Quadratic homology
Hans-Joachim
Baues
429-475
Abstract: We describe axioms for a `quadratic homology theory' which generalize the classical axioms of homology. As examples we consider quadratic homology theories induced by 2-excisive homotopy functors in the sense of Goodwillie and the homology of a space with coefficients in a square group which generalizes the homology of a space with coefficients in an abelian group.
Global character formulae for compact Lie groups
A.
H.
Dooley;
N.
J.
Wildberger
477-495
Abstract: We introduce the concept of a modulator, which leads to a family of character formulae, each generalizing the Kirillov formula. For a suitable choice of modulator, this enables one to understand the Plancherel measure of a compact Lie group as arising from a partition of the identity on the dual of its Lie algebra.
On some Banach space properties sufficient for weak normal structure and their permanence properties
Brailey
Sims;
Michael
A.
Smyth
497-513
Abstract: We consider Banach space properties that lie between conditions introduced by Bynum and Landes. These properties depend on the metric behavior of weakly convergent sequences. We also investigate the permanence properties of these conditions.
The stability of the equilibrium of reversible systems
Bin
Liu
515-531
Abstract: In this paper, we consider the system \begin{displaymath}\dot x=a(t)y^{2m+1}+f_1(x,y,t),\quad\dot y=-b(t)x^{2n+1}+f_2(x,y,t),\end{displaymath} where $m,n\in \mathbf Z_+$, $m+n\ge 1$, $a(t)$ and $b(t)$ are continuous, even and 1-periodic in the time variable $t$; $f_1$ and $f_2$ are real analytic in a neighbourhood of the origin $(0,0)$ of $(x,y)$-plane and continuous 1-periodic in $t$. We also assume that the above system is reversible with respect to the involution $G\colon(x,y)\mapsto(-x,y)$. A sufficient and necessary condition for the stability in the Liapunov sense of the equilibrium $(x,y)=(0,0)$ is given.
Homogeneous projective varieties with degenerate secants
Hajime
Kaji
533-545
Abstract: The secant variety of a projective variety $X$ in $\mathbb{P}$, denoted by $\operatorname{Sec}X$, is defined to be the closure of the union of lines in $\mathbb{P}$ passing through at least two points of $X$, and the secant deficiency of $X$ is defined by $\delta := 2 \dim X + 1 - \dim \operatorname{Sec}X$. We list the homogeneous projective varieties $X$ with $\delta > 0$ under the assumption that $X$ arise from irreducible representations of complex simple algebraic groups. It turns out that there is no homogeneous, non-degenerate, projective variety $X$ with $\operatorname{Sec}X \not = \mathbb{P}$ and $\delta > 8$, and the $E_{6}$-variety is the only homogeneous projective variety with largest secant deficiency $\delta = 8$. This gives a negative answer to a problem posed by R. Lazarsfeld and A. Van de Ven if we restrict ourselves to homogeneous projective varieties.
The nilpotence height of $P_t^s$ for odd primes
Ismet
Karaca
547-558
Abstract: K. G. Monks has recently shown that the element $P^{s}_{t}$ has nilpotence height $2[\frac{s}{t}] + 2$ in the mod $2$ Steenrod algebra. Here the method and result are generalized to show that for an odd prime $p$ the element $P^{s}_{t}$ has nilpotence height $p[\frac{s}{t}] + p$ in the mod $p$ Steenrod algebra.
Regularity of solutions to the measurable Livsic equation
M.
Pollicott;
M.
Yuri
559-568
Abstract: In this note we give generalisations of Livsic's result that a priori measurable solutions to cocycle equations must in fact be more regular. We go beyond the original continuous hyperbolic examples of Livsic to consider examples of this phenomenon in the context of: (a) $\beta$-transformations; (b) rational maps; and (c) planar maps with indifferent periodic points. Such examples are not immediately covered by Livsic's original approach either due to a lack of continuity or hyperbolicity.
Windows of given area with minimal heat diffusion
Jochen
Denzler
569-580
Abstract: For a bounded Lipschitz domain $\Omega$, we show the existence of a measurable set $D\subset \partial\Omega$ of given area such that the first eigenvalue of the Laplacian with Dirichlet conditions on $D$ and Neumann conditions on $\partial \Omega \setminus D$ becomes minimal. If $\Omega$ is a ball, $D$ will be a spherical cap.
Golubev series for solutions of elliptic equations
Ch.
