Let $A$ and $B$ be C*-algebras, with $A$ separable and $B$$\sigma$-unital and stable. It is shown that there are natural isomorphisms

$$ E(A,B) = KK(SA,Q(B)) = [SA,Q(B) \otimes K], $$

where $SA=C_0(0, 1) \otimes A$, $[\cdot, \cdot]$ denotes the set of homotopy classes of *-homomorphisms, $Q(B) = M(B) / B$ is the generalized Calkin algebra, and $K$ denotes the C*-algebra of compact operators of an infinite-dimensional separable Hilbert space.

In this paper we obtain optimal bounds for the length of the longest arithmetic progression in various kinds of sum-sets. As an application, we derive a sharp estimate for the number of sets $A$ of residues modulo a prime $n$ such that no subsum of $A$ equals $x$ modulo $n$, where $x$ is a fixed residue modulo $n$.

In this paper we determine the possible Hilbert functions of a Cohen--Macaulay local ring of dimension $d$ and multiplicity $e$, in the case where the embedding dimension $v$ satisfies $v=e+d-3$ and the Cohen--Macaulay type is less than or equal to $e-3$.

1991 Mathematics Subject Classification: primary 13D40; secondary 13P99.

The concept of generalized functions taking values in a differentiable manifold is extended to a functorial theory. We establish several characterization results which allow a global intrinsic formulation both of the theory of manifold-valued generalized functions and of generalized vector bundle homomorphisms. As a consequence, a characterization of equivalence that does not resort to derivatives (analogous to scalar-valued cases of Colombeau's construction) is achieved. These results are employed to derive a point value description of all types of generalized functions valued in manifolds and to show that composition can be carried out unrestrictedly. Finally, a new concept of association adapted to the present setting is introduced.

A theorem of J. Hersch (1970) states that for any smooth metric on $S^2$, with total area equal to $4\pi$, the first non-zero eigenvalue of the Laplace operator acting on functions is less than or equal to 2 (this being the value for the standard round metric). For metrics invariant under the standard $S^1$-action on $S^2$, one can restrict the Laplace operator to the subspace of $S^1$-invariant functions and consider its spectrum there. The corresponding eigenvalues will be called invariant eigenvalues, and the purpose of this paper is to analyse their possible values.

We first show that there is no general analogue of Hersch's theorem, by exhibiting explicit families of $S^1$-invariant metrics with total area $4\pi$, where the first invariant eigenvalue ranges through any value between 0 and $\infty$. We then restrict ourselves to $S^1$-invariant metrics that can be embedded in $\mathbb{R}^3$ as surfaces of revolution. For this subclass we are able to provide optimal upper bounds for all invariant eigenvalues. As a consequence, we obtain an analogue of Hersch's theorem with an optimal upper bound (greater than 2 and geometrically interesting). This subclass of metrics on $S^2$ includes all $S^1$-invariant metrics with non-negative Gauss curvature.

One of the key ideas in the proofs of these results comes from symplectic geometry, and amounts to the use of the moment map of the $S^1$-action as a coordinate function on $S^2$.

2000 Mathematical Subject Classification: primary 35P15; secondary 58J50, 53D20.

In this paper we consider Fourier multipliers for $L^p$ $(p>1)$ on Chébli-Trimèche hypergroups and establish a version of Hörmander's multiplier theorem. As applications we give some results concerning the Riesz potentials and oscillating multipliers.

1991 Mathematics Subject Classification: 43A62, 43A15, 43A32.

The aim of the paper is to construct a calculus of pseudodifferential operators (PDOs) on a smooth manifold $M$ without using local coordinate systems. Instead we deal with linear connections $\Gamma$ of $M$.

The fact that a linear connection $\Gamma$ is a global object enables one to associate with a PDO its full symbol, which is a function on the cotangent bundle $T^*M$ (depending on the choice of $\Gamma\,$). This idea was put forward by H. Widom. He has suggested a method of defining full symbols of PDOs on a manifold with a linear connection and constructed a version of symbolic calculus. However, H. Widom defined the classes of PDOs in local coordinates, using standard local phase functions.

On the contrary, in the paper the PDOs are defined in a coordinate-free way, with use of invariant oscillatory integrals over $T^*M\,$. The invariant approach allows us to define $\tau$-symbols of pseudodifferential operators acting on a manifold; in particular, we introduce the Weyl symbols. In fact, it is a kind of quantization on a manifold provided with a linear connection.

The results are applied to study of semi-elliptic differential operators and functions of the Laplacian on a closed Riemannian manifold.

1991 Mathematical Subject Classification: 35S05, 58G30.

We show that if $G$ is a group of exponent 5, and if $G$ satisfies the Engel-4 identity $[x,y,y,y,y]=1$, then $G$ is locally finite. By a result of Traustason, this implies that Engel-4 5-groups are locally finite. We also show that a group of exponent 5 is locally finite if and only if it satsifies the identity

$$[x,[y,z,z,z,z],[y,z,z,z,z]]=1.$$

This result implies that a group of exponent 5 is locally finite if its three generator subgroups are finite.

