A construction of the supercuspidal representations of ${\rm GL}\sb n(F),\;F\;p$-adic
Lawrence
Corwin
1-58
Abstract: Let $F$ be a nondiscrete, locally compact, non-Archimedean field. In this paper, we construct all irreducible supercuspidal representations of $G = {\text{GL}_n}(F)$ For each such representation $ \pi$ (which we may as well assume is unitary), we give a subgroup $ J$ of $G$ that is compact mod the center $ Z$ of $G$ and a (finite-dimensional) representation $\sigma$ of $J$ such that inducing $\sigma$ to $G$ gives $\pi$. The proof that all supercuspidals have been constructed appeals to a theorem (the Matching Theorem) that has been proved by global methods.
Propagation of singularities, Hamilton-Jacobi equations and numerical applications
Eduard
Harabetian
59-71
Abstract: We consider applications of Hamilton-Jacobi equations for which the initial data is only assumed to be in ${L^\infty }$. Such problems arise for example when one attempts to describe several characteristic singularities of the compressible Euler equations such as contact and acoustic surfaces, propagating from the same discontinuous initial front. These surfaces represent the level sets of solutions to a Hamilton-Jacobi equation which belongs to a special class. For such Hamilton-Jacobi equations we prove the existence and regularity of solutions for any positive time and convergence to initial data along rays of geometrical optics at any point where the gradient of the initial data exists. Finally, we present numerical algorithms for efficiently capturing singular fronts with complicated topologies such as corners and cusps. The approach of using Hamilton-Jacobi equations for capturing fronts has been used in [14] for fronts propagating with curvature-dependent speed.
Quantization of K\"ahler manifolds. II
Michel
Cahen;
Simone
Gutt;
John
Rawnsley
73-98
Abstract: We use Berezin's dequantization procedure to define a formal $ \ast$-product on a dense subalgebra of the algebra of smooth functions on a compact homogeneous Kähler manifold $M$. We prove that this formal $ \ast$-product is convergent when $M$ is a hermitian symmetric space.
An example of a two-term asymptotics for the ``counting function'' of a fractal drum
Jacqueline
Fleckinger-Pellé;
Dmitri G.
Vassiliev
99-116
Abstract: In this paper we study the spectrum of the Dirichlet Laplacian in a bounded domain $\Omega \subset {\mathbb{R}^n}$ with fractal boundary $ \partial \Omega$. We construct an open set $ \mathcal{Q}$ for which we can effectively compute the second term of the asymptotics of the "counting function" $N(\lambda ,\mathcal{Q})$, the number of eigenvalues less than $\lambda$. In this example, contrary to the M. V. Berry conjecture, the second asymptotic term is proportional to a periodic function of In $ \lambda$, not to a constant. We also establish some properties of the $ \zeta$-function of this problem. We obtain asymptotic inequalities for more general domains and in particular for a connected open set $\mathcal{O}$ derived from $ \mathcal{Q}$. Analogous periodic functions still appear in our inequalities. These results have been announced in $[{\text{FV}}]$.
Divisors on symmetric products of curves
Alexis
Kouvidakis
117-128
Abstract: For a curve with general moduli, the Neron-Severi group of its symmetric products is generated by the classes of two divisors $x$ and $\theta$. In this paper we give bounds for the cones of effective and ample divisors in the $ x\theta$-plane.
Minimal hypersurfaces of ${\bf R}\sp {2m}$ invariant by ${\rm SO}(m)\times {\rm SO}(m)$
Hilário
Alencar
129-141
Abstract: Let $G = {\text{SO}}(m) \times {\text{SO}}(m)$ act in the standard way on $ {{\mathbf{R}}^m} \times {{\mathbf{R}}^m}$. We describe all complete minimal hypersurfaces of ${{\mathbf{R}}^m}\backslash \{ 0\}$ which are invariant under $G$ for $m = 2$, $3$ . We also show that the unique minimal hypersurface of $ {{\mathbf{R}}^{2m}}$ which is invariant under $G$ and passes through the origin of ${{\mathbf{R}}^{2m}}$ is the minimal quadratic cone.