Dorschfeldt;
N.
N.
Tarkhanov
581-594
Abstract: Let $P$ be an elliptic system with real analytic coefficients on an open set $X\subset {\Bbb R}^{n},$ and let $\Phi$ be a fundamental solution of $P.$ Given a locally connected closed set $\sigma \subset X,$ we fix some massive measure $m$ on $\sigma$. Here, a non-negative measure $m$ is called massive, if the conditions $s \subset \sigma$ and $m(s)=0$ imply that $\overline{\sigma \setminus s} = \sigma .$ We prove that, if $f$ is a solution of the equation $Pf =0$ in $X \setminus \sigma ,$ then for each relatively compact open subset $U$ of $X$ and every $1<p<\infty$ there exist a solution $f_{e}$ of the equation in $U$ and a sequence $f_{\alpha }$ ($\alpha \in {\Bbb N}^{n}_{0}$) in $L^{p} (\sigma \cap U, m)$ satisfying $\| \alpha ! f_{\alpha } \|^{1/|\alpha|}_{L^{p} (\sigma \cap U,m)} \rightarrow 0$ such that $f(x) = f_{e} (x) +\sum _{\alpha}\int _{\sigma \cap U} D^{\alpha }_{y} \Phi (x,y) f_{\alpha } (y) dm(y)$ for $x \in U \setminus \sigma .$ This complements an earlier result of the second author on representation of solutions outside a compact subset of $X.$
Maximal ideals in modular group algebras of the finitary symmetric and alternating groups
Alexander
Baranov;
Alexander
Kleshchev
595-617
Abstract: The main result of the paper is a description of the maximal ideals in the modular group algebras of the finitary symmetric and alternating groups (provided the characteristic $p$ of the ground field is greater than 2). For the symmetric group there are exactly $p-1$ such ideals and for the alternating group there are $(p-1)/2$ of them. The description is obtained in terms of the annihilators of certain systems of the `completely splittable' irreducible modular representations of the finite symmetric and alternating groups. The main tools used in the proofs are the modular branching rules (obtained earlier by the second author) and the `Mullineux conjecture' proved recently by Ford-Kleshchev and Bessenrodt-Olsson. The results obtained are relevant to the theory of PI-algebras. They are used in a later paper by the authors and A. E. Zalesskii on almost simple group algebras and asymptotic properties of modular representations of symmetric groups.
Gap estimates of the spectrum of Hill's equation and action variables for KdV
T.
Kappeler;
B.
Mityagin
619-646
Abstract: Consider the Schrödinger equation $-y'' + Vy = \lambda y$ for a potential $V$ of period 1 in the weighted Sobolev space $(N \in \mathbb{Z}_{\ge 0}, \omega \in \mathbb{R}_{\ge 0})$ \begin{displaymath}H^{N, \omega}(S^1; \mathbb{C}) := \{ f(x) = \sum^{\infty}_{k= - \infty} \Hat {\Hat f}(k) e^{i 2 \pi kx} \bigg | \parallel f \parallel _{N, \omega} < \infty \}\end{displaymath} where $\Hat {\Hat f}(k) (k \in \mathbb{Z})$ denote the Fourier coefficients of $f$ when considered as a function of period 1, \begin{displaymath}\parallel f \parallel _{N, \omega} := \bigg ( \sum _k (1+| k|)^{2N} e^{2 \omega | k |} | \ \Hat{\Hat{f}} (k) |^2 \bigg )^{^{1}/2} < \infty ,\end{displaymath} and where $S^1$ is the circle of length 1. Denote by $\lambda _k \equiv \lambda _k (V) (k \ge 0)$ the periodic eigenvalues of $- \frac{d^2}{dx^2} + V$ when considered on the interval $[0,2],$ with multiplicities and ordered so that $Re \lambda _j \le Re \lambda _{j+1} (j \ge 0).$ We prove the following result. Theorem. For any bounded set ${\mathcal B} \subseteq H^{N, \omega} (S^1; \mathbb{C}),$ there exist $n_0 \ge 1$ and $M \ge 1$ so that for $k \ge n_0$ and $V \in {\mathcal B}$, the eigenvalues $\lambda _{2k}, \lambda _{2k -1}$ are isolated pairs, satisfying (with $\{ \lambda _{2k}, \lambda _{2k-1} \} = \{ \lambda^+_k , \lambda^-_k \})$ (i) $\sum _{k \ge n_0} (1+k)^{2N} e^{2 \omega k} | \lambda _k^+ - \lambda^-_k |^2 \le M$, (ii) $\sum _{k \ge n_0} (1 + k)^{2 N+1} e^{2 \omega k} \bigg | (\lambda^+_k - \lambda^-_k) -2 \sqrt{\Hat{\Hat{V}} (k) \Hat{\Hat{V}}(-k)} \bigg |^2 \le M$.
