Realization of square-integrable representations of unimodular Lie groups on $L\sp{2}$-cohomology spaces
Jonathan
Rosenberg
1-32
Abstract: An analogue of the ``Langlands conjecture'' is proved for a large class of connected unimodular Lie groups having square-integrable representations (modulo their centers). For nilpotent groups, it is shown (without restrictions on the group or the polarization) that the $ {L^2}$-cohomology spaces of a homogeneous holomorphic line bundle, associated with a totally complex polarization for a flat orbit, vanish except in one degree given by the ``deviation from positivity'' of the polarization. In this degree the group acts irreducibly by a square-integrable representation, confirming a conjecture of Moscovici and Verona. Analogous results which improve on theorems of Satake are proved for extensions of a nilpotent group having square-integrable representations by a reductive group, by combining the theorem for the nilpotent case with Schmid's proof of the Langlands conjecture. Some related results on Lie algebra cohomology and the ``Harish-Chandra homomorphism'' for Lie algebras with a triangular decomposition are also given.
Lie cohomology of representations of nilpotent Lie groups and holomorphically induced representations
Richard
Penney
33-51
Abstract: Let U be a locally injective, Moore-Wolf square integrable representation of a nilpotent Lie group N. Let $ (\mathcal{H},\,\lambda )$ be a complex, maximal subordinate pair corresponding to U and let $ {\mathcal{H}_0}\, = \,\ker \,\lambda \, \cap \,\mathcal{H}$. The space ${C^\infty }(U)$ of differentiable vectors for U is an ${\mathcal{H}_0}$ module. In this work we compute the Lie algebra cohomology $ {H^p}({\mathcal{H}_0},\,{C^\infty }(U))$ of this Lie module. We show that the cohomology is zero for all but one value of p and that for this specific value the cohomology is one dimensional. These results, when combined with earlier results of ours, yield the existence and irreducibility of holomorphically induced representations for arbitrary (nonpositive), totally complex polarizations.
On random Fourier series
Jack
Cuzick;
Tze Leung
Lai
53-80
Abstract: Motivated by Riemann's ${R_1}$ summation method for i.i.d. random variables ${X_1},\,{X_2},\, \ldots$, this paper studies random Fourier series of the form $\sum\nolimits_1^\infty {{a_n}{X_n}\,\sin (nt\, + \,{\Phi _n})}$, where $ \{ {a_n}\}$ is a sequence of constants and $ \{ {\Phi _n}\}$ is a sequence of independent random variables which are independent of $\{ {X_n}\}$. Questions of continuity and of unboundedness are analyzed through the interplay between the asymptotic properties of $\{ {a_n}\}$ and the tail distribution of $ {X_1}$. A law of the iterated logarithm for the local behavior of the series is also obtained and extends the classical result for Brownian motion to a general class of random Fourier series.
Free modular lattices
Ralph
Freese
81-91
Abstract: It is shown that the word problem for the free modular lattice on five generators is recursively unsolvable.
On a simplicial complex associated to the monodromy
Gerald Leonard
Gordon
93-101
Abstract: Suppose we have a complex analytic family, ${V_t}$, $\left\vert t \right\vert\, \leqslant \,1$, such that the generic fibre is a nonsingular complex manifold of complex dimension n. Let T denote the monodromy induced from going once around the singular fibre and let I denote the identity map. We shall associate to the singular fibre a simplicial complex $\Gamma$, which is at most n-dimensional. Then under certain conditions on the family ${V_t}$ (which are satisfied for the Milnor fibration of an isolated singularity or if the $ {V_t}$ are compact Kähler), there is an integer $N\, > \,0$ such that ${({T^N}\, - \,I)^k}{H_k}({V_t})\, = \,0$ if and only if ${H_k}(\Gamma )\, = \,0$.