Formal moduli of modules over local $k$-algebras
Allan
Adler;
Pradeep
Shukla
143-158
Abstract: We determine explicitly the formal moduli space of certain complete topological modules over a topologically finitely generated local $k$-algebra $R$, not necessarily commutative, where $ k$ is a field. The class of topological modules we consider include all those of finite rank over $k$ and some of infinite rank as well, namely those with a Schauder basis in the sense of $\S1$. This generalizes the results of [Sh], where the result was obtained in a different way in case the ring $R$ is the completion of the local ring of a plane curve singularity and the module is ${k^n}$. Along the way, we determine the ring of infinite matrices which correspond to the endomorphisms of the modules with Schauder bases. We also introduce functions called "growth functions" to handle explicit epsilonics involving the convergence of formal power series in noncommuting variables evaluated at endomorphisms of our modules. The description of the moduli space involves the study of a ring of infinite series involving possibly infinitely many variables and which is different from the ring of power series in these variables in either the wide or the narrow sense. Our approach is beyond the methods of [Sch] which were used in [Sh] and is more conceptual.
Residual finiteness of color Lie superalgebras
Yu. A.
Bahturin;
M. V.
Zaicev
159-180
Abstract: A (color) Lie superalgebra $L$ over a field $K$ of characteristic $\ne 2, 3$ is called residually finite if any of its nonzero elements remains nonzero in a finite-dimensional homomorphic image of $L$. In what follows we are looking for necessary and sufficient conditions under which all finitely generated Lie superalgebras satisfying a fixed system of identical relations are residually finite. In the case $\operatorname{char}\;K = 0$ we show that a variety $ V$ satisfies this property if and only if $V$ does not contain all center-by-metabelian algebras and every finitely generated algebra of $ V$ has nilpotent commutator subalgebra.
Triangulations in M\"obius geometry
Feng
Luo
181-193
Abstract: We prove that a conformally flat closed manifold of dimension at least three adimts a hyperbolic, spherical or similarity structure in the conformally flat class if and only if the manifold has a smooth triangulation so that all codimension one Simplexes are in some codimension one spheres.
Symmetrization with respect to a measure
Friedmar
Schulz;
Virginia
Vera de Serio
195-210
Abstract: In this paper we study the spherical symmetric rearrangement ${u^\ast}$ of a nonnegative measurable function $u$ on $ {\mathbb{R}^n}$ with respect to a measure given by a nonhomogeneous density distribution $p$. Conditions on $u$ are given which guarantee that $ {u^\ast}$ is continuous, of bounded variation, or absolutely continuous on lines, i.e., Sobolev regular. The energy inequality is proven in $n = 2$ dimensions by employing a Carleman type isoperimetric inequality if $\log p$ is subharmonic. The energy equality is settled via a reduction to the case of a homogeneous mass density.
Classification of all parabolic subgroup-schemes of a reductive linear algebraic group over an algebraically closed field
Christian
Wenzel
211-218
Abstract: Let $G$ be a reductive linear algebraic group over an algebraically closed field $K$. The classification of all parabolic subgroups of $G$ has been known for many years. In that context subgroups of $G$ have been understood as varieties, i.e. as reduced schemes. Also several nontrivial nonreduced subgroup schemes of $G$ are known, but until now nobody knew how many there are and what there structure is. Here I give a classification of all parabolic subgroup schemes of $G$ in $\operatorname{char}(K) > 3$ .