Exact categories and vector space categories
Peter
Dräxler;
Idun
Reiten;
Sverre
O.
Smal\o;
Øyvind
Solberg;
with an appendix by
B. Keller
647-682
Abstract: In a series of papers additive subbifunctors $F$ of the bifunctor $\operatorname{Ext}_{\Lambda} ( , )$ are studied in order to establish a relative homology theory for an artin algebra $\Lambda$. On the other hand, one may consider the elements of $F(X,Y)$ as short exact sequences. We observe that these exact sequences make $\operatorname{mod}\Lambda$ into an exact category if and only if $F$ is closed in the sense of Butler and Horrocks. Concerning the axioms for an exact category we refer to Gabriel and Roiter's book. In fact, for our general results we work with subbifunctors of the extension functor for arbitrary exact categories. In order to study projective and injective objects for exact categories it turns out to be convenient to consider categories with almost split exact pairs, because many earlier results can easily be adapted to this situation. Exact categories arise in representation theory for example if one studies categories of representations of bimodules. Representations of bimodules gained their importance in studying questions about representation types. They appear as domains of certain reduction functors defined on categories of modules. These reduction functors are often closely related to the functor $\operatorname{Ext}_{\Lambda}( , )$ and in general do not preserve at all the usual exact structure of $\operatorname{mod}\Lambda$. By showing the closedness of suitable subbifunctors of $\operatorname{Ext}_{\Lambda}( , )$ we can equip $\operatorname{mod}\Lambda$ with an exact structure such that some reduction functors actually become `exact'. This allows us to derive information about the projective and injective objects in the respective categories of representations of bimodules appearing as domains, and even show that almost split sequences for them exist. Examples of such domains appearing in practice are the subspace categories of a vector space category with bonds. We provide an example showing that existence of almost split sequences for them is not a general fact but may even fail if the vector space category is finite.
The heat kernel weighted Hodge Laplacian on noncompact manifolds
Edward
L.
Bueler
683-713
Abstract: On a compact orientable Riemannian manifold, the Hodge Laplacian $\triangle$ has compact resolvent, therefore a spectral gap, and the dimension of the space $\mathcal{H}^{p} = \ker \triangle ^{p}$ of harmonic $p$-forms is a topological invariant. By contrast, on complete noncompact Riemannian manifolds, $\triangle$ is known to have various pathologies, among them the absence of a spectral gap and either ``too large'' or ``too small'' a space $\mathcal{H}^{p}$. In this article we use a heat kernel measure $d\mu$ to determine the space of square-integrable forms and to construct the appropriate Laplacian $\triangle _{\mu }$. We recover in the noncompact case certain results of Hodge's theory of $\triangle$ in the compact case. If the Ricci curvature of a noncompact connected Riemannian manifold $M$ is bounded below, then this ``heat kernel weighted Laplacian'' $\triangle _{\mu }$ acts on functions on $M$ in precisely the manner we would wish, that is, it has a spectral gap and a one-dimensional kernel. We prove that the kernel of $\triangle _{\mu }$ on $n$-forms is zero-dimensional on $M$, as we expect from topology, if the Ricci curvature is nonnegative. On Euclidean space, there is a complete Hodge theory for $\triangle _{\mu }$. Weighted Laplacians also have a duality analogous to Poincaré duality on noncompact manifolds. Finally, we show that heat kernel-like measures give desirable spectral properties (compact resolvent) in certain general cases. In particular, we use measures with Gaussian decay to justify the statement that every topologically tame manifold has a strong Hodge decomposition.
