The Minakshisundaram-Pleijel coefficients for the vector-valued heat kernel on compact locally symmetric spaces of negative curvature
Roberto J.
Miatello
1-33
Abstract: We use harmonic analysis on semisimple Lie groups to determine the Minakshisundaram-Pleijel asymptotic expansion for the trace of the heat kernel on natural vector bundles over compact, locally symmetric spaces of strictly negative curvature.
Homological algebra on a complete intersection, with an application to group representations
David
Eisenbud
35-64
Abstract: Let R be a regular local ring, and let $A\, = \,R/(x)$, where x is any nonunit of R. We prove that every minimal free resolution of a finitely generated A-module becomes periodic of period 1 or 2 after at most $\operatorname{dim} \, A$ steps, and we examine generalizations and extensions of this for complete intersections. Our theorems follow from the properties of certain universally defined endomorphisms of complexes over such rings.
A relation between the coefficients in the recurrence formula and the spectral function for orthogonal polynomials
Jeffrey S.
Geronimo
65-82
Abstract: A relation is found between the rate of convergence of the coefficients in the recurrence formula for polynomials orthogonal on a segment of the real line and certain properties of the spectral function. The techniques of Banach algebras and scattering theory are used. The close connection between polynomials orthogonal on the unit circle and polynomials orthogonal on the real line is exploited.
Principal $2$-blocks of the simple groups of Ree type
Peter
Landrock;
Gerhard O.
Michler
83-111
Abstract: The decomposition numbers in characteristic 2 of the groups of Ree type are determined, as well as the Loewy and socle series of the indecomposable projective modules. Moreover, we describe the Green correspondents of the simple modules. As an application of this and similar works on other simple groups with an abelian Sylow 2-subgroup, all of which have been classified apart from those considered in the present paper, we show that the Loewy length of an indecomposable projective module in the principal block of any finite group with an abelian Sylow 2-subgroup of order ${2^n}$ is bounded by $\max \{ 2n\, + \,1,\,{2^n}\}$. This bound is the best possible.
On Harish-Chandra's $\mu $-function for $p$-adic groups
Allan J.
Silberger
113-121
Abstract: The Harish-Chandra $ \mu$-function is, up to known constant factors, the Plancherel's measure associated to an induced series of representations. In this paper we show that, when the series is induced from special representations lifted to a parabolic subgroup, the $ \mu$-function is a quotient of translated $\mu$-functions associated to series induced from supercuspidal representations. It is now known, in both the real and p-adic cases, that the $\mu$-function is always an Euler factor.
Harmonically induced representations on nilpotent Lie groups and automorphic forms on nilmanifolds
Richard C.
Penney
123-145
Abstract: It is shown that the irreducible ``discrete series'' representations of certain nilpotent Lie groups may be realized in square integrable $\bar \partial $ cohomology spaces. This theory is applied to obtain a concept of automorphic forms on nilmanifolds which generalizes the niltheta functions of Cartier and Auslander-Tolimieri. We also use the automorphic cohomology to solve certain holomorphic difference equations on ${{\textbf{C}}^n}$.
Nonstandard extensions of transformations between Banach spaces
D. G.
Tacon
147-158
Abstract: Let X and Y be (infinite-dimensional) Banach spaces and denote their nonstandard hulls with respect to an ${\aleph _1}$-saturated enlargement by $\hat X$ and $\hat Y$ respectively. If ${\mathcal{B}}\,(X,\,Y)$ denotes the space of bounded linear transformations then a subset S of elements of $ {\mathcal{B}}\,(X,\,Y)$ extends naturally to a subset $\hat S$ of ${\mathcal{B}}\,(\hat X,\,\hat Y)$. This paper studies the behaviour of various kinds of transformations under this extension and introduces, in this context, the concepts of super weakly compact, super strictly singular and socially compact operators. It shows that $({\mathcal{B}}\,(X,\,Y)\hat )\,\mathop \subset \limits_ \ne \,{\mathcal{B}}\,(\hat X,\,\hat Y)$ provided X and Y are infinite dimensional and contrasts this with the inclusion ${\mathcal{K}}(\hat H)\,\mathop \subset \limits_ \ne \,({\mathcal{K}}(H)\hat )$ where ${\mathcal{K}}(H)$ denotes the space of compact operators on a Hilbert space.
Shellable and Cohen-Macaulay partially ordered sets
Anders
Björner
159-183
Abstract: In this paper we study shellable posets (partially ordered sets), that is, finite posets such that the simplicial complex of chains is shellable. It is shown that all admissible lattices (including all finite semimodular and supersolvable lattices) and all bounded locally semimodular finite posets are shellable. A technique for labeling the edges of the Hasse diagram of certain lattices, due to R. Stanley, is generalized to posets and shown to imply shellability, while Stanley's main theorem on the Jordan-HÖlder sequences of such labelings remains valid. Further, we show a number of ways in which shellable posets can be constructed from other shellable posets and complexes. These results give rise to several new examples of Cohen-Macaulay posets. For instance, the lattice of subgroups of a finite group G is Cohen-Macaulay (in fact shellable) if and only if G is supersolvable. Finally, it is shown that all the higher order complexes of a finite planar distributive lattice are shellable.
Distinguished subfields
James K.
Deveney;
John N.
Mordeson
185-193
Abstract: Let L be a finitely generated nonalgebraic extension of a field K of characteristic $p\, \ne \,0$. A maximal separable extension D of K in L is distinguished if $ L\, \subseteq \,{K^{{p^{ - \,n}}}}(D)$ for some n. Let d be the transcendence degree of L over K. If every maximal separable extension of K in L is distinguished, then every set of d relatively p-independent elements is a separating transcendence basis for a distinguished subfield. Conversely, if $K({L^p})$ is separable over K, this condition is also sufficient. A number of properties of such fields are determined and examples are presented illustrating the results.
