Représentations -adiques et normes universelles I. Le cas cristallin

Let be a crystalline -adic representation of the absolute Galois group of an finite unramified extension of , and let be a lattice of stable by . We prove the following result: Let be the maximal sub-representation of with Hodge-Tate weights strictly positive and . Then, the projective limit of is equal up to torsion to the projective limit of . So its rank over the Iwasawa algebra is .

     

Parametrization of local CR automorphisms by finite jets and applications

For any real-analytic hypersurface , which does not contain any complex-analytic subvariety of positive dimension, we show that for every point the local real-analytic CR automorphisms of fixing can be parametrized real-analytically by their jets at . As a direct application, we derive a Lie group structure for the topological group . Furthermore, we also show that the order of the jet space in which the group embeds can be chosen to depend upper-semicontinuously on . As a first consequence, it follows that given any compact real-analytic hypersurface in , there exists an integer depending only on such that for every point germs at of CR diffeomorphisms mapping into another real-analytic hypersurface in are uniquely determined by their -jet at that point. Another consequence is the following boundary version of H. Cartan's uniqueness theorem: given any bounded domain with smooth real-analytic boundary, there exists an integer depending only on such that if is a proper holomorphic mapping extending smoothly up to near some point with the same -jet at with that of the identity mapping, then necessarily .

Our parametrization theorem also holds for the stability group of any essentially finite minimal real-analytic CR manifold of arbitrary codimension. One of the new main tools developed in the paper, which may be of independent interest, is a parametrization theorem for invertible solutions of a certain kind of singular analytic equations, which roughly speaking consists of inverting certain families of parametrized maps with singularities.

     

Homogeneous Hilbert scheme

Let be a locally noetherian scheme and an -graded -algebra of finite type. We say that is a homogeneous variety over . In this paper we prove that the functor

is representable by an -scheme that is a disjoint union of locally projective schemes over . The proof is very simple, and it only makes use of the theory of graded modules and standard flatness criteria. From this, one obtains an elementary construction (which does not make use of cohomology) of the ordinary Hilbert scheme of a locally projective -scheme.

     

Norming algebras and automatic complete boundedness of isomorphisms of operator algebras

We combine the notion of norming algebra introduced by Pop, Sinclair and Smith with a result of Pisier to show that if and are operator algebras, then any bounded epimorphism of onto is completely bounded provided that contains a norming -subalgebra. We use this result to give some insights into Kadison's Similarity Problem: we show that every faithful bounded homomorphism of a -algebra on a Hilbert space has completely bounded inverse, and show that a bounded representation of a -algebra is similar to a -representation precisely when the image operator algebra -norms itself. We give two applications to isometric isomorphisms of certain operator algebras. The first is an extension of a result of Davidson and Power on isometric isomorphisms of CSL algebras. Secondly, we show that an isometric isomorphism between subalgebras of -diagonals () satisfying extends uniquely to a -isomorphism of the -algebras generated by and ; this generalizes results of Muhly-Qiu-Solel and Donsig-Pitts.

     

Optimal constants in the exceptional case of Sobolev inequalities on Riemannian manifolds

Let be a Riemannian compact -manifold. We know that for any , there exists such that for any , , being the smallest constant possible such that the inequality remains true for any . We call the ``first best constant'. We prove in this paper that it is possible to choose and keep a finite constant. In other words we prove the existence of a ``second best constant' in the exceptional case of Sobolev inequalities on compact Riemannian manifolds.

     

Semi-invariants of quivers and saturation for Littlewood-Richardson coefficients

Let be a quiver without oriented cycles. For a dimension vector let be the set of representations of with dimension vector . The group acts on . In this paper we show that the ring of semi-invariants is spanned by special semi-invariants associated to representations of . From this we show that the set of weights appearing in is saturated. In the case of triple flag quiver this reduces to the results of Knutson and Tao on the saturation of the set of triples of partitions for which the Littlewood-Richardson coefficient is nonzero.

     

On a correspondence between cuspidal representations of

Let be an irreducible, automorphic, self-dual, cuspidal representation of , where is the adele ring of a number field . Assume that has a pole at and that . Given a nontrivial character of , we construct a nontrivial space of genuine and globally -generic cusp forms on -the metaplectic cover of . is invariant under right translations, and it contains all irreducible, automorphic, cuspidal (genuine) and -generic representations of , which lift (``functorially, with respect to ") to . We also present a local counterpart. Let be an irreducible, self-dual, supercuspidal representation of , where is a -adic field. Assume that has a pole at . Given a nontrivial character of , we construct an irreducible, supercuspidal (genuine) -generic representation of , such that has a pole at , and we prove that is the unique representation of satisfying these properties.

     

Measurable rigidity of actions on infinite measure homogeneous spaces, II

We consider the problems of measurable isomorphisms and joinings, measurable centralizers and quotients for certain classes of ergodic group actions on infinite measure spaces. Our main focus is on systems of algebraic origin: actions of lattices and other discrete subgroups on homogeneous spaces where is a sufficiently rich unimodular subgroup in a semi-simple group . We also consider actions of discrete groups of isometries of a pinched negative curvature space , acting on the space of horospheres . For such systems we prove that the only measurable isomorphisms, joinings, quotients, etc., are the obvious algebraic (or geometric) ones. This work was inspired by the previous work of Shalom and Steger but uses completely different techniques which lead to more general results.

