Locally finite dimensional shift-invariant spaces in

We prove that a locally finite dimensional shift-invariant linear space of distributions must be a linear subspace of some shift-invariant space generated by finitely many compactly supported distributions. If the locally finite dimensional shift-invariant space is a subspace of the Hölder continuous space or the fractional Sobolev space , then the superspace can be chosen to be or , respectively.

     

Hausdorff measures and dimension on

We consider the Hausdorff measures , , defined on with the topology induced by the metric

for all . We study its properties, their relation to the ``Lebesgue measure" defined on by R. Baker in 1991, and the associated Hausdorff dimension. Finally, we give some examples.

     

Local automorphisms and derivations on

Let be an algebra. A mapping is called a -local automorphism if for every there is an automorphism , depending on and , such that and (no linearity, surjectivity or continuity of is assumed). Let be an infinite-dimensional separable Hilbert space, and let be the algebra of all linear bounded operators on . Then every -local automorphism is an automorphism. An analogous result is obtained for derivations.

     

On the structure of

We show that for with its geometrically induced structure maps is not an Hopf algebroid because neither the augmentation nor the coproduct are multiplicative. As a consequence the algebra structure of is slightly different from what was supposed to be the case. We give formulas for and and show that the inversion of the formal group of is induced by an antimultiplicative involution . Some consequences for multiplicative and antimultiplicative automorphisms of for are also discussed.

     

Finite dimensional representations of

Structures of the finite dimensional simple weight -modules are studied in detail for both the generic and the roots of 1 cases.

     

Local complete intersections in and Koszul syzygies

We study the syzygies of a codimension two ideal . Our main result is that the module of syzygies vanishing (scheme-theoretically) at the zero locus is generated by the Koszul syzygies iff is a local complete intersection. The proof uses a characterization of complete intersections due to Herzog. When is saturated, we relate our theorem to results of Weyman and Simis and Vasconcelos. We conclude with an example of how our theorem fails for four generated local complete intersections in and we discuss generalizations to higher dimensions.

     

and applications

We study an analogue of the classical theory of weights in without assuming that the underlying measure is doubling. Then, we obtain weighted norm inequalities for the (centered) Hardy-Littlewood maximal function and corresponding weighted estimates for nonclassical Calderón-Zygmund operators. We also consider commutators of those Calderón- Zygmund operators with bounded mean oscillation functions (), extending the main result from R. Coifman, R. Rochberg, and G. Weiss, Factorization theorems for Hardy spaces in several variables, Ann. of Math. 103 (1976), 611-635. Finally, we study self-improving properties of Poincaré-B.M.O. type inequalities within this context; more precisely, we show that if is a locally integrable function satisfying for all cubes , then it is possible to deduce a higher integrability result for , assuming a certain simple geometric condition on the functional .     

Local Rankin-Selberg convolutions for : explicit conductor formula

Let be a non-Archimedean local field and , positive integers. For , let and let be an irreducible supercuspidal representation of . Jacquet, Piatetskii-Shapiro and Shalika have defined a local constant to the and an additive character of . This object is of central importance in the study of the local Langlands conjecture. It takes the form

where is an integer. The irreducible supercuspidal representations of have been described explicitly by Bushnell and Kutzko, via induction from open, compact mod centre, subgroups of . This paper gives an explicit formula for in terms of the inducing data for the . It uses, on the one hand, the alternative approach to the local constant due to Shahidi, and, on the other, the general theory of types along with powerful existence theorems for types in , developed by Bushnell and Kutzko.

     

Lévy group action and invariant measures on

For let be defined by . We investigate permutations of , which satisfy as with for (i.e. is in the Lévy group , or for in the subspace of Cesàro-summable sequences. Our main interest are -invariant means on or equivalently -invariant probability measures on . We show that the adjoint of maps measures supported in onto a weak*-dense subset of the space of -invariant measures. We investigate the dynamical system and show that the support set of invariant measures on is the closure of the set of almost periodic points and the set of non-topologically transitive points in . Finally we consider measures which are invariant under .

     

Shintani functions on

In this paper, in analogy to the real case, we give a formulation of the Shintani functions on , which have been studied by Murase and Sugano within the theory of automorphic -functions. Also, we obtain the multiplicity one theorem for these functions and an explicit formula in a special case.

     

On and discriminants

We modify the -conjecture for number fields in order to make the support (like the height) well-behaved under field extensions. We show further that the exponent 1$"> of the absolute value of the discriminant cannot be replaced by , and even that an arbitrarily large power of must be present.

     

Certain invariant subspace structure of

In this note, we study certain structure of an invariant subspace of . Considering the largest -invariant (resp. -invariant) subspace in the wandering subspace of with respect to the shift operator , we give an alternative characterization of Beurling-type invariant subspaces. Furthermore, we consider a certain class of invariant subspaces.

     

Mirror symmetry and

We show that counting functions of covers of are equal to sums of integrals associated to certain `Feynman' graphs. This is an analogue of the mirror symmetry for elliptic curves.

     

Exotic smooth structures on

We construct exotic and as a corollary of recent results of I. Dolgachev and C. Werner concerning a numerical Godeaux surface. We also construct another exotic using the surgery techniques of R. Fintushel and R. J. Stern. We show that these 4-manifolds are irreducible by computing their Seiberg-Witten invariants.

     

Locally analytic distributions and

In this paper we study continuous representations of locally -analytic groups in locally convex -vector spaces, where is a finite extension of and is a spherically complete nonarchimedean extension field of . The class of such representations includes both the smooth representations of Langlands theory and the finite dimensional algebraic representations of , along with interesting new objects such as the action of on global sections of equivariant vector bundles on -adic symmetric spaces. We introduce a restricted category of such representations that we call ``strongly admissible' and we show that, when is compact, our category is anti-equivalent to a subcategory of the category of modules over the locally analytic distribution algebra of . As an application we prove the topological irreducibility of generic members of the -adic principal series for . Our hope is that our definition of strongly admissible representation may be used as a foundation for a general theory of continuous -valued representations of locally -analytic groups.

     

representations and combinatorial identities

For various probability measures on the space of the infinite standard Young tableaux we study the probability that in a random tableau, the entry equals a given number . Beside the combinatorics of finite standard tableaux, the main tools here are from the Vershik-Kerov character theory of . The analysis of these probabilities leads to many explicit combinatorial identities, some of which are related to hypergeometric series.

     

harmonic approximation on closed sets

In this paper the harmonic approximation ( ) on relatively closed subsets of a domain in the complex plane is characterized under the same conditions given by S. Gardiner for the uniform case. Thus, the result of P. Paramonov on harmonic polynomial approximation for compact subsets is extended to closed sets. Moreover, the problem of uniform approximation with continuous extension to the boundary for harmonic functions and similar questions in harmonic approximation are also studied.

     

Tensor product varieties and crystals: case

A geometric theory of tensor product for -crystals is described. In particular, the role of Spaltenstein varieties in the tensor product is explained, and thus a direct (non-combinatorial) proof of the fact that the number of irreducible components of a Spaltenstein variety is equal to a Littlewood-Richardson coefficient (i.e. certain tensor product multiplicity) is obtained.