On se propose d'étendre un résultat de Rorhlich [5] concernant les moyennes galoisiennes des valeurs en s = 1 des fonctions L associées à une forme modulaire de poids 2, tordue par certains caractères de Dirichlet. On considérera une représentation automorphe parabolique π de GL(n), n≦2, sur un corps de nombres K, et l'on s'intéressera aux valeurs des fonctions L(S,πχ) et de leurs dérivées L(m)(s, πχ), m ≧1, où χ parcourt certains caractères de Hecke de K, d'ordre fini, et où s appartient à la bande a < Re s < 1 — a, où a mesure la déviation de n par rapport à la conjecture de Petersson généralisée.

Let D stand for a division ring (or skewfield), let G stand for an ordered abelian group with positive infinity adjoined, and let ω: D → G. We call to a valuation of D with value group G, if ω is an onto mapping from D to G such that

(i) ω(x) = ∞ if and only if x = 0,

(ii) ω(x1 + x2) = min(ω (x1), ω (x2)), and

(iii) ω (x1 x2) = ω (x1) + ω (x2).

Associated to the valuation ω are its valuation ring

R = ﹛x ∈ Dω(x) ≧ 0﹜,

its maximal ideal

J = ﹛x ∈ |ω(x) > 0﹜, and its residue division ring D = R/J.The invertible elements of the ring R are called valuation units. Clearly R and, hence, J are preserved under conjugation so that 1 + J is also preserved under conjugation. The latter is thus a normal subgroup of the multiplicative group Dm of D and hence, the quotient group D˙/1 + J makes sense (the residue group of ω). It enlarges in a natural way the residue division ring D (0 excluded, and addition “forgotten“).

The main result of this paper is the following:

Theorem A. Let H and N be finite groups with coprime orders andsuppose that H acts nontrivially on N via automorphisms. Assume that Hfixes every nonlinear irreducible character of N. Then the derived subgroup ofN is nilpotent and so N is solvable of nilpotent length≦ 2.

Why might one be interested in a situation like this? There has been considerable interest in the question of what one can deduce about a group Gfrom a knowledge of the set

cd(G) = ﹛x(l)lx ∈ Irr(G) ﹜

of irreducible character degrees of G.Recently, attention has been focused on the prime divisors of the elements of cd(G). For instance, in [9], O. Manz and R. Staszewski consider π-separable groups (for some set π of primes) with the property that every element of cd(G) is either a 77-number or a π'-number.

It was not always clear that there could exist a continuous function which was differentiable at no point. (Such functions are now known as nowhere differentiable continuous functions. By “differentiable” we mean having a finite derivative.) In fact in 1806 M. Ampere [2] even tried to show that no such function could exist but his reasonings were later discovered to be fallacious. Of the early attempts at constructing a nowhere differentiable continuous function mention must be made of B. Bolzano. In a manuscript dated around 1830, (see [21]) he constructed a continuous function on an interval and showed that it was not differentiable on a dense set of points. (It was later shown by K. Rychlik [21] that this function was in fact nowhere differentiable.)

The main result of this paper concerns the eigenvalues of an operator in the Hilbert space l2that is represented by a matrix having zeros everywhere except in a neighborhood of the main diagonal. Write (c)+ for the positive part of a real number c, i.e., put (c+ = cif c≧ 0 and (c)+=0 otherwise. Then this result can be formulated as follows. Theorem 1.1. Let k ≧ 1 be an integer, and consider the operator S on l2 such that

This paper has been motivated by earlier work of the first two authors (see [3] ), where distinct Nielsen classes of generating systems for a Fuchsian group have been established and, in the case of odd and pairwise relative prime exponents π(i),classified. As a consequence they could distinguish nonisotopic Heegaard decompositions of Seifert fibred 3-manifolds. In proving that these decompositions are actually non-homeomorphic (see [3], Section 2), they investigated the question whether the different Nielsen classes of generating systems for G remain distinct, if one passes over to the weaker notion of “Nielsen equivalence up to automorphisms” (see [12], p. 3.5, 4.11 a-c): this means that the automorphisms of G are added to the Nielsen equivalence relations on the generators.

We are concerned in this paper with the study of homomorphisms between different algebras of continuous functions, especially the algebras of real functions which are either weakly continuous on bounded sets or weakly uniformly continuous on bounded sets on a Banach space (see definitions below).

These spaces of weakly [uniformly] continuous functions appeared in relation with some questions in Infinite-dimensional Approximation Theory (see [4], [6], [11], [12], [13] and [16]); and since the structure of these function spaces is closely related with properties of different weak topologies (the bounded-weak and bounded-weak* topologies, respectively) and with the structure of Banach spaces on which they are defined, their study also presents interest from the point of view of Banach space theory, as can be seen in [2], [12] or [17].

For homogeneous spaces of a (real) Lie group one of the fundamental results concerning ends (in the sense of Freudenthal [8] ) is due to A. Borel [6]. He showed that if X = G/H is the homogeneous space of a connected Lie group G by a closed connected subgroup H, then X has at most two ends. And if X does have two ends, then it is diffeomorphic to the product of R with the orbit of a maximal compact subgroup of G.

In the setting of homogeneous complex manifolds the basic idea should be to find conditions which imply that the space has at most two ends and then, when the space has exactly two ends, to display the ends via bundles involving C* and compact homogeneous complex manifolds. An analytic condition which ensures that a homogeneous complex manifold X has at most two ends is that X have non-constant holomorphic functions and the structure of such a space with exactly two ends is determined, namely, it fibers over an affine homogeneous cone with its vertex removed with the fiber being compact [9], [13].

We write formally (C, p) indicating that the integral is summable (C, p), i.e.,if this limit exists. We note here that all integrals over a finite range are taken in the Lebesgue sense, and all inversions of such iterated integrals are justifiable by Fubini's Theorem.