New identities of differential operators from orbital integrals on

We derive identities of differential operators on complex general linear groups which appear in the differential equations satisfied by weighted orbital integrals. These identities stem from and have applications to comparisons of metaplectic coverings.

     

Low-dimensional unitary representations of

We characterize all simple unitarizable representations of the braid group on complex vector spaces of dimension . In particular, we prove that if and denote the two generating twists of , then a simple representation (for ) is unitarizable if and only if the eigenvalues of are distinct, satisfy and 0$"> for , where the are functions of the eigenvalues, explicitly described in this paper.

     

Completely isometric representations of

Let be a locally compact group. It is shown that there exists a natural completely isometric representation of the completely bounded Fourier multiplier algebra , which is dual to the representation of the measure algebra , on . The image algebras of and in are intrinsically characterized, and some commutant theorems are proved. It is also shown that for any amenable group , there is a natural completely isometric representation of on , which can be regarded as a duality result of Neufang's completely isometric representation theorem for .

     

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.

     

Mod representations of elliptic curves

Explicit equations are given for the elliptic curves (in characteristic ) with mod representation isomorphic to that of a given one.

     

Mod Galois representations of solvable image

It is proved that, for a number field and a prime number , there exist only finitely many isomorphism classes of continuous semisimple Galois representations of into of fixed dimension and bounded Artin conductor outside which have solvable images. Some auxiliary results are also proved.

     

The characters of supercuspidal representations as weighted orbital integrals

Weighted orbital integrals are the terms which occur on the geometric side of the trace formula. We shall investigate these distributions on ap-adic group. We shall evaluate the weighted orbital integral of a supercuspidal matrix coefficient as a multiple of the corresponding character. Supported in part by NSERC Operating Grant A3483.     

Self-contragredient supercuspidal representations of

Let be a non-archimedean local field of residual characteristic . Then has tamely ramified self-contragredient supercuspidal representations if and only if or is even. When such representations exist, they do so in abundance.

     

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.

     

New rapidly convergent series representations for

We give three series representations for the values of the Riemann zeta function at positive odd integers. One representation extends Ewell's result for [Amer. Math. Monthly 97 (1990), 219-220] and is considerably simpler than the two generalisations proposed earlier. The second representation is even simpler:

where the coefficients for a fixed are rational in and are explicitly given by the finite sum involving the Bernoulli numbers. The third representation is obtained from the second by the Kummer transformation. We demonstrate the rapid convergence of this series using several examples.

     

Generalized principal series representations of

We consider certain induced representations of the group
realized on line bundles over the projective space of . We calculate the intertwining operators in the compact picture. We find all the unitarizable representations and determine the invariant norm.

     

Induction formula for the Artin conductors of mod Galois representations

A formula is given to describe how the Artin conductor of a mod Galois representation behaves with respect to induction of the representation.

     

Abstract competitive systems and orbital stability in

Competitive autonomous systems in have the remarkable property of verifying an analogue of the Poincaré-Bendixon theorem for planar equations. This fact allows us to prove the existence of orbitally stable closed orbits for those systems under easily checkable hypothesis. Our aim is to introduce, by changing the ordering in , a new class of autonomous systems for which the preceding results directly extend. As a consequence we shall reinterpret some of the results of R. A. Smith in terms of the theory of monotone systems.

     

Distinguished representations and quadratic base change for

Let be a quadratic extension of number fields. Suppose that every real place of splits in and let be the unitary group in 3 variables. Suppose that is an automorphic cuspidal representation of . We prove that there is a form in the space of such that the integral of over is non zero. Our proof is based on earlier results and the notion, discussed in this paper, of Shalika germs for certain Kloosterman integrals.

     

A criterion on weighted boundedness for rough multilinear oscillatory singular integrals

In this paper the authors give a criterion on the weighted boundedness of the multilinear oscillatory singular integral operators with rough kernels.

     

On the correspondence of representations between and division algebras

For a division algebra over a -adic field we prove that depth is preserved under the correspondence of discrete series representations of and irreducible representations of by proving that an explicit relation holds between depth and conductor for all such representations. We also show that this relation holds for many (possibly all) discrete series representations of

     

boundedness of localization operators associated to left regular representations

We prove an boundedness result for localization operators associated to left regular representations of locally compact and Hausdorff groups and give an application to wavelet multipliers.

     

Oscillatory singular integrals on and Hardy spaces

We consider boundedness properties of oscillatory singular integrals on and Hardy spaces. By constructing a phase function, we prove that boundedness may fail while boundedness holds for all . This shows that the theory and theory for such operators are fundamentally different.

     

Blowup for

In this note we consider the global regularity of smooth solutions to the vector-valued Cauchy problem

We show that if , the gradient-blowup phenomenon occurs in finite time for suitably chosen vanishing at infinity. We also present a simple example of the -blowup solutions for for any 0$">, if .