In this paper we show that for a Grothendieck category $\mathcal{A}$ and a complex $E$ in $\mathbf{C}(\mathcal{A})$ there is an associated localization endofunctor $\ell$ in $\mathbf{D}(\mathcal{A})$ . This means that $\ell$ is idempotent (in a natural way) and that the objects that go to 0 by $\ell$ are those of the smallest localizing (= triangulated and stable for coproducts) subcategory of $\mathbf{D}(\mathcal{A})$ that contains $E$ . As applications, we construct $\text{K}$ -injective resolutions for complexes of objects of $\mathcal{A}$ and derive Brown representability for $\mathbf{D}(\mathcal{A})$ from the known result for $\mathbf{D}(R-\mathbf{mod})$ , where $R$ is a ring with unit.

The nonlinear Sturm-Liouville equation

$$-{{\left( p{y}' \right)}^{\prime }}\,+\,qy\,=\,\text{ }\!\!\lambda\!\!\text{ }\left( 1-f \right)ry\,on\,[0,\,1]$$

is considered subject to the boundary conditions

$$\left( {{\text{a}}_{j}}\text{ }\lambda \text{ }\text{+}{{\text{b}}_{j}} \right)y\left( j \right)=\left( {{c}_{j}}\text{ }\lambda \text{ }+{{d}_{j}} \right)\left( p{y}' \right)\left( j \right),j=0,1$$

Here ${{\text{a}}_{0}}\,=\,0\,=\,{{c}_{0}}$ and $p,\,r\,>\,0$ and $q$ are functions depending on the independent variable $x$ alone, while $f$ depends on $x,\,y\,and\,{y}'$ . Results are given on existence and location of sets of $(\lambda ,\,y)$ bifurcating from the linearized eigenvalues, and for which $y$ has prescribed oscillation count, and on completeness of the $y$ in an appropriate sense.

We study $K$ -orbits in $G/P$ where $G$ is a complex connected reductive group, $P\,\subseteq \,G$ is a parabolic subgroup, and $K\,\subseteq \,G$ is the fixed point subgroup of an involutive automorphism $\theta$ . Generalizing work of Springer, we parametrize the (finite) orbit set $K\,\backslash \,G/P$ and we determine the isotropy groups. As a consequence, we describe the closed (resp. affine) orbits in terms of $\theta$ -stable (resp. $\theta$ -split) parabolic subgroups. We also describe the decomposition of any $(K,\,P)$ -double coset in $G$ into $(K,\,B)$ -double cosets, where $B\,\subseteq \,P$ is a Borel subgroup. Finally, for certain $K$ -orbit closures $X\,\subseteq \,G/B$ , and for any homogeneous line bundle $\mathcal{L}$ on $G/B$ having nonzero global sections, we show that the restriction map $\text{re}{{\text{s}}_{X}}\,:\,{{H}^{0}}\,\left( G\,/\,B,\,\mathcal{L} \right)\,\to \,{{H}^{0}}\,\left( X,\,\mathcal{L} \right)$ is surjective and that ${{H}^{i}}\,\left( X,\mathcal{L} \right)\,=\,0$ for $i\,\ge \,1$ . Moreover, we describe the $K$ -module ${{H}^{0}}\left( X,L \right)$ . This gives information on the restriction to $K$ of the simple $G$ -module ${{H}^{0}}\,\left( G\,/\,B,\mathcal{L} \right)$ . Our construction is a geometric analogue of Vogan and Sepanski’s approach to extremal $K$ -types.

In this article, using 3-orbifolds singular along a knot with underlying space a homology sphere ${{Y}^{3}}$ , the question of existence of non-trivial and non-abelian $\text{SU(2)}$ -representations of the fundamental group of cyclic branched covers of ${{Y}^{3}}$ along a knot is studied. We first use Floer Homology for knots to derive an existence result of non-abelian $\text{SU(2)}$ -representations of the fundamental group of knot complements, for knots with a non-vanishing equivariant signature. This provides information on the existence of non-trivial and non-abelian $\text{SU(2)}$ -representations of the fundamental group of cyclic branched covers. We illustrate the method with some examples of knots in ${{S}^{3}}$ .