On the group of volume-preserving diffeomorphisms of ${\bf R}\sp{n}$
Dusa
McDuff
103-113
Abstract: The group of all diffeomorphisms of $ {\textbf{R}^n}$ which preserve a given volume form is shown to be perfect when $n\, \geqslant \,3$. Some useful factorizations of such diffeomorphisms are also obtained.
Binary sequences which contain no $BBb$
Earl D.
Fife
115-136
Abstract: A (one-sided) sequence or (two-sided) bisequence is irreducible provided it contains no block of the form BBb, where b is the initial symbol of the block B. Gottschalk and Hedlund [Proc. Amer. Math. Soc. 15 (1964), 70-74] proved that the set of irreducible binary bisequences is the Morse minimal set M. Let ${M^ + }$ denote the one-sided Morse minimal set, i.e. ${M^ + }\, = \,\{ {x_0}{x_1}{x_2}\, \ldots : \ldots \,{x_{ - 1}}{x_0}{x_1}\, \ldots \, \in \,M\}$. Let ${P^ + }$ denote the set of all irreducible binary sequences. We establish a method for generating all $x\, \in \,{P^ + }$. We also determine ${P^ + }\, - \,{M^ + }$. Considering $ {P^ + }$ as a one-sided symbolic flow, ${P^ + }$ is not the countable union of transitive flows, thus ${P^ + }$ is considerably larger than ${M^ + }$. However ${M^ + }$ is the $\omega $-limit set of each $x\, \in \,{P^ + }$, and in particular ${M^ + }$ is the nonwandering set of $ {P^ + }$.
Critical points of harmonic functions on domains in ${\bf R}\sp{3}$
Robert
Shelton
137-158
Abstract: It is shown that the critical point relations of Morse theory, together with the maximum principle, comprise a complete set of critical point relations for harmonic functions of three variables. The proof proceeds by first constructing a simplified example and then developing techniques to modify this example to realize all admissible possibilities. Techniques used differ substantially from those used by Morse in his solution of the analogous two-variable problem.
The essential norm of an operator and its adjoint
Sheldon
Axler;
Nicholas
Jewell;
Allen
Shields
159-167
Abstract: We consider the relationship between the essential norm of an operator T on a Banach space X and the essential norm of its adjoint $T^{\ast}$. We show that these two quantities are not necessarily equal but that they are equivalent if $ X^{\ast}$ has the bounded approximation property. For an operator into the sequence space ${c_0}$, we give a formula for the distance to the compact operators and show that this distance is attained. We introduce a property of a Banach space which is useful in showing that operators have closest compact approximants and investigate which Banach spaces have this property.
Uniform approximation on unbounded sets by harmonic functions with logarithmic singularities
P. M.
Gauthier;
M.
Goldstein;
W. H.
Ow
169-183
Abstract: This paper deals with the qualitative theory of uniform approximation by harmonic functions. The theorems of Brelot and Deny on Runge- and Walsh-type approximation on compact sets are extended to unbounded closed sets.
Isotopy types of knots of codimension two
M. Š.
Farber
185-209
Abstract: In this paper the classification of n-dimensional knots in ${S^{n + 2}}$, bounding r-connected manifolds, where $3r\, \geqslant \,n\, + \,1\, \geqslant \,6$, in terms of stable homotopy theory is suggested.
Hardy spaces and rearrangements
Burgess
Davis
211-233
Abstract: Let f be an integrable valued function on the unit circle in the complex plane, and let g be the rearrangement of f satisfying $ g({e^{i\theta }})\, \geqslant \,g({e^{i\varphi }})$ if $0\, \leqslant \,\theta \, < \,\varphi \, < \,2\pi$. Define $\displaystyle G(\theta )\, = \,\int_{ - \theta }^\theta {g({e^{i\varphi }})} \,d\varphi$ . It is shown that some rearrangement of f is in $\operatorname{Re} \,{H^1}$, that is, the distribution of f is the distribution of a function in $\operatorname{Re} \,{H^1}$, if and only if $\int_0^\pi {\vert G(\theta )/\theta \vert} \,d\theta \, < \,\infty$, and that, if any rearrangement of f is in $\operatorname{Re} \,{H^1}$, then g is. The existence and form of rearrangements minimizing the ${H^1}$ norm are investigated. It is proved that f is in $\operatorname{Re} \,{H^1}$ if and only if some rotation of f is in the space dyadic $ {H^1}$ of martingales. These results are extended to other ${H^p}$ spaces.