Cohomological dimension and metrizable spaces
Jerzy
Dydak
219-234
Abstract: The purpose of this paper is to address several problems posed by V. I. Kuzminov [Ku] regarding cohomological dimension of noncompact spaces. In particular, we prove the following results: Theorem A. Suppose $X$ is metrizable and $G$ is the direct limit of the direct system $\{ {G_s},{h_{s\prime ,s}},S\}$ of abelian groups. Then, $\displaystyle {\dim _G}X \leq \max \{ {\dim _{{G_s}}}X\vert s \in S\}$ . Theorem B. Let $X$ be a metrizable space and let $ G$ be an abelian group. Let $l = \{ p\vert p \cdot (G/\operatorname{Tor}G) \ne G/\operatorname{Tor}G\}$. (a) If $G = \operatorname{Tor}G$, then ${\dim _G}X = \max \{ {\dim _H}X\vert H \in \sigma (G)\}$, (b) ${\dim _G}X = \max \{ {\dim _{\operatorname{Tor}G}}X,{\dim _{G/\operatorname{Tor}G}}X\}$, (c) ${\dim _G}X \geq {\dim _\mathbb{Q}}X$ if $ G \ne \operatorname{Tor}G$, (d) ${\dim _G}X \geq {\dim _{{{\hat{\mathbb{Z}}}_l}}}X$, where $ {\hat{\mathbb{Z}}_l}$ is the group of $l$-adic integers, (e) $\max ({\dim _G}X,{\dim _\mathbb{Q}}X + 1) \geq \max \{ {\dim _H}X\vert H \in \sigma (G)\}$, (f) ${\dim _G}X \leq \,{\dim _{{\mathbb{Z}_l}}}X \leq \,{\dim _G}X + 1$ if $G \ne 0$ is torsion-free. Theorem B generalizes a well-known result of M. F. Bockstein [B].
Algebraic convergence of Schottky groups
Richard D.
Canary
235-258
Abstract: A discrete faithful representation of the free group on $g$ generators ${F_g}$ into $\operatorname{Isom}_ + ({{\mathbf{H}}^3})$ is said to be a Schottky group if $ ({{\mathbf{H}}^3} \cup {D_\Gamma })/\Gamma$ is homeomorphic to a handlebody $ {H_g}$ (where ${D_\Gamma }$ is the domain of discontinuity for $ \Gamma$'s action on the sphere at infinity for $ {{\mathbf{H}}^3}$). Schottky space $ {\mathcal{S}_g}$, the space of all Schottky groups, is parameterized by the quotient of the Teichmüller space $\mathcal{T}({S_g})$ of the closed surface of genus $ g$ by ${\operatorname{Mod} _0}({H_g})$ where $ {\operatorname{Mod} _0}({H_g})$ is the group of (isotopy classes of) homeomorphisms of ${S_g}$ which extend to homeomorphisms of $ {H_g}$ which are homotopic to the identity. Masur exhibited a domain $\mathcal{O}({H_g})$ of discontinuity for $ {\operatorname{Mod} _0}({H_g})$'s action on $PL({S_g})$ (the space of projective measured laminations on ${S_g}$), so $\mathcal{B}({H_g}) = \mathcal{O}({H_g})/{\operatorname{Mod} _0}({H_g})$ may be appended to ${\mathcal{S}_g}$ as a boundary. Thurston conjectured that if a sequence $ \{ {\rho _i}:{F_g} \to \operatorname{Isom}_ + ({{\mathbf{H}}^3})\}$ of Schottky groups converged into $\mathcal{B}({H_g})$, then it converged as a sequence of representations, up to subsequence and conjugation. In this paper, we prove Thurston's conjecture in the case where ${H_g}$ is homeomorphic to $S \times I$ and the length ${l_{{N_i}}}({(\partial S)^\ast})$ in ${N_i} = {{\mathbf{H}}^3}/{\rho _i}({F_g})$ of the closed geodesic(s) in the homotopy class of the boundary of $S$ is bounded above by some constant $K$.
Nonwandering structures at the period-doubling limit in dimensions $2$ and $3$
Marcy M.
Barge;
Russell B.
