Let ${\cal M} =\langle M,<,\ldots\rangle$ be alinearly ordered structure. We define ${\cal M}$ to be{\em o-minimal} if every definable subset of $M$ is a finite union of intervals. Classical examples are ordered divisible abelian groups and real closed fields. We prove a trichotomy theorem for the structure that an arbitraryo-minimal ${\cal M}$ can induce on a neighbourhood of any $a$in $M$. Roughly said, one of the following holds: \begin{enumerate}\item[(i)] $a$ is trivial (technical term), {\em or}\item[(ii)] $a$ has a convex neighbourhood on which ${\cal M}$ induces the structure of an ordered vector space, {\em or}\item[(iii)] $a$ is contained in an open interval on which ${\cal M}$ induces the structure of an expansion of a real closed field. \end{enumerate} The proof uses ‘geometric calculus’ which allows one to recover a differentiable structure by purely geometric methods.

1991 Mathematics Subject Classification: primary 03C45; secondary 03C52, 12J15, 14P10.

We investigate definability in $\mathcal{R}$, the recursively enumerable Turing degrees, using codings of standard models of arithmetic (SMAs) as a tool. First we show that an SMA can be interpreted in $\mathcal{R}$ without parameters. Building on this, we prove that the recursively enumerable $T$-degrees satisfy a weak form of the bi-interpretability conjecture which implies that all jump classes $\mathrm{Low}_n$ and $\mathrm{High}_{n-1}$$(n\ge 2)$ are definable in $\mathcal{R}$ without parameters and, more generally, that all relations on $\mathcal{R}$ that are definable in arithmetic and invariant under the double jump are actually definable in $\mathcal{R}$. This partially answers Soare's Question 3.7 (R. Soare, {\emRecursively enumerable sets and degrees} (Springer, Berlin, 1987), Chapter XVI).

1991 Mathematics Subject Classification: primary 03D25, 03D35; secondary 03D30.

The mod $p$ representation associated to an elliptic curve is called split or non-split dihedral if its image lies in the normaliser of a split or non-split Cartan subgroup of $\GL_2(\f_p)$, respectively. Let $\xsplit$ and $\xnonsplit$ denote the modular curves which classify elliptic curves with split and non-split dihedral mod $p$ representation, respectively. We call such curves split and non-split{\it Cartan modular curves}. The curve $\xsplit$ is isomorphic to the curve $X_0^+(p^2)$. Using the Selberg trace formula for Hecke operators, we verify that the jacobian of $\xnonsplit$ is isogenous to the new part of the jacobian of $X_0^+(p^2)$.

1991 Mathematics Subject Classification: primary 11G18; secondary 11F72.

Let $G$ be a geometrically finite Kleinian group with parabolic elements and let $p$ be any parabolic fixed point of $G$. For each positive real $\tau$, let ${\cal W}_{p}(\tau)$ denote the set of limit points of $G$ for which the inequality $$ | x-g(p) | \leq |g^\prime(0)|^{\tau}$$ is satisfied for infinitely many elements $g$ in $G$. This subset of the limit set is precisely the analogue of the set of $\tau$-well approximable numbers in the classical theory of metric Diophantine approximation. In this paper we consider the following question. What is the `size' of the set ${\cal W}_{p}(\tau)$ expressed in terms of its Hausdorff dimension? We provide a complete answer, namely that for $\tau \geq 1$,$$ \dim {\cal W}_{p} (\tau) = \min \left\{ \frac{\delta + \mbox{rk}(p) (\tau - 1) }{2 \tau - 1}, \, \frac{\delta}{\tau}\right\},$$ where $\mbox{rk}(p)$ denotes the rank of the parabolic fixed point $p$.

1991 Mathematics Subject Classification: 11K55, 11K60, 11F99, 58D20.

Let $S(x,y)$ be the set $S(x,y)=\{ 1 \leq n \leq x : P(n)\leq y\}$, where $P(n)$ denotes the largest prime factor of $n$. We study $E_f(x,y;\theta)=\sum_{n\inS(x,y)}f(n)e^{2\pi in\theta}$, where $f$ is a multiplicative function. When $f=1$ and when $f=\mu$, we widen the domain of uniform approximation using the method of Fouvry and Tenenbaum and making explicit the contribution of the Siegel zero.