Quadratic forms and the Birman-Craggs homomorphisms
Dennis
Johnson
235-254
Abstract: Let ${\mathcal{M}_g}$ be the mapping class group of a genus g orientable surface M, and ${\mathcal{J}_g}$ the subgroup of those maps acting trivially on the homology group ${H_1}(M,\,Z)$. Birman and Craggs produced homomorphisms from $ {\mathcal{J}_g}$ to $ {Z_2}$ via the Rochlin invariant and raised the question of enumerating them; in this paper we answer their question. It is shown that the homomorphisms are closely related to the quadratic forms on $ {H_1}(M,\,{Z_2})$ which induce the intersection form; in fact, they are in 1-1 correspondence with those quadratic forms of Arf invariant zero. Furthermore, the methods give a description of the quotient of $ {\mathcal{J}_g}$ by the intersection of the kernels of all these homomorphisms. It is a ${Z_2}$-vector space isomorphic to a certain space of cubic polynomials over ${H_1}(M,\,{Z_2})$. The dimension is then computed and found to be $\left( {\begin{array}{*{20}{c}} {2g} 3 \end{array} } \right)\, + \,\left( {\begin{array}{*{20}{c}} {2g} 2 \end{array} } \right)$ . These results are also extended to the case of a surface with one boundary component, and in this situation the linear relations among the various homomorphisms are also determined.
Existentially complete abelian lattice-ordered groups
A. M. W.
Glass;
Keith R.
Pierce
255-270
Abstract: The theory of abelian totally ordered groups has a model completion. We show that the theory of abelian lattice-ordered groups has no model companion. Indeed, the Archimedean property can be captured by a first order $\forall\exists\forall$ sentence for existentially complete abelian lattice-ordered groups, and distinguishes between finitely generic abelian lattice-ordered groups and infinitely generic ones. We then construct (by sheaf techniques) the model companions of certain classes of discrete abelian lattice-ordered groups.
A spectral sequence for group presentations with applications to links
Selma
Wanna
271-285
Abstract: A spectral sequence is associated with any presentation of a group G. It turns out that this spectral sequence is independent of the chosen presentation. In particular if G is the fundamental group of a link L in ${R^3}$ the spectral sequence leads to invariants that compare to the Milnor invariants of L.
Transforms of measures on quotients and spline functions
Alan
MacLean
287-296
Abstract: Let G be a LCA group with closed subgroup H and let $v\, \in \,M(G/H)$. A general procedure is established for constructing a large family of measures in $M(G)$ whose Fourier transforms interpolate $ \hat v$. This method is used to extend a theorem of Shepp and Goldberg by showing that if $ v\, \in \,M([0,\,2\pi ))$, then each even order cardinal spline function which interpolates the sequence $(\hat v(n))_{n\, = \, - \,\infty }^\infty$ Fourier transform of a bounded Borel measure on R.
Vanishing theorems and K\"ahlerity for strongly pseudoconvex manifolds
Vo Van
Tan
297-302
Abstract: A precise vanishing theorem of Kodaira-Nakano type for strongly pseudoconvex manifolds and Nakano semipositive vector bundles is established. This result answers affirmatively a question posed by Grauert and Riemenschneider. However an analogous version of vanishing theorem of Akizuki-Nakano type for strongly pseudoconvex manifolds and Nakano semipositive line bundles does not hold in general. A counterexample for this fact is explicitly constructed. Furthermore we prove that any strongly pseudoconvex manifold with 1-dimensional exceptional subvariety is Kählerian; in particular any strongly pseudoconvex surface is Kählerian.