1991 Mathematics Subject Classification: 20D15, 20F45, 20F50.

We prove that the zeta function and normal zeta function of a virtually abelian group have meromorphic continuation to the whole complex plane. We do this by relating the functions to classical L-functions of arithmetic orders considered by Hey, Solomon, Bushnell and Reiner. We calculate the zeta functions and normal zeta functions of the plane crystallographic groups. As a corollary of these calculations we produce \begin{enumerate} \item[(1)] examples of two isospectral residually finite groups with non-isomorphic profinite completions and even distinct lattices of subgroups; and \item[(2)] examples of non-nilpotent residually finite groups whose zeta functions enjoy an Euler product.

1991 Mathematics Subject Classification: 11M41, 20H15.

Let $L(V)$ be the free Lie algebra on a finite-dimensional vector space $V$ over a field $K$, with homogeneous components $L^n(V)$ for $n \geq 1$. If $G$ is a group and $V$ is a $KG$-module, the action of $G$ extends naturally to $L(V)$, and the $L^n(V)$ become finite-dimensional $KG$-modules, called the Lie powers of $V$. In the decomposition problem, the aim is to identify the isomorphism types of indecomposable $KG$-modules, with their multiplicities, in unrefinable direct decompositions of the Lie powers. This paper is concerned with the case where $G$ has prime order $p$, and $K$ has characteristic $p$. As is well known, there are $p$ indecomposables, denoted here by $J_1,\dots,J_p$, where $J_r$ has dimension $r$. A theory is developed which provides information about the overall module structure of $L(V)$ and gives a recursive method for finding the multiplicities of $J_1,\dots,J_p$ in the Lie powers $L^n(V)$. For example, the theory yields decompositions of $L(V)$ as a direct sum of modules isomorphic either to $J_1$ or to an infinite sum of the form $J_r \oplus J_{p-1} \oplus J_{p-1} \oplus \ldots $ with $r \geq 2$. Closed formulae are obtained for the multiplicities of $J_1,\dots,J_p$ in $L^n(J_p)$ and $L^n(J_{p-1})$. For $r < p-1$, the indecomposables which occur with non-zero multiplicity in $L^n(J_r)$ are identified for all sufficiently large $n$.

2000 Mathematical Subject Classification: 17B01, 20C20.

We study ergodic random Schrödinger operators on a covering manifold, where the randomness enters both via the potential and the metric. We prove measurability of the random operators, almost sure constancy of their spectral properties, the existence of a self-averaging integrated density of states and a Pastur–šubin type trace formula.

Dedicated to the memory of Professor A. I. Kostrikin

The main problem under discussion is to determine, for quasi-simple groups of Lie type G, irreducible representations $\phi$ of G that remain irreducible under reduction modulo the natural prime p. The method is new. It works only for p >3 and for representations $\phi$ that can be realized over an unramified extension of ${\mathbb Q}_p$, the field of p -adic numbers. Under these assumptions, the main result says that the trivial and the Steinberg representations of G are the only representations in question provided G is not of type A1. This is not true for G=SL(2, p). The paper contains a result of independent interest on infinitesimally irrreducible representations $\rho$ of G over an algebraically closed field of characteristic p. Assuming that G is not of rank 1 and $G\neq G_2(5)$, it is proved that either the Jordan normal form of a root element contains a block of size d with 1<d<p -1 or the highest weight of $\rho$ is equal to p -1 times the sum of the fundamental weights.

2000 Mathematical Subject Classification: 20C33, 20G15.

In this paper a practical algorithm is given to find all binary forms with rational coefficients of given degree with discriminant divisible by a given finite set of rational primes, up to an obvious equivalence relation.

This is done by adapting the finiteness result of Evertse and Gy\H{o}ry. A technical assumption that all fields used have class number $1$ is made to aid the exposition.

All binary forms of degree less than or equal to $6$ with $2$-power discriminant are then calculated. This is then used to complete the table of curves of genus $2$ which had previously been computed by Merriman and the author. Some ranks of the associated Jacobian varieties are also computed.

1991 Mathematics Subject Classification: primary 11E76, 11G30; secondary 11Y40, 11D61.

Let $\varGamma =\varGamma _{\tau ,z}$ be one of the $N^2$-dimensional bicovariant first order differential calculi for the quantum groups $\mathrm{GL}_q(N)$, $\mathrm{SL}_q(N)$, $\mathrm{SO}_q(N)$, or $\mathrm{Sp}_q(N)$, where $q$ is a transcendental complex number and $z$ is a regular parameter. It is shown that the de Rham cohomology of Woronowicz' external algebra $\varGamma^\land $ coincides with the de Rham cohomologies of its left-coinvariant, its right-coinvariant and its (two-sided) coinvariant subcomplexes. In the cases $\mathrm{GL}_q(N)$ and $\mathrm{SL}_q(N)$ the cohomology ring is isomorphic to the coinvariant external algebra $\varGamma ^\land _{\scriptscriptstyle{\mathrm{Inv}}}$ and to the vector space of harmonic forms. We prove a Hodge decomposition theorem in these cases. The main technical tool is the spectral decomposition of the quantum Laplace-Beltrami operator. 2000 Mathematical Subject Classification: 46L87, 58A12, 81R50.