Walker
259-277
Abstract: A Cantor set supporting an adding machine is the simplest nonwandering structure that can occur at the conclusion of a sequence of perioddoubling bifurcations of plane homeomorphisms. In some families this structure is persistent. In this manuscript it is shown that no plane homeomorphism has nonwandering Knaster continua on which the homeomorphism is semiconjugate to the adding machine. Using a theorem of M. Brown, a three-space homeomorphism is constructed which has an invariant set, $ \Lambda$, the product of a Knaster continuum and a Cantor set. $\Lambda$ is chainable, supports positive entropy but contains only power-of-two periodic orbits. And the homeomorphism restricted to $\Lambda$ is semiconjugate to the adding machine. Lastly, a zero topological entropy ${C^\infty }$ disk diffeomorphism is constructed which has large nonwandering structures over a generalized adding machine on a Cantor set.
The rectifiable metric on the set of closed subspaces of Hilbert space
Lawrence G.
Brown
279-289
Abstract: Consider the set of selfadjoint projections on a fixed Hilbert space. It is well known that the connected components, under the norm topology, are the sets $\{ p:{\text{rank}}\;p = \alpha ,{\text{rank}}(1 - p) = \beta \}$, where $\alpha$ and $\beta$ are appropriate cardinal numbers. On a given component, instead of using the metric induced by the norm, we can use the rectifiable metric $ {d_r}$ which is defined in terms of the lengths of rectifiable paths or, equivalently in this case, the lengths of $\varepsilon$-chains. If $\left\Vert {p - q} \right\Vert < 1$, then ${d_r}(p,q) = {\sin ^{ - 1}}(\left\Vert {p - q} \right\Vert)$, but if $\left\Vert {p - q} \right\Vert = 1$, ${d_r}(p,q)$ can have any value in $\left[ {\frac{\pi } {2},\pi } \right]$ (assuming $\alpha$ and $\beta$ are infinite). If ${d_r}(p,q) \ne \frac{\pi } {2}$, a minimizing path joining $p$ and $q$ exists; but if ${d_r}(p,q) = \frac{\pi } {2}$, a minimizing path exists if and only if $ {\text{rank}}(p \wedge (1 - q)) = {\text{rank}}(q \wedge (1 - p))$.
Symmetries of homotopy complex projective three spaces
Mark
Hughes
291-304
Abstract: We study symmetry properties of six-dimensional, smooth, closed manifolds which are homotopy equivalent to ${\mathbf{C}}{P^3}$. There are infinitely differentiably distinct such manifolds. It is known that if $m$ is an odd prime, infinitely many homotopy $ {\mathbf{C}}{P^3}$'s admit $ {{\mathbf{Z}}_m}$-actions whereas only the standard ${\mathbf{C}}{P^3}$ admits an action of the group $ {{\mathbf{Z}}_m} \times {{\mathbf{Z}}_m} \times {{\mathbf{Z}}_m}$. We study the intermediate case of ${{\mathbf{Z}}_m} \times {{\mathbf{Z}}_m}$-actions and show that infinitely many homotopy ${\mathbf{C}}{P^3}$'s do admit $ {{\mathbf{Z}}_m} \times {{\mathbf{Z}}_m}$-actions for a fixed prime $m$. The major tool involved is equivariant surgery theory. Using a transversality argument, we construct normal maps for which the relevant surgery obstructions vanish allowing the construction of $ {{\mathbf{Z}}_m} \times {{\mathbf{Z}}_m}$-actions on homotopy ${\mathbf{C}}{P^3}$'s which are $ {{\mathbf{Z}}_m} \times {{\mathbf{Z}}_m}$-homotopy equivalent to a specially chosen linear action on $ {\mathbf{C}}{P^3}$. A key idea is to exploit an extra bit of symmetry which is built into our set-up in a way that forces the signature obstruction to vanish. By varying the parameters of our construction and calculating Pontryagin classes, we may construct actions on infinitely many differentiably distinct homotopy $ {\mathbf{C}}{P^3}$'s as claimed.
