A_2. Assuming a conjecture of Serre on the Congruence Subgroup Problem, we thereby prove a conjecture of Larsen and Lubotzky on lattices in higher-rank semisimple groups for algebraic groups of type A_2 defined over number fields. Comment: 62 pages

R.~Lawrence has conjectured that for rational homology spheres, the series of Ohtsuki's invariants converges p-adicly to the SO(3) Witten-Reshetikhin-Turaev invariant. We prove this conjecture for Seifert rational homology spheres. We also derive it for manifolds constructed by a surgery on a knot in S^3. Our derivation is based on a conjecture about the colored Jones polynomial that we have formulated in our previous paper. We also present numerical examples of p-adic convergence for some simple manifolds.

A rational function f(z) of degree d (at least 2) with coefficients in an algebraically closed field is post-critically finite (PCF) if all of its critical points have finite forward orbit under iteration. We show that the collection of PCF rational functions is a set of bounded height in the moduli space of rational functions over the complex numbers, once the well-understood family known as flexible Lattes maps is excluded. As a consequence, there are only finitely many conjugacy classes of non-Lattes PCF rational maps of a given degree defined over any given number field. The key ingredient of the proof is a non-archimedean version of Fatou's classical result that every attracting cycle of a rational function over the complex numbers attracts a critical point.

Let $F$ be a non-Archimedean locally compact field, let $G$ be a split connected reductive group over $F$. For a parabolic subgroup $Q\subset G$ and a ring $L$ we consider the $G$-representation on the $L$-module$$(*)\quad\quad\quad\quad C^{\infty}(G/Q,L)/\sum_{Q'\supsetneq Q}C^{\infty}(G/Q',L).$$Let $I\subset G$ denote an Iwahori subgroup. We define a certain free finite rank $L$-module ${\mathfrak M}$ (depending on $Q$; if $Q$ is a Borel subgroup then $(*)$ is the Steinberg representation and ${\mathfrak M}$ is of rank one) and construct an $I$-equivariant embedding of $(*)$ into $C^{\infty}(I,{\mathfrak M})$. This allows the computation of the $I$-invariants in $(*)$. We then prove that if $L$ is a field with characteristic equal to the residue characteristic of $F$ and if $G$ is a classical group, then the $G$-representation $(*)$ is irreducible. This is the analog of a theorem of Casselman (which says the same for $L={\mathbb C}$); it had been conjectured by Vign\'eras. Herzig (for $G={\rm GL}_n(F)$) and Abe (for general $G$) have given classification theorems for irreducible admissible modulo $p$ representations of $G$ in terms of supersingular representations. Some of their arguments rely on the present work.

We study the properties of Eisenstein-Kronecker numbers, which are related to special values of Hecke $L$-function of imaginary quadratic fields. We prove that the generating function of these numbers is a reduced (normalized or canonical in some literature) theta function associated to the Poincare bundle of an elliptic curve. We introduce general methods to study the algebraic and $p$-adic properties of reduced theta functions for CM abelian varieties. As a corollary, when the prime $p$ is ordinary, we give a new construction of the two-variable $p$-adic measure interpolating special values of Hecke $L$-functions of imaginary quadratic fields, originally constructed by Manin-Vishik and Katz. Our method via theta functions also gives insight for the case when $p$ is supersingular. The method of this paper will be used in subsequent papers to study the precise $p$-divisibility of critical values of Hecke $L$-functions associated to Hecke characters of quadratic imaginary fields for supersingular $p$, as well as explicit calculation in two-variables of the $p$-adic elliptic polylogarithm for CM elliptic curves.

We prove the vanishing of the geometric Bloch-Kato Selmer group for the adjoint representation of a Galois representation associated to regular algebraic polarized cuspidal automorphic representations under an assumption on the residual image. Using this, we deduce that the localization and completion of a certain universal deformation ring for the residual representation at the characteristic zero point induced from the automorphic representation is formally smooth of the correct dimension. We do this by employing the Taylor-Wiles-Kisin patching method together with Kisin's technique of analyzing the generic fibre of universal deformation rings. Along the way we give a characterization of smooth closed points on the generic fibre of Kisin's potentially semistable local deformation rings in terms of their Weil-Deligne representations.

We study the vertex algebras associated with modular invariant representations of affine Kac-Moody algebras at fractional levels, whose simple highest weight modules are classified by Joseph's characteristic varieties. We show that an irreducible highest weight representation of a non-twisted affine Kac-Moody algebra at an admissible level k is a module over the associated simple affine vertex algebra if and only if it is an admissible representation whose integral root system is isomorphic to that of the vertex algebra itself. This in particular proves the conjecture of Adamovic and Milas on the rationality of admissible affine vertex algebras in the category O.