Let $G$ be a group, $F$ a field of prime characteristic $p$ and $V$ a finite-dimensional $FG$-module. Let $L(V)$ denote the free Lie algebra on $V$ regarded as an $FG$-submodule of the free associative algebra (or tensor algebra) $T(V)$. For each positive integer $r$, let $L^r (V)$ and $T^r (V)$ be the $r$th homogeneous components of $L(V)$ and $T(V)$, respectively. Here $L^r (V)$ is called the $r$th Lie power of $V$. Our main result is that there are submodules $B_1$, $B_2$, ... of $L(V)$ such that, for all $r$, $B_r$ is a direct summand of $T^r(V)$ and, whenever $m \geqslant 0$ and $k$ is not divisible by $p$, the module $L^{p^mk} (V)$ is the direct sum of $L^{p^m} (B_k)$, $L^{p^{m - 1}} (B_{pk})$, ..., $L^1 (B_{p^mk})$. Thus every Lie power is a direct sum of Lie powers of $p$-power degree. The approach builds on an analysis of $T^r (V)$ as a bimodule for $G$ and the Solomon descent algebra.

Let $F$ be an algebraically closed field of characteristic $p \geq 0$ and $A_n$ be the alternating group on $n$ letters. The main goal of this paper is to describe the pairs $(G, E)$, where $E$ is an irreducible $FA_n$-module and $G < A_n$ is a proper subgroup of $A_n$ such that the restriction $E{\downarrow_G}$ is irreducible. We are able to give a list of all such pairs, provided $p > 3$. The case $p = 0$ has been treated by J. Saxl. The problem is important for the classification of maximal subgroups in finite classical groups.

2000 Mathematical Subject Classification: 20C20, 20C30, 20B35, 20B20.

Representations of Hecke and $q$-Schur algebras are closely related to those of finite general linear groups $G$ in non-describing characteristics. Such a relationship can be described by certain functors. Using these functors, we determine the Harish-Chandra vertices and sources of certain indecomposable $G$-modules. The Green correspondence is investigated in this context. As a further application of our theory, we establish Steinberg's tensor product theorems for irreducible representations of $G$ in non-describing characteristics.

1991 Mathematics Subject Classification: 20C20, 20C33, 20G05, 20G40.

We determine strong constraints on the generalized Euler invariants of Seifert bundles over non-orientable base orbifolds which may embed as topologically locally flat submanifolds of $S^4$. In particular, a circle bundle over a non-orientable surface $F$ embeds if and only if it embeds as the boundary of a regular neighbourhood of an embedding of $F$ in $S^4$, and we show that precisely thirteen geometric 3-manifolds with elementary amenable fundamental groups embed. With the exception of the Poincaré homology sphere, each member of the latter class may be obtained by 0-framed surgery on a link which is the union of two slice links, and so embeds smoothly in $S^4$.

1991 Mathematics Subject Classification: 57N13.

We study a $q$-deformation for the crossed product of the symmetric group $S_n$ with the Clifford algebra on $n$ anticommuting generators. We obtain a $q$-version of the projective analogue for the classical Young symmetrizer introduced by the second author. Our main tool is an analogue of the Hecke algebra of complex-valued functions on the group $\mbox{GL}_n$ over a $p$-adic field relative to the Iwahori subgroup.

In this paper a $T(1)$ theorem for Calder\'on--Zygmund operators (CZOs) valid for non-doubling measures with atoms is proved. In the classical Calder\'on--Zigmund theory an essential assumption is the doubling property of the underlying measure $\mu$ in ${\mathbb R}^n$. Some recent results have shown that a big part of the classical theory can be extended to the case of non-doubling measures, with only some mild `growth' condition on the measure $\mu$. In the particular case of the Cauchy transform $Cf(x) = \int f(y)/(y-x) \, d\mu(y)$, the main result of the paper can be stated easily: the Cauchy transform is bounded in $L^2(\mu)$ if and only if it is bounded in $L^2(\mu)$ over characteristic functions of squares. The proof of the main result is based on the use of the Haar basis with random dyadic lattices, following an idea of Nazarov, Treil and Volberg (`Cauchy integral and Calder\'on--Zygmund operators on nonhomogeneous spaces', {\em Internat. Math. Res. Notices} (1997) 15, 703--726), where a $T(1)$ theorem for non-doubling continuous measures was obtained. The case of a measure $\mu$ with atoms studied in the present paper is quite different from the case of continuous measures. For example, if $\mu$ has atoms, the $L^2(\mu)$ boundedness of a CZO does not imply its $L^p(\mu)$ boundedness for some $p\neq2$. The differences arise because a measure $\mu$ with atoms does not satisfy the essential `growth' condition that is involved in the arguments for proving the $L^2$ boundedness of CZOs for continuous measures. 2000 Mathematics Subject Classification: 42B20.