The canonical compactification of a finite group of Lie type
Mohan S.
Putcha;
Lex E.
Renner
305-319
Abstract: Let $G$ be a finite group of Lie type. We construct a finite monoid $ \mathcal{M}$ having $ G$ as the group of units. $\mathcal{M}$ has properties analogous to the canonical compactification of a reductive group. The complex representation theory of $\mathcal{M}$ yields Harish-Chandra's philosophy of cuspidal representations of $G$. The main purpose of this paper is to determine the irreducible modular representations of $\mathcal{M}$. We then show that all the irreducible modular representations of $G$ come (via the 1942 work of Clifford) from the one-dimensional representations of the maximal subgroups of $ \mathcal{M}$. This yields a semigroup approach to the modular representation theory of $G$, via the full rank factorizations of the 'sandwich matrices' of $ \mathcal{M}$. We then determine the irreducible modular representations of any finite monoid of Lie type.
Some geometric properties on the Fourier and Fourier-Stieltjes algebras of locally compact groups, Arens regularity and related problems
Anthony To Ming
Lau;
Ali
Ülger
321-359
Abstract: Let $G$ be a locally compact topological group and $ A(G)\;[B(G)]$ be, respectively, the Fourier and Fourier-Stieltjes algebras of $G$. It is one of the purposes of this paper to investigate the $ {\text{RNP}}$ (= Radon-Nikodym property) and some other geometric properties such as weak $RNP$, the Dunford-Pettis property and the Schur property on the algebras $A(G)$ and $B(G)$, and to relate these properties to the properties of the multiplication operator on the group $ {C^\ast}$-algebra ${C^\ast}(G)$. We also investigate the problem of Arens regularity of the projective tensor products $ {C^\ast}(G)\hat \otimes A$, when $ B(G) = {C^\ast}{(G)^\ast}$ has the $ {\text{RNP}}$ and $ A$ is any $ {C^\ast}$-algebra. Some related problems on the measure algebra, the group algebra and the algebras ${A_p}(G)$, $P{F_p}(G)$, $P{M_p}(G)\;(1 < p < \infty )$ are also discussed.
The Martin boundary in non-Lipschitz domains
Richard F.
Bass;
Krzysztof
Burdzy
361-378
Abstract: The Martin boundary with respect to the Laplacian and with respect to uniformly elliptic operators in divergence form can be identified with the Euclidean boundary in ${C^\gamma }$ domains, where $\displaystyle \gamma (x) = bx\log \log (1/x)/\log \log \log (1/x),$ $b$ small. A counterexample shows that this result is very nearly sharp.
Characterization of completions of unique factorization domains
Raymond C.
Heitmann
379-387
Abstract: It is shown that a complete local ring is the completion of a unique factorization domain if and only if it is a field, a discrete valuation ring, or it has depth at least two and no element of its prime ring is a zerodivisor. It is also shown that the Normal Chain Conjecture is false and that there exist local noncatenary UFDs.
On the resolution of certain graded algebras
M. P.
Cavaliere;
M. E.
Rossi;
G.
Valla
389-409
Abstract: Let $A = R/I$ be a graded algebra over the polynomial ring $R = k[{X_0}, \ldots ,{X_n}]$. Some properties of the numerical invariants in a minimal free resolution of $A$ are discussed in the case $A$ is a "Short Graded Algebra". When $ A$ is the homogeneous coordinate ring of a set of points in generic position in the projective space, several result are obtained on the line traced by some conjectures proposed by Green and Lazarsfeld in [GL] and Lorenzini in [L1]
Conjugate loci of totally geodesic submanifolds of symmetric spaces
J. M.
Burns
411-425
Abstract: The conjugate and cut loci of fixed point sets of involutions which fix the origin of a compact symmetric space are studied. The first conjugate locus is described in terms of roots and weights of certain representations. When the first conjugate locus and the cut locus agree, we study Morse functions which give a simple decomposition of the symmetric space. We describe for some examples the topological implications of our results.