This paper expresses the character of certain depth-zero supercuspidal representations of the rank-2 symplectic group as the Fourier transform of a finite linear combination of regular elliptic orbital integrals—an expression which is ideally suited for the study of the stability of those characters. Building on work of F. Murnaghan, our proof involves Lusztig’s Generalised Springer Correspondence in a fundamental way, and also makes use of some results on elliptic orbital integrals proved elsewhere by the author using Moy-Prasad filtrations of $p$ -adic Lie algebras. Two applications of the main result are considered toward the end of the paper.

Suppose $G$ is a countable, Abelian group with an element of infinite order and let $\text{ }\!\!\chi\!\!\text{ }$ be a mixing rank one action of $G$ on a probability space. Suppose further that the Følner sequence $\{{{F}_{n}}\}$ indexing the towers of $\text{ }\!\!\chi\!\!\text{ }$ satisfies a “bounded intersection property”: there is a constant $p$ such that each $\{{{F}_{n}}\}$ can intersect no more than $p$ disjoint translates of $\{{{F}_{n}}\}$ . Then $\text{ }\!\!\chi\!\!\text{ }$ is mixing of all orders. When $G\,=\,\mathbf{Z}$ , this extends the results of Kalikow and Ryzhikov to a large class of “funny” rank one transformations. We follow Ryzhikov’s joining technique in our proof: the main theorem follows from showing that any pairwise independent joining of $k$ copies of $\text{ }\!\!\chi\!\!\text{ }$ is necessarily product measure. This method generalizes Ryzhikov’s technique.

Nous étudions les polynômes $F\,\in \,\mathbb{C}\,\{{{S}_{\mathcal{T}}}\}\,\left[ Y \right]$ à coefficients dans l’anneau de germes de fonctions holomorphes au point spécial d’une variété torique affine. Nous généralisons à ce cas la paramétrisation classique des singularités quasi-ordinaires. Cela fait intervenir d’une part une généralization de l’algorithme de Newton-Puiseux, et d’autre part une relation entre le polyèdre de Newton du discriminant de $F$ par rapport à $Y$ et celui de $F$ au moyen du polytope-fibre de Billera et Sturmfels. Cela nous permet enfin de calculer, sous des hypothèses de non dégénérescence, les sommets du polyèdre de Newton du discriminant a partir de celui de $F$ , et les coefficients correspondants à partir des coefficients des exposants de $F$ qui sont dans les arêtes de son polyèdre de Newton.

It is shown by a combination of analytic and computational techniques that for any positive fundamental discriminant $D\,>\,3705$ , there is always at least one prime $p\,<\,\sqrt{D}/2$ such that the Kronecker symbol $(D/P)\,=\,-1$ .

${{H}^{p}}$ estimate for the multilinear operators which are finite sums of pointwise products of singular integrals and fractional integrals is given. An application to Sobolev space and some examples are also given.

The main new results in this paper are contained in the geometric Theorems 1 and 2 of Section 0.1 below and they are related to previous results of M. Gromov and of myself (cf. [11], [29]). These results are used to prove some general potential theoretic estimates on Lie groups (cf. Section 0.3) that are related to my previous work in the area (cf. [28], [34]) and to some deep recent work of G. Alexopoulos (cf. [3], [4]).

In the first part of this paper generalizations of Hesselink’s $q$ -analog of Kostant’smultiplicity formula for the action of a semisimple Lie group on the polynomials on its Lie algebra are given in the context of the Kostant-Rallis theorem. They correspond to the cases of real semisimple Lie groups with one conjugacy class of Cartan subgroup. In the second part of the paper a $q$ -analog of the Kostant-Rallis theorem is given for the real group $\text{SL(4,}\,\mathbb{R}\text{)}$ (that is $\text{SO}(4)$ acting on symmetric 4 × 4 matrices). This example plays two roles. First it contrasts with the examples of the first part. Second it has implications to the study of entanglement of mixed 2 qubit states in quantum computation.