We show that the zero locus of a normal function on a smooth complex algebraic variety S is algebraic provided that the normal function extends to a admissible normal function on a smooth compactification of S with torsion singularity. This result generalizes our previous result for admissible normal functions on curves [arxiv:math/0604345 [math.AG]]. It has also been obtained by M. Saito using a different method in a recent preprint [arXiv:0803.2771v2].

This paper is the first in a series that describe a conjectural analog of the geometric Satake isomorphism for an affine Kac-Moody group. In this paper we construct a model for the singularities of some would-be Schubert varieties in the affine Grassmannian for an affine Kac-Moody group. We formulate a conjecture describing the (local) intersection cohomology of these varieties in terms of integrable representations of the Langlands dual affine Kac-Moody group and check this conjecture in a number of cases. Roughly speaking the above singularities are constructed by looking at the Uhlenbeck space of instantons on the quotient of the affine plane by a finite cyclic subgroup of SL(2). Comment: Section 7 has been rewritten to correct previous mistakes

Assuming a certain "purity" conjecture, we derive a formula for the (complex) cohomology groups of the affine Springer fiber corresponding to any unramified regular semi-simple element. We use this calculation to present a complex analog of the fundamental lemma for function fields. We show that the "kappa" orbital integral which arises in the fundamental lemma is equal to the Lefschetz trace of Frobenius acting on the etale cohomology of a related variety.

We give a cohomological classification of vector bundles on smooth affine threefolds over algebraically closed fields having characteristic unequal to 2. As a consequence, if A is a smooth affine algebra of dimension 3 over an algebraically closed field having characteristic unequal to 2, we deduce that cancellation holds for arbitrary rank projective modules. The proofs of these results involve three main ingredients. First, we give a description of the second unstable A^1-homotopy sheaf of the general linear group. Second, these computations can be used in concert with F. Morel's A^1-homotopy classification of vector bundles on smooth affine schemes and obstruction theoretic techniques (stemming from a version of the Postnikov tower in A^1-homotopy theory) to reduce the classification results to cohomology vanishing statements. Third, we prove the required vanishing statements.

We prove a Hitchin-Kobayashi correspondence for affine vortices generalizing a result of Jaffe-Taubes for the action of the circle on the affine line. Namely, suppose a compact Lie group K has a Hamiltonian action on a Kaehler manifold X which is either compact or convex at infinity with a proper moment map, and so that stable=semistable for the action of the complexified Lie group G. Then, for some sufficiently divisible integer n, there is a bijection between gauge equivalence classes of K-vortices with target X modulo gauge and isomorphism classes of maps from the weighted projective line P(1,n) to X/G that map the stacky point at infinity P(n) to the semistable locus in X. The results allow the construction and partial computation of the quantum Kirwan map in Woodward, and play a role in the conjectures of Dimofte, Gukov, and Hollande relating vortex counts to knot invariants.

We describe vector valued conjugacy equivariant functions on a group K in two cases -- K is a compact simple Lie group, and K is an affine Lie group. We construct such functions as weighted traces of certain intertwining operators between representations of K. For a compact group $K$, Peter-Weyl theorem implies that all equivariant functions can be written as linear combinations of such traces. Next, we compute the radial parts of the Laplace operators of $K$ acting on conjugacy equivariant functions and obtain a comple- tely integrable quantum system with matrix coefficients, which in a special case coincides with the trigonometric Calogero-Sutherland-Moser multi-particle system. In the affine Lie group case, we prove that the space of equivariant functions having a fixed homogeneity degree with respect to the action of the center of the group is finite-dimensional and spanned by weighted traces of intertwining operators. This space coincides with the space of Wess-Zumino-Witten conformal blocks on an elliptic curve. We compute the radial part of the second order Laplace operator on the affine Lie group acting on equivariant functions, and find that it is a certain parabolic partial differential operator, which degenerates to the elliptic Calogero-Sutherland-Moser hamiltonian as the central charge tends to minus the dual Coxeter number (the critical level). Quantum integrals of this hamiltonian are obtained as radial part of the higher Sugawara operators which are central at the critical level.

We prove that the stably free modules over a smooth affine threefold over an algebraically closed field of characteristic different from 2 are free.

We express the Poisson brackets of local fields of the affine Toda field theories in terms of the Drinfeld-Sokolov dressing operator. For this, we introduce a larger space of fields, containing ``half screening charges'' and ``half integrals of motions''. In addition to local terms, the Poisson brackets contain nonlocal terms related to trigonometric r-matrices.