Soit $S(x,y)$ l'ensemble $S(x,y)=\{ 1 \leq n \leq x : P(n)\leq y\}$, o\`u$P(n)$ d\'esigne le plus grand facteur premier de $n$. Nous étudions $E_f(x,y;\theta)=\sum_{n\in S(x,y)}f(n)e^{2\pi in\theta}$, lorsque $f$ est une fonction multiplicative. Quand $f=1$ et quand $f=\mu$, nous élargissons le domaine d'approximation uniforme enutilisant la méthode d\'eveloppée par Fouvry et Tenenbaum et en explicitant la contribution du zéro de Siegel.

1991 Mathematics Subject Classification: 11N25, 11N99.

Bushnell and Kutzko gave a complete and effective classification of the smooth dual of $\mbox{GL}(N,F)$, where $F$ is a non-archimedean local field. Similarly, Zink gave a classification of the smooth dual of $D^{\times}$, where $D$ is a division algebra with centre $F$, in terms of non-canonical objects and under the restrictive hypothesis that $F$ has characteristic $0$. In this paper, we extend part of Bushnell and Kutzko's formalism to $D^{\times}$ and obtain a complete classification of the smooth dual working for any characteristic. The crucial point of this work is to define a good way of splitting the algebra $D$ so that the important notion of {\it simple stratum}, and its properties, can be translated to $D^{\times}$ by some descent arguments.

1991 Mathematics Subject Classification: 12E15, 20G05, 20G25, 22E50.

This paper describes parabolic induction, for smooth representations of finite length of the general linear group $\mbox{GL}(N, F)$ of a non-archimedean local field $F$, in terms of a functor between categories of modules over certain affine Hecke algebras, using the categorical equivalences developed in the Bushnell-Kutzko classification of the admissible dual of $\mbox{GL}(N, F)$. This result is used to prove that the Zelevinsky automorphism, which is an involutive automorphism on the representation ring of the category of smooth representations of finite length of $\mbox{GL}(N, F)$, preserves the irreducible representations. The result also implies a simplification of the study of multiplicities of composition factors of induced representations.

1991 Mathematics Subject Classification: 11S37, 22E50.

In this paper we use the Hecke algebra of type $\bf B$ to define a new algebra $\mathcal S$ which is an analogue of the $q$-Schur algebra. We show that $\mathcal S$ has ‘generic’ basis which is independent of the choice of ring and the parameters $q$ and $Q$. We then construct Weyl modules for $\mathcal S$ and obtain, as factor modules, a family of irreducible $\mathcal S$-modules defined over any field.

1991 Mathematics Subject Classification: 16G99, 20C20, 20G05.

We prove the quantum version - for Hecke algebras $H(A_n)$ of type $A$ at roots of unity - of Kleshchev's modular branching rule for symmetric groups. This result describes the socle of the restriction of an irreducible $H(A_n)$-module to the subalgebra $H(A_{n-1})$. As a consequence, we describe the quantum version of the Mullineux involution describing the irreducible module obtained on twisting an irreducible module with the sign representation.