Topological properties of $q$-convex sets
Guido
Lupacciolu
427-435
Abstract: We discuss the topological properties of a certain class of compact sets in a $q$-complete complex manifold $M$. These sets--which we call $ q$-convex in $ M$--include, for $ q = 0$, the $\mathcal{O}(M)$-convex compact sets in a Stein manifold. Then we show applications of the topological results to the subjects of removable singularities for ${\bar \partial _b}$.
Taylor series with limit-points on a finite number of circles
Emmanuel S.
Katsoprinakis
437-450
Abstract: Let $ S(z):\sum\nolimits_{n = 0}^\infty {{a_n}{z_n}}$ be a power series with complex coefficients. For each $z$ in the unit circle $T = \{ z \in \mathbb{C}:\vert z\vert = 1\}$ we denote by $L(z)$ the set of limit-points of the sequence $\{ {s_n}(z)\} $ of the partial sums of $ S(z)$. In this paper we examine Taylor series for which the set $L(z)$, for $z$ in an infinite subset of $T$, is the union of a finite number, uniformly bounded in $z$, of concentric circles. We show that, if in addition $ \lim \inf \vert{a_n}\vert\; > 0$, a complete characterization of these series in terms of their coefficients is possible (see Theorem 1).
Polynomial identities in graded group rings, restricted Lie algebras and $p$-adic analytic groups
Aner
Shalev
451-462
Abstract: Let $G$ be any finitely generated group, and let $K$ be a field of characteristic $ p > 0$. It is shown that the graded group ring $\operatorname{gr}(KG)$ satisfies a nontrivial polynomial identity if and only if the pro-$p$ completion of $G$ is $p$-adic analytic, i.e. can be given the structure of a Lie group over the $p$-adic field $ {\mathbb{Q}_p}$. The proof applies theorems of Lazard, Quillen and Passman, as well as results on Engel Lie algebras and on dimension subgroups in positive characteristic.
Algebraic cycles and approximation theorems in real algebraic geometry
J.
Bochnak;
W.
Kucharz
463-472
Abstract: Let $M$ be a compact ${C^\infty }$ manifold. A theorem of Nash-Tognoli asserts that $M$ has an algebraic model, that is, $ M$ is diffeomorphic to a nonsingular real algebraic set $X$. Let $ H_{{\text{alg}}}^k(X,\mathbb{Z}/2)$ denote the subgroup of ${H^k}(X,\mathbb{Z}/2)$ of the cohomology classes determined by algebraic cycles of codimension $ k$ on $X$. Assuming that $M$ is connected, orientable and $\dim\,M \geq 5$, we prove in this paper that a subgroup $G$ of $ {H^2}(M,\mathbb{Z}/2)$ is isomorphic to $ H_{{\text{alg}}}^2(X,\mathbb{Z}/2)$ for some algebraic model $X$ of $M$ if and only if ${w_2}(TM)$ is in $G$ and each element of $G$ is of the form $ {w_2}(\xi )$ for some real vector bundle $\xi$ over $M$, where ${w_2}$ stands for the second Stiefel-Whitney class. A result of this type was previously known for subgroups $G$ of $ {H^1}(M,\mathbb{Z}/2)$.
Composition algebras over algebraic curves of genus zero
Holger P.
Petersson
473-493
Abstract: We rephrase the classical theory of composition algebras over fields, particularly the Cayley-Dickson Doubling Process and Zorn's Vector Matrices, in the setting of locally ringed spaces. Fixing an arbitrary base field, we use these constructions to classify composition algebras over (complete smooth) curves of genus zero. Applications are given to composition algebras over function fields of genus zero and polynomial rings.