Let $G$ be a reductive $p$-adic group. Consider the category of smooth (complex) representations of $G$ in which a (fixed) closed cocompact subgroup of the centre acts by a (fixed) character. It is well known that the supercuspidal representations in this category are both injective and projective. It is shown that, conversely, an admissible injective or projective object is necessarily supercuspidal.

We show that given an affine algebraic group G over a field K and a finite subgroup scheme H of G there exists a finite dimensional G-module V such that V[mid ]H is free. The problem is raised in the recent paper by Kuzucuo˘glu and Zalesskiiˇ [15] which contains a treatment of the special case in which K is the algebraic closure of a finite field and H is reduced. Our treatment is divided into two parts, according to whether K has zero or positive characteristic. The essence of the characteristic 0 case is a proof that, for given n, there exists a polynomial GLn(ℚ)-module V of dimension 2nΠpp(n2), where the product is over all primes less than or equal to n+1, such that V is free as a ℚH-module for every finite subgroup H of GLn(ℚ). The module V is the tensor product of the exterior algebra Λ*(E), on the natural GLn(ℚ)-module E, and Steinberg modules Stp, one for each prime not exceeding n+1. The Steinberg modules also play the major role in the case in which K has characteristic p>0 and the key point in our treatment is to show that for a finite subgroup scheme H of a general linear group scheme (or universal Chevalley group scheme) G over K, the Steinberg module Stpn for G is injective (and projective) on restriction to H for n[Gt ]0. A curious consequence of this is that, despite the wild behaviour of the modular representation theory of finite groups, one has the following. Let H be a finite group and V a finite dimensional vector space. Then there exists a (well-understood) faithful rational representation π: GL(V)→GL(W) such that, for each faithful representation ρ: H→GL(V), the composite πορ: H→GL(W) is free, in particular all representations πορ are equivalent.

Let $T=U(1)$ and $M$ be a Hamiltonian $T$-space with proper moment map $\mu\,{:}\,M\longrightarrow \rr$. When 0 is not a regular value of $\mu$, the symplectic quotient $X=\mu^{-1}(0)/T$ is a singular stratified space. A description is provided of the middle perversity intersection cohomology of $X$ as a subspace of the equivariant cohomology $H^*_T(\mu^{-1}(0))$. The approach is sheaf theoretic.

In this paper we study the abstract quasi-linear evolution equation of second order

formula here

in a general Banach space Z. It is well-known that the abstract quasi-linear theory due to Kato [10, 11] is widely applicable to quasi-linear partial differential equations of second order and that his theory is based on the theory of semigroups of class (C0). (For example, see the work of Hughes et al. [9] and Heard [8].) However, even in the special case where A(t, w, v) = A is independent of (t, w, v), it is found in [2] and [14] that there exist linear partial differential equations of second order for which Cauchy problems are not solvable by the theory of semigroups of class (C0) but fit into the mould of well-posed problems where the solution and its derivative depend continuously on the initial data if the initial condition is measured in the graph norm of a suitable power of A. (See also work by Krein and Khazan [13] and Fattorini [6, Chapter 8].) This kind of Cauchy problem has recently been studied extensively, using the theory of integrated semigroups or regularized semigroups. The theory of integrated semigroups was studied intensively by Arendt [1] and that of regularized semigroups was initiated by Da Prato [3] and renewed by Davies and Pang [4]. For the theory of regularized semigroups we refer the reader to [5] and [16].

Weakly almost periodic compactifications have been seriously studied for over 30 years. In the pioneering papers of de Leeuw and Glicksberg [4] and [5], the approach adopted was operator-theoretic. The current definition is more likely to be created from the perspective of universal algebra (see [1, Chapter 3]). For a discrete group or semigroup S, the weakly almost periodic compactification wS is the largest compact semigroup which (i) contains S as a dense subsemigroup, and (ii) has multiplication continuous in each variable separately (where largest means that any other compact semigroup with the properties (i) and (ii) is a quotient of wS). A third viewpoint is to envisage wS as the Gelfand space of the C*-algebra of bounded weakly almost periodic functions on S (for the definition of such functions, see below).

In this paper, we are concerned only with the simplest semigroup (ℕ, +). The three approaches described above give three methods of obtaining information about wℕ. An early striking result about wℕ, that it contains more than one idempotent, was obtained by T. T. West using operator theory [13]. He considered the weak operator closure of the semigroup {T, T2, T3, …} of iterates of a single operator T on the Hilbert space L2(μ) for a particular measure μ on [0, 1]. Brown and Moran, in a series of papers culminating in [2], used sophisticated techniques from harmonic analysis to produce measures μ that permitted the detection of further structure in wℕ; in particular, they found 2[cfr ] distinct idempotents. However, for many years, no other way of showing the existence of more than one idempotent in wℕ was found.