1991 Mathematics Subject Classification: 20C05, 20G05.

The ground field $F$ is an algebraically closed field of zero characteristic, and ${\Bbb N}$ is the set of natural numbers. Let $L$ be a locally finite-dimensional (or {\em locally finite}, for brevity) Lie algebra. Assume for simplicity that $L$ has countable dimension. Then one can choose finite-dimensional subalgebras $L_i$ ($i\in{\Bbb N}$) of $L$ in such a way that $L_{i}\subset L_{i+1}$ for all $i$ and $L=\bigcup_{i\in{\Bbb N}}L_i$.The set $\{L_i\mid i\in{\Bbb N}\}$ is called a{\em local system} of $L$. Assume that $L$ is simple. Then all $L_i$ can be chosen to be perfect (that is, $[L_i,L_i]=L_i$). We shall call such local systems {\em perfect}. Let $S_i=S_i^1\oplus\ldots\oplus S_i^{n_i}$ be a Levi subalgebra of $L_i$, where $S_i^1,\ldots,S_i^{n_i}$ are simple components of $S_i$, and let $V_i^k$ be the standard $S_i^k$-module. Since $L_i$ is perfect, for every $k$ there exists a unique irreducible $L_i$-module${\cal V}_i^k$ such that the restriction ${\cal V}_i^k{\downarrow} S_i^k$ is isomorphic to $V_i^k$. An embedding $L_i\to L_{i+1}$ is called {\em diagonal} if, for each $l$, any non-trivial composition factor of the restriction ${\cal V}_{i+1}^l{\downarrow} L_i$ is isomorphic to ${\cal V}_i^k$ or its dual ${\cal V}_i^{k^*}$ for some $k$. We prove that a simple locally finite Lie algebra $L$ can be embedded into a locally finite associative algebra if and only if there exists a perfect local system $\{L_i\}_{i\in{\Bbb N}}$ of $L$ such that all embeddings $L_i\to L_{i+1}$ are diagonal. Note that this result can be considered as a version of Ado's theorem for simple locally finite Lie algebras. Let $V$ be a vector space. An element $x\in \frak{gl}(V)$ is called {\em finitary}if $\dim xV<\infty$. The finitary transformations of $V$ form an ideal $\frak{fgl}(V)$ of $\frak{gl}(V)$, and any subalgebra of $\frak{fgl}(V)$ is called a {\em finitary Lie algebra}. We classify all finitary simple Lie algebras of countable dimension. It turns out that there are only three: $\frak{sl}_\infty(F)$, $\frak{so}_\infty(F)$, and $\frak{sp}_\infty(F)$. The author has classified all finitary simple Lie algebras and will describe this in a subsequent paper.The analogous problem in group theory has been recently solved by J.I. Hall. He classified simple locally finite groups of finitary linear transformations.

1991 Mathematics Subject Classification: 17B35, 17B65.

Here, $F$ denotes a non-Archimedean local field (with finite residue field) and $G$ the group of $F$-points of a connected reductive algebraic group defined over $F$. Let $\frak R(G)$ denote the category of smooth, complex representations of $G$. Let ${\cal B}(G)$ be the set of pairs $(L,\sigma)$, where $L$ is an $F$-Levi subgroup of $G$ and $\sigma$ is an irreducible supercuspidal representation of $L$, taken modulo the equivalence relation generated by twisting with unramified quasi characters and $G$-conjugacy. To $\frak s\in {\cal B}(G)$, one can attach a full (abelian) sub-category $\frak R^\frak s(G)$ of $\frak R(G)$; the theory of the Bernstein centre shows that $\frak R(G)$ is the direct product of these $\frak R^\frak s(G)$. The object of the paper is to give a general method for describing these factor categories via representations of compact open subgroups within a uniform framework. Fix $\frak s\in {\cal B}(G)$. Let $K$ be a compact open subgroup of $G$ and $\rho$ an irreducible smooth representation of $K$. The pair $(K,\rho)$ is an $\frak s$-type if it has the following property: an irreducible representation $\pi$ of $G$ contains $\rho$ if and only if $\pi\in \frak R^\frak s(G)$. Let ${\cal H}(G,\rho)$ be the Hecke algebra of compactly supported $\rho$-spherical functions on $G$; if $(K,\rho)$ is an $\frak s$-type, then the category $\frak R^\frak s(G)$ is canonically equivalent to the category ${\cal H}(G,\rho) \text{-Mod}$ of ${\cal H}(G,\rho)$-modules. Let $M$ be a Levi subgroup of $G$; there is a canonical map ${\cal B}(M)\to {\cal B}(G)$. Take $\frak t\in {\cal B}(M)$ with image $\frak s\in {\cal B}(G)$. The choice of a parabolic subgroup of $G$ with Levi component $M$ gives functors of {\it parabolic induction\/} and {\it Jacquet restriction\/} connecting $\frak R^\frak t(M)$ with $\frak R^\frak s(G)$. We assume given a $\frak t$-type $(K_M,\rho_M)$ in $M$; the paper concerns a general method of constructing from this data an $\frak s$-type $(K,\rho)$ in $G$. One thus obtains a description of these induction and restriction functors in terms of an injective ring homomorphism ${\cal H}(M,\rho_M) \to {\cal H}(G,\rho)$. The method applies in a wide variety of cases, and subsumes much previous work. Under further conditions, observed in certain particularly interesting cases, one can go some distance to describing ${\cal H}(G,\rho)$ explicitly. This enables one to isolate cases in which the map on Hecke algebras is an isomorphism, and this in turn implies powerful intertwining theorems for the types.