We study the equivariant K-group of the affine flag manifold with respect to the Borel group action. We prove that the structure sheaf of the (infinite-dimensional) Schubert variety in the K-group is represented by a unique polynomial, which we call the affine Grothendieck polynomial.

We introduce certain raising and lowering operators for Macdonald polynomials (of type $A_{n-1}$) by means of Dunkl operators. The raising operators we discuss are a natural $q$-analogue of raising operators for Jack polynomials introduced by L.Lapointe and L.Vinet. As an application we prove the integrality of double Kostka coefficients. Double analog of the multinomial coefficients are introduced.

This article presents a formula for products of Schubert classes in the quantum cohomology ring of the Grassmannian. We introduce a generalization of Schur symmetric polynomials for shapes that are naturally embedded in a torus. Then we show that the coefficients in the expansion of these toric Schur polynomials, in terms of the regular Schur polynomials, are exactly the 3-point Gromov-Witten invariants; which are the structure constants of the quantum cohomology ring. This construction implies that the Gromov-Witten invariants of the Grassmannian are invariant with respect to the action of a twisted product of the groups S_3, (Z/nZ)^2, and Z/2Z. The last group gives a certain strange duality of the quantum cohomologythat inverts the quantum parameter q. Our construction gives a solution to a problem posed by Fulton and Woodward about the characterization of the powers of the quantum parameter q that occur with nonzero coefficients in the quantum product of two Schubert classes. The strange duality switches the smallest such power of q with the highest power. We also discuss the affine nil-Temperley-Lieb algebra that gives a model for the quantum cohomology.

Let $G$ be a connected reductive group over $\CC$ and let $G^{\vee}$ be the Langlands dual group. Crystals for $G^{\vee}$ were introduced by Kashiwara as certain ``combinatorial skeletons'' of finite-dimensional representations of $G^{\vee}$. For every dominant integral weight of $G^{\vee}$ Kashiwara constructed a canonical crystal. Other (independent) constructions of those crystals were given by Lusztig and Littelmann. It was also shown by Kashiwara and Joseph that the above family of crystals is unique if certain reasonable conditions are imposed. The purpose of this paper is to give another (rather simple) construction of these crystals using the geometry of the affine Grassmannian $\calG_G=G(\calK)/G(\calO)$ of the group $G$, where $\calK=\CC((t))$ is the field of Laurent power series and $\calO=\CC[[t]]$ is the ring of Taylor series. We check that the crystals we construct satisfy the conditions of the uniqueness theorem mentioned above, which shows that our crystals coincide with those constructed in {\it loc. cit}. It would be interesting to find these isomorphisms directly (cf., however, \cite{Lus3}).

We prove that the Poisson deformation functor of an affine (singular) symplectic variety is unobstructed. As a corollary, we prove the following result. For an affine symplectic variety X with a good C -action (where its natural Poisson structure is positively weighted), the following are equivalent. (1) X has a crepant projective resolution. (2) X has a smoothing by a Poisson deformation. A typical example is (the normalization) of a nilpotent orbit closure in a complex simple Lie algebra. By the theorem, one can see which orbit closure has a smoothing by a Poisson deformation.

The canonical basis for quantized universal enveloping algebras associated to the finite--dimensional simple Lie algebras, was introduced by Lusztig. The principal technique is the explicit construction (via the braid group action) of a lattice over $\bz[q^{-1}]$. This allows the algebraic characterization of the canonical basis as a certain bar-invariant basis of $\cl$. Here we present a similar algebraic characterization of the affine canonical basis. Our construction is complicated by the need to introduce basis elements to span the ``imaginary'' subalgebra which is fixed by the affine braid group. Once the basis is found we construct a PBW-type basis whose $\bz[q^{-1}]$-span reduces to a ``crystal'' basis at $q=\infty,$ with the imaginary component given by the Schur functions.

Let ${\mathfrak g}$ be a simple Lie algebra. For a level $\kappa$ (thought of as a symmetric ${\mathfrak g}$-invariant form of ${\mathfrak g}$), let $\hat{\mathfrak g}_\kappa$ be the corresponding affine Kac-Moody algebra. Let $Gr_G$ be the affine Grassmannian of ${\mathfrak g}$, and let $D_\kappa(Gr_G)-mod$ be the category of $\kappa$-twisted right D-modules on $Gr_G$. By taking global sections of a D-module, we obtain a functor $\Gamma:D_\kappa(Gr_G)-mod\to {\mathfrak g}_\kappa-mod$. It is known that this functor is exact and faithful when $\kappa$ is negative or irrational. In this paper, we show that the functor $\Gamma$ is exact and faithful also when $\kappa$ is the critical level.