Galois coverings of selfinjective algebras by repetitive algebras
Andrzej
Skowronski;
Kunio
Yamagata
715-734
Abstract: In the representation theory of selfinjective artin algebras an important role is played by selfinjective algebras of the form $\widehat {B}/G$ where $\widehat {B}$ is the repetitive algebra of an artin algebra $B$ and $G$ is an admissible group of automorphisms of $\widehat {B}$. If $B$ is of finite global dimension, then the stable module category $\underline{\operatorname{mod}} \widehat {B}$ of finitely generated $\widehat {B}$-modules is equivalent to the derived category $D^{b} (\operatorname{mod} B)$ of bounded complexes of finitely generated $B$-modules. For a selfinjective artin algebra $A$, an ideal $I$ and $B=A/I$, we establish a criterion for $A$ to admit a Galois covering $F: \widehat {B}\to \widehat {B}/G=A$ with an infinite cyclic Galois group $G$. As an application we prove that all selfinjective artin algebras $A$ whose Auslander-Reiten quiver $\Gamma _{A}$ has a non-periodic generalized standard translation subquiver closed under successors in $\Gamma _{A}$ are socle equivalent to the algebras $\widehat {B}/G$, where $B$ is a representation-infinite tilted algebra and $G$ is an infinite cyclic group of automorphisms of $\widehat{B}$.
On Siegel modular forms of half-integral weights and Jacobi forms
Koichi
Takase
735-780
Abstract: We will establish a bijective correspondence between the space of the cuspidal Jacobi forms and the space of the half-integral weight Siegel cusp forms which is compatible with the action of the Hecke operators. This correspondence is based on a bijective correspondence between the irreducible unitary representations of a two-fold covering group of a symplectic group and a Jacobi group (that is, a semidirect product of a symplectic group and a Heisenberg group). The classical results due to Eichler-Zagier and Ibukiyama will be reconsidered from our representation theoretic point of view.
The nonarchimedean theta correspondence for $\mathrm{GSp}(2)$ and $\mathrm{GO}(4)$
Brooks
Roberts
781-811
Abstract: In this paper we consider the theta correspondence between the sets $\operatorname{Irr} (\operatorname{GSp} (2,k))$ and $\operatorname{Irr} (\operatorname{GO} (X))$ when $k$ is a nonarchimedean local field and $\dim _{k} X =4$. Our main theorem determines all the elements of $\operatorname{Irr} (\operatorname{GO} (X))$ that occur in the correspondence. The answer involves distinguished representations. As a corollary, we characterize all the elements of $\operatorname{Irr} (\operatorname{O} (X))$ that occur in the theta correspondence between $\operatorname{Irr} (\operatorname{Sp} (2,k))$ and $\operatorname{Irr} (\operatorname{O} (X))$. We also apply our main result to prove a case of a new conjecture of S.S. Kudla concerning the first occurrence of a representation in the theta correspondence.
Projective structures with discrete holonomy representations
Hiroshige
Shiga;
Harumi
Tanigawa
813-823
Abstract: Let $K(X)$ denote the set of projective structures on a compact Riemann surface $X$ whose holonomy representations are discrete. We will show that each component of the interior of $K(X)$ is holomorphically equivalent to a complex submanifold of the product of Teichmüller spaces and the holonomy representation of every projective structure in the interior of $K(X)$ is a quasifuchsian group.
Conjugacy classes of $SU(h,\mathcal O_S)$ in $SL(2,\mathcal O_S)$
Donald
G.
James
825-835
Abstract: Let $K$ be a quadratic extension of a global field $F$, of characteristic not two, and $\mathcal{O}_S$ the integral closure in $K$ of a Dedekind ring of $S$-integers $\mathfrak{O}_S$ in $F$. Then $PSL(2, \mathcal{O}_S)$ is isomorphic to the spinorial kernel $O'(L)$ for an indefinite quadratic $\mathfrak{O}_S$-lattice $L$ of rank 4. The isomorphism is used to study the conjugacy classes of unitary groups $PSU(h,\mathcal{O}_S)$ of primitive odd binary hermitian matrices $h$ under the action of $PSL(2, \mathcal{O}_S)$.
Distribution semigroups and abstract Cauchy problems
Peer
Christian
Kunstmann
837-856
Abstract: We present a new definition of distribution semigroups, covering in particular non-densely defined generators. We show that for a closed operator $A$ in a Banach space $E$ the following assertions are equivalent: (a) $A$ generates a distribution semigroup; (b) the convolution operator $\delta'\otimes I-\delta\otimes A$ has a fundamental solution in ${\mathcal D}'(L(E,D))$ where $D$ denotes the domain of $A$ supplied with the graph norm and $I$ denotes the inclusion $D\to E$; (c) $A$ generates a local integrated semigroup. We also show that every generator of a distribution semigroup generates a regularized semigroup.