1991 Mathematics Subject Classification: 22E50.

We describe an algorithm to determine whether a matrix group over a finite field, generated by a given set of matrices, contains one of the classical groups or the special linear group. The algorithm is based on finding elements with certain properties by independent random selection from the group. Very few matrix groups other than the classical groups contain subsets of elements with the required properties, and a classification of such matrix groups is the principal theoretical basis of the algorithm. The classification, which is proved here, builds on work of Guralnick, Penttila, Saxl and the second author, and depends on the finite simple group classification. Moreover, probability estimates for finding such elements in classical groups are derived, and these are used to give a theoretical analysis of the performance of the algorithm. The algorithm has been implemented in the computer systems {\sf GAP} and{\scMagma}.

1991 Mathematics Subject Classification: primary 20G40, 11E57, 20-04, 60B99; secondary 15A30.

In this paper we study the notion of joint functional calculus associated with a couple of resolvent commuting sectorial operators on a Banach space $X$. We present some positive results when $X$ is, for example, a Banach lattice or a quotient of subspaces of a $B$-convex Banach lattice. Furthermore, we develop a notion of a generalized $H^\infty$-functional calculus associated with the extension to $\Lambda(H)$ of a sectorial operator on a $B$-convex Banach lattice $\Lambda$, where $H$ is a Hilbert space. We apply our results to a new construction of operators with a bounded $H^\infty$-functional calculus and to the maximal regularity problem.

1991 Mathematics Subject Classification: 47A60, 47D06, 46C15.

We study densities $\rho$ on the unit ball in euclidean space which satisfy a Harnack type inequality and a volume growth condition for the measure associated with $\rho$. For these densities a geometric theory can be developed which captures many features of the theory of quasiconformal mappings. For example, we prove generalizations of the Gehring-Hayman theorem, the radial limit theorem and find analogues of compression and expansion phenomena on the boundary.

1991 Mathematics Subject Classification: 30C65.

A famous theorem of Donaldson describes a correspondence between $\mbox{SU}(2)$ monopoles over three-dimensional Euclidean space and maps from $\Bbb{CP}^1$ to itself. This paper generalises this to monopoles with arbitrary gauge group $G,$ and new maps from $\Bbb{CP}^1$ to flag manifolds $G/H$ are produced. This is in line with a conjecture of Atiyah and Murray, following a similar result of Atiyah's on hyperbolic monopoles.

Donaldson's approach, also followed by Hurtubise and Murray in previous proofs of many important cases of the current result, depends on a description of monopoles in terms of a system of ordinary differential equations known as Nahm's equations. In contrast, our approach is more direct, and the bulk of the paper is concerned with describing the rational map associated to a particular framed monopole, via solutions to a scattering equation (first introduced by Hitchin) along parallel lines.

A subsidiary section of the paper analyses rational maps into flag manifolds, constructing canonical lifts into larger flag manifolds, and into the (complexified) Lie group. The main result is dependent upon additional work in a companion paper, ‘Construction of Euclidean monopoles’, to appear in the same journal, where a procedure is described for recovering a monopole from its rational map.

1991 Mathematics Subject Classification: 53C80, 58D27, 58E15, 58G11.

In this paper we investigate how the volume of a hyperbolic manifold increases under the process of removing a curve, that is, Dehn drilling. If the curve we remove is a geodesic, we show that for a certain family of manifolds the volume increase is bounded above by $\pi \cdot l$ where $l$ is the length of the geodesic drilled. We further construct examples to show that there is no lower bound to the volume increase in terms of a monotonic unbounded function of length. In particular, this shows that volume increase is not bounded below linearly in length.