The breakthrough came in 1991, and it was made by Ruppert [11]. In his paper, he created a direct construction of a family of weakly almost periodic functions which could detect 2[cfr ] different idempotents in wℕ. His method was very ingenious (he used a unique variant of the p-adic expansion of integers) and rather complicated. Our main aim in this paper is to construct weakly almost periodic functions which are easy to describe and so appear more ‘natural’ than Ruppert's. We also show that there are enough functions of our type to distinguish 2[cfr ] idempotents in wℕ.

Finite-dimensional algebras over an algebraically closed field are divided into two disjoint classes, called tame and wild respectively, by Drozd's tame and wild dichotomy (see [5] and [2]). A tame algebra, roughly speaking, has its n-dimensional indecomposable modules parametrized by finitely many one-parameter families, for all natural numbers n, but a wild algebra has more indecomposable modules and it is considered hopeless to classify them. In [2], Crawley-Boevey showed that all but finitely many n-dimensional indecomposable modules over a tame algebra are τ-invariant, for all natural numbers n, and conjectured that the converse would be true, where τ := DTr is the Auslander–Reiten translation (see [1]) and we call an indecomposable module X τ-invariant if X ≅ = τX.

The close relationship between the notions of positive forms and representations for a C*-algebra A is one of the most basic facts in the subject. In particular the weak containment of representations is well understood in terms of positive forms: given a representation π of A in a Hilbert space H and a positive form φ on A, its associated representation π φ is weakly contained in π (that is, ker π φ ⊃ ker π) if and only if φ belongs to the weak* closure of the cone of all finite sums of coefficients of π. Among the results on the subject, let us recall the following ones. Suppose that A is concretely represented in H. Then every positive form φ on A is the weak* limit of forms of the type x [map ] [sum ]ki=1 〈ξi, xξi〉 with the ξi in H; moreover if A is a von Neumann subalgebra of ℒ(H) and φ is normal, there exists a sequence (ξi)i [ges ] 1 in H such that φ (x) = [sum ]i [ges ] 1 〈ξi, xξi〉 for all x.

A version of interpolation inequalities for derivatives in logarithmic Orlicz spaces is obtained where the first gradient of $u$ is estimated in terms of $u$ and its second gradient. One of the Orlicz functions considered is supposed to be ${\backslash\lambda^p}$. The motivation, examples and applications are discussed.

The paper is concerned with rings of polynomial invariants of finite groups. In particular, it will be shown that these rings are isomorphic as modules over the Steenrod algebra [Pscr ]* if and only if the group representations are pointwise conjugate. An application to cohomology is the construction of classifying spaces of finite groups which are not homotopy equivalent, but where the cohomology rings are isomorphic as unstable modules over the (topological) Steenrod algebra.

If the sum of a series of positive numbers is not convergent and the series satisfies certain regularity conditions, a probability measure can be found whose support contains no interval, yet whose Fourier series decreases faster than the given series.

The paper is concerned with the uniqueness and existence problem for a multidimensional reacting and convection system with the vanishing viscosity method. The uniqueness theorem is obtained from the stability with respect to the initial data. To solve the existence problem, the uniqueness and existence of solutions to a viscous system are first proved, and then the L1-modulus of continuity of solutions independent of the small viscosity is obtained.

The Kronecker quiver $K$ is considered, and the relations for the specialisation at $q=0$ of the generic composition algebra are given, as well as those for Reineke's composition monoid. As a corollary, it is deduced that the composition monoid is a proper factor of the specialisation of the composition algebra. A normal form is also obtained for the varieties occurring in the composition monoid in terms of Schur roots.

It is proved that the word problem of the direct product of two free groups of rank 2 can be recognised by a 2-tape real-time but not by a 1-tape real-time Turing machine. It is also proved that the Baumslag–Solitar groups $B$(1, $r$) have the 5-tape real-time word problem for all $r\neq 0$.

A formula of Poisson summation type is established for sums involving relative norms in an extension of number fields.

A new proof is given of a theorem of Menšov which states that a measurable function can be changed on a set of arbitrarily small measure to obtain a function with uniformly convergent Fourier series. Other results of Menšov on the representation of functions by trigonometric series are obtained by using a related result.