On the topology of simply connected algebraic surfaces
Richard
Mandelbaum;
Boris
Moishezon
195-222
Abstract: Suppose x is a smooth simply-connected compact 4-manifold. Let $ p\, = \,{\textbf{C}}{P^2}$ and $ Q\, = \, - {\textbf{C}}{P^2}$ be the complex projective plane with orientation opposite to the usual. We shall say that X is completely decomposable if there exist integers a, b such that X is diffeomorphic to $ aP\,{\text{\char93 }}\,bQ$. By a result of Wall [W1] there always exists an integer k such that $X\,\char93 \,(k\, + \,1)P\,\char93 kQ$ is completely decomposable. If $X\,\char93 \,P$ is completely decomposable we shall say that X is almost completely decomposable. In [MM] we demonstrated that any nonsingular hypersurface of $ {\textbf{C}}{P^3}$ is almost completely decomposable. In this paper we generalize this result in two directions as follows: Theorem 3.5. Suppose W is a simply-connected nonsingular complex projective 3-fold. Then there exists an integer $ {m_0}\, \geqslant \,1$ such that any hypersurface section ${V_m}$ of W of degree $m\, \geqslant \,{m_0}$ which is nonsingular will be almost completely decomposable. Theorem 5.3. Let V be a nonsingular complex algebraic surface which is a complete intersection. Then V is almost completely decomposable.
Extending combinatorial piecewise linear structures on stratified spaces. II
Douglas R.
Anderson;
Wu Chung
Hsiang
223-253
Abstract: Let X be a stratified space and suppose that both the complement of the n-skeleton and the n-stratum have been endowed with combinatorial piecewise linear (PL) structures. In this paper we investigate the problem of ``fitting together'' these separately given PL structures to obtain a single combinatorial PL structure on the complement of the $ (n\, - \,1)$-skeleton. The first main result of this paper reduces the geometrically given ``fitting together'' problem to a standard kind of obstruction theory problem. This is accomplished by introducing a tangent bundle for the n-stratum and using immersion theory to show that the ``fitting together'' problem is equivalent to reducing the structure group of the tangent bundle of the n-stratum to an appropriate group of PL homeomorphisms. The second main theorem describes a method for computing the homotopy groups arising in the obstruction theory problem via spectral sequence methods. In some cases, the spectral sequences involved are fairly small and the first few differentials are described. This paper is an outgrowth of earlier work by the authors on this problem.
Stability theorems for holomorphic foliations
T.
Duchamp;
M.
Kalka
255-266
Abstract: Here we investigate topological stability in the space of holomorphic foliations on a compact manifold. We show that under certain conditions nearby holomorphic foliations are topologically equivalent. We then present examples of foliations which are stable as holomorphic foliations but unstable as smooth foliations.
A representation theorem and applications to topological groups
J.-M.
Belley
267-279
Abstract: We show that, given a set S dense in a compact Hausdorff space X and a complex-valued bounded linear functional $ \Lambda$ on the space $ C(X)$ of continuous complex-valued functions on X with uniform norm, there exist an algebra $ {\mathcal{A}}$ of sets in S and a unique bounded finitely additive set function $\mu :\,{\mathcal{A}}\, \to \,{\textbf{C}}$ which is inner and outer regular with respect to the zero and cozero sets respectively and such that $\int_s {f\left\vert S \right.\,d\mu }$ exists and is equal to $\Lambda (f)$ for all $ f\, \in \,C(X)$. In the context of topological groups, this theorem permits us to obtain (1) a concrete representation theorem for bounded complex-valued linear functionals on the space of almost periodic functions with uniform norm, (2) a representation theorem for (not necessarily continuous) positive definite functions, (3) a characterization of the space B of finite linear combinations of positive definite functions, and (4) a necessary and sufficient condition to have a linear transformation from B to B.
Moduli for analytic left algebraic groups
Andy R.
Magid
281-291
Abstract: This paper classifies left algebraic group structures on faithfully representable complex analytic groups by establishing the existence of an algebraic variety whose complex points correspond to such structures on a given analytic group.
Disconjugacy and integral inequalities
Achim
Clausing
293-307
Abstract: The basic data in this paper are a disconjugate differential operator and an associated two-point boundary value problem. These define in a natural way a cone of functions satisfying a differential inequality with respect to the operator. By using a result of P. W. Bates and G. B. Gustafson on the monotonicity properties of Green's kernels it is shown that such a cone has a compact convex base which is a Bauer simplex. This result is used to derive a variety of integral inequalities which include known inequalities of Frank and Pick, Levin and Steckin, Karlin and Ziegler, as well as several new ones.
A characterization of periodic automorphisms of a free group
James
McCool
309-318
Abstract: Let $\theta$ be an automorphism of finite order of a free group X. We characterise the action of $\theta$ on X by showing that X has a free basis which is the disjoint union of finite subsets ${S_j}$, where if ${S_j}\, = \,\{ {u_0},\,{u_1},\, \ldots ,\,{u_k}\}$ then $ {u_i}\theta \, = \,{u_{i\, + \,1}}\,(0\, \leqslant \,1\, < \,k)$ and $ {u_k}\theta \, = \,{A_j}u_0^e{B_j}$ for some ${A_j}$, ${B_j}$ in X and $\varepsilon \, = \, \pm \,1$. As an application of this result, we obtain a list of the conjugacy classes of periodic automorphisms of the free group of rank three.