1991 Mathematics Subject Classification: primary 51M10, 51M20; secondary 51M25, 52C25.

The invariantly harmonic functions in the unit ball ${\Bbb B}^n$ in ${\Bbb C}^n$ are those annihilated by the Bergman Laplacian $\Delta$. The Poisson-Szeg\"o kernel $P(z,\zeta)$ solves the Dirichlet problem for $\Delta$: if $f\in C(S^n)$, the Poisson-Szeg\"o transform of $f$, $P[f](z)=\int_{S^n}P(z,\zeta) f(\zeta)\,d\sigma(\zeta),$ where $d\sigma$ is the normalized Lebesgue measure on $S^n$, is the unique invariantly harmonic function $u$ in ${\Bbb B}^n$, continuous up to the boundary, such that $u=f$ on $S^n$. The Poisson-Szeg\"o transform establishes, loosely speaking, a one-to-one correspondence between function theory in $S^n$ and invariantly harmonic function theory in ${\Bbb B}^n$. When $n\geq 2$, it is natural to consider on $S^n$ function spaces related to its natural non-isotropic metric, for these are the spaces arising from complex analysis. In the paper, different characterizations of such spaces of smooth functions are given in terms of their invariantly harmonic extensions, using maximal functions and area integrals, as in the corresponding Euclidean theory. Particular attention is given to characterization in terms of purely radial or purely tangential derivatives. The smoothness is measured in two different scales: that of Sobolev spaces and that of Lipschitz spaces, including BMO and Besov spaces.

1991 Mathematics Subject Classification: 32A35, 32A37, 32M15, 42B25.

This paper describes a procedure for the construction of monopoles on three-dimensional Euclidean space, starting from their rational maps. A companion paper, ‘Euclidean monopoles and rational maps’, to appear in the same journal, describes the assignment to a monopole of a rational map, from $\Bbb{CP}^1$ to a suitable flag manifold. In describing the reverse direction, this paper completes the proof of the main theorem therein.

A construction of monopoles from solutions to Nahm's equations (a system of ordinary differential equations) has been well-known for certain gauge groups for some time. These solutions are hard to construct however, and the equations themselves become increasingly unwieldy when the gauge group is not $\mbox{SU}(2).$

Here, in contrast, a rational map is the only initial data. But whereas one can be reasonably explicit in moving from Nahm data to a monopole, here the monopole is only obtained from the rational map after solving a partial differential equation.

A non-linear flow equation, essentially just the path of steepest descent down the Yang-Mills-Higgs functional, is set up. It is shown that, starting from an ‘approximate monopole’ - constructed explicitly from the rational map - a solution to the flow must exist, and converge to an exact monopole having the desired rational map.

1991 Mathematics Subject Classification: 53C07, 53C80, 58D27, 58E15, 58G11.

Let $M$ be a compact, smooth, $2$-connected, $2m$-dimensional manifold with $\partial M$ simply connected. If $M$ has the homotopytype of an $m$-dimensional $CW$-complex, then it supports a smooth, self-indexed function, maximal, constant and regular on $\partial M$ with at most $\mbox{cat}(M)+2$ critical points, all of which are of a certain ‘reasonable’ type. To such a critical point there corresponds, homotopically, the attachment of a cone. Conversely, to a cone attachment we may associate, under certain dimensionality and connectivity conditions, a ‘reasonable’ critical point.

1991 Mathematics Subject Classification: primary 55M30, 57R70; secondary 58E05, 55P50, 55P62.

Green's function $G(x)$ of a zero mean random walk on the $N$-dimensional integer lattice ($N\geq 2$) is expanded in powers of $1/|x|$ under suitable moment conditions. In particular, we find minimal moment conditions for $G(x)$ to behave like a constant times the Newtonian potential (or logarithmic potential in two dimensions) for large values of $|x|$. Asymptotic estimates of $G(x)$ in dimensions $N\geq 4$, which are valid even when these moment conditions are violated, are computed. Such estimates are applied to determine the Martin boundary of the random walk. If $N= 3$ or $4$ and the random walk has zero mean and finite second moment,the Martin boundary consists of one point, whereas if $N\geq 5$, this is not the case, because non-harmonic functions arise as Martin boundary points for a large class of such random walks. A criterion for when this happens is provided.

1991 Mathematics Subject Classification: 60J15, 60J45, 31C20.