     

On the equation and application to control of phases

The main result is the following. Let be a bounded Lipschitz domain in , . Then for every with , there exists a solution of the equation div in , satisfying in addition on and the estimate

where depends only on . However one cannot choose depending linearly on .

Our proof is constructive, but nonlinear--which is quite surprising for such an elementary linear PDE. When there is a simpler proof by duality--hence nonconstructive.

     

Simple homogeneous models

Geometrical stability theory is a powerful set of model-theoretic tools that can lead to structural results on models of a simple first-order theory. Typical results offer a characterization of the groups definable in a model of the theory. The work is carried out in a universal domain of the theory (a saturated model) in which the Stone space topology on ultrafilters of definable relations is compact. Here we operate in the more general setting of homogeneous models, which typically have noncompact Stone topologies. A structure equipped with a class of finitary relations is strongly -homogeneous if orbits under automorphisms of have finite character in the following sense: Given an ordinal and sequences , from , if and have the same orbit, for all and , then for some automorphism of . In this paper strongly -homogeneous models in which the elements of induce a symmetric and transitive notion of independence with bounded character are studied. This notion of independence, defined using a combinatorial condition called ``dividing', agrees with forking independence when is saturated. A concept central to the development of stability theory for saturated structures, namely parallelism, is also shown to be well-behaved in this setting. These results broaden the scope of the methods of geometrical stability theory.

     

Integral motives and special values of zeta functions

For each field , we define a category of rationally decomposed mixed motives with -coefficients. When is finite, we show that the category is Tannakian, and we prove formulas relating the behaviour of zeta functions near integers to certain groups.

     

Minkowski's conjecture, well-rounded lattices and topological dimension

Let be the diagonal subgroup, and identify with the space of unimodular lattices in . In this paper we show that the closure of any bounded orbit

meets the set of well-rounded lattices. This assertion implies Minkowski's conjecture for and yields bounds for the density of algebraic integers in totally real sextic fields.

The proof is based on the theory of topological dimension, as reflected in the combinatorics of open covers of and .

     

Convex solids with planar midsurfaces

We show that the boundary of an -dimensional closed convex set , possibly unbounded, is a convex quadric surface if and only if the middle points of every family of parallel chords of lie in a hyperplane. To prove this statement, we show that the boundary of is a convex quadric surface if and only if there is a point such that all sections of by 2-dimensional planes through are convex quadric curves. Generalizations of these statements that involve boundedly polyhedral sets are given.

     

Curvature and injectivity radius estimates for Einstein 4-manifolds

Let denote an Einstein -manifold with Einstein constant, , normalized to satisfy . For , a metric ball, we prove a uniform estimate for the pointwise norm of the curvature tensor on , under the assumption that the -norm of the curvature on is less than a small positive constant, which is independent of , and which in particular, does not depend on a lower bound on the volume of . In case , we prove a lower injectivity radius bound analogous to that which occurs in the theorem of Margulis, for compact manifolds with negative sectional curvature, . These estimates provide key tools in the study of singularity formation for -dimensional Einstein metrics. As one application among others, we give a natural compactification of the moduli space of Einstein metrics with negative Einstein constant on a given .

     

Convexity and the Exterior Inverse Problem of Potential Theory

Let and be bounded solid domains such that their associated volume potentials agree outside . Under the assumption that one of the domains is convex, it is deduced that .

     

Kazhdan-Lusztig polynomials and character formulae for the Lie superalgebra

We compute the characters of the finite dimensional irreducible representations of the Lie superalgebra , and determine 's between simple modules in the category of finite dimensional representations. We formulate conjectures for the analogous results in category . The combinatorics parallels the combinatorics of certain canonical bases over the Lie algebra .

     

Multivariable cochain operations and little -cubes

In this paper we construct a small chain operad which acts naturally on the normalized cochains of a topological space. We also construct, for each , a suboperad which is quasi-isomorphic to the normalized singular chains of the little -cubes operad. The case leads to a substantial simplification of our earlier proof of Deligne's Hochschild cohomology conjecture.

     

On the Cachazo-Douglas-Seiberg-Witten conjecture for simple Lie algebras

We prove a part of the Cachazo-Douglas-Seiberg-Witten conjecture uniformly for any simple Lie algebra . The main ingredients in the proof are: Garland's result on the Lie algebra cohomology of ; Kostant's result on the `diagonal' cohomolgy of and its connection with abelian ideals in a Borel subalgebra of ; and a certain deformation of the singular cohomology of the infinite Grassmannian introduced by Belkale-Kumar.

     

Long arithmetic progressions in sumsets: Thresholds and bounds

For a set of integers, the sumset consists of those numbers which can be represented as a sum of elements of :

Closely related and equally interesting notion is that of , which is the collection of numbers which can be represented as a sum of different elements of :

The goal of this paper is to investigate the structure of and , where is a subset of . As application, we solve two conjectures by Erdös and Folkman, posed in 1960s.