Paley--Wiener theorems for the Dunkl transform
Marcel
de Jeu
4225-4250
Abstract: We conjecture a geometrical form of the Paley-Wiener theorem for the Dunkl transform and prove three instances thereof, by using a reduction to the one-dimensional even case, shift operators, and a limit transition from Opdam's results for the graded Hecke algebra, respectively. These Paley-Wiener theorems are used to extend Dunkl's intertwining operator to arbitrary smooth functions. Furthermore, the connection between Dunkl operators and the Cartan motion group is established. It is shown how the algebra of radial parts of invariant differential operators can be described explicitly in terms of Dunkl operators. This description implies that the generalized Bessel functions coincide with the spherical functions. In this context of the Cartan motion group, the restriction of Dunkl's intertwining operator to the invariants can be interpreted in terms of the Abel transform. We also show that, for certain values of the multiplicities of the restricted roots, the Abel transform is essentially inverted by a differential operator.
Whitney towers and gropes in 4--manifolds
Rob
Schneiderman
4251-4278
Abstract: Many open problems and important theorems in low-dimensional topology have been formulated as statements about certain 2-complexes called gropes. This paper describes a precise correspondence between embedded gropes in 4-manifolds and the failure of the Whitney move in terms of iterated `towers' of Whitney disks. The `flexibility' of these Whitney towers is used to demonstrate some geometric consequences for knot and link concordance connected to $n$-solvability, $k$-cobordism and grope concordance. The key observation is that the essential structure of gropes and Whitney towers can be described by embedded unitrivalent trees which can be controlled during surgeries and Whitney moves. It is shown that a Whitney move in a Whitney tower induces an IHX (Jacobi) relation on the embedded trees.
Root invariants in the Adams spectral sequence
Mark
Behrens
4279-4341
Abstract: Let $E$ be a ring spectrum for which the $E$-Adams spectral sequence converges. We define a variant of Mahowald's root invariant called the `filtered root invariant' which takes values in the $E_1$ term of the $E$-Adams spectral sequence. The main theorems of this paper are concerned with when these filtered root invariants detect the actual root invariant, and explain a relationship between filtered root invariants and differentials and compositions in the $E$-Adams spectral sequence. These theorems are compared to some known computations of root invariants at the prime $2$. We use the filtered root invariants to compute some low-dimensional root invariants of $v_1$-periodic elements at the prime $3$. We also compute the root invariants of some infinite $v_1$-periodic families of elements at the prime $3$.
Functional distribution of $L(s, \chi_d)$ with real characters and denseness of quadratic class numbers
Hidehiko
Mishou;
Hirofumi
Nagoshi
4343-4366
Abstract: We investigate the functional distribution of $L$-functions $L(s, \chi_d)$ with real primitive characters $\chi_d$ on the region $1/2 < \operatorname{Re} s <1$ as $d$ varies over fundamental discriminants. Actually we establish the so-called universality theorem for $L(s, \chi_d)$ in the $d$-aspect. From this theorem we can, of course, deduce some results concerning the value distribution and the non-vanishing. As another corollary, it follows that for any fixed $a, b$ with $1/2< a< b<1$ and positive integers $r', m$, there exist infinitely many $d$ such that for every $L^{(r)} (s, \chi_d)$ has at least $m$ zeros on the interval $[a, b]$ in the real axis. We also study the value distribution of $L(s, \chi_d)$ for fixed $s$ with $\operatorname{Re} s =1$ and variable $d$, and obtain the denseness result concerning class numbers of quadratic fields.
Frankel's theorem in the symplectic category
Min
Kyu
Kim
4367-4377
Abstract: We prove that if an $(n-1)$-dimensional torus acts symplectically on a $2n$-dimensional symplectic manifold, then the action has a fixed point if and only if the action is Hamiltonian. One may regard it as a symplectic version of Frankel's theorem which says that a Kähler circle action has a fixed point if and only if it is Hamiltonian. The case of $n=2$ is the well-known theorem by McDuff.
$\mathbf{h}$-principles for hypersurfaces with prescribed principle curvatures and directions
Mohammad
Ghomi;
Marek
Kossowski
4379-4393
Abstract: We prove that any compact orientable hypersurface with boundary immersed (resp. embedded) in Euclidean space is regularly homotopic (resp. isotopic) to a hypersurface with principal directions which may have any prescribed homotopy type, and principal curvatures each of which may be prescribed to within an arbitrary small error of any constant. Further we construct regular homotopies (resp. isotopies) which control the principal curvatures and directions of hypersurfaces in a variety of ways. These results, which we prove by holonomic approximation, establish some h-principles in the sense of Gromov, and generalize theorems of Gluck and Pan on embedding and knotting of positively curved surfaces in 3-space.
A preparation theorem for Weierstrass systems
Daniel
J.
Miller
4395-4439
Abstract: It is shown that Lion and Rolin's preparation theorem for globally subanalytic functions holds for the collection of definable functions in any expansion of the real ordered field by a Weierstrass system.
Transplantation and multiplier theorems for Fourier-Bessel expansions
Óscar
Ciaurri;
Krzysztof
Stempak
4441-4465
Abstract: Proved are weighted transplantation inequalities for Fourier-Bessel expansions. These extend known results on this subject by considering the largest possible range of parameters, allowing more weights and admitting a shift. The results are then used to produce a fairly general multiplier theorem with power weights for considered expansions. Also fractional integral results and conjugate function norm inequalities for these expansions are proved.
Approximation and regularization of Lipschitz functions: Convergence of the gradients
Marc-Olivier
Czarnecki;
Ludovic
Rifford
4467-4520
Abstract: We examine the possible extensions to the Lipschitzian setting of the classical result on $C^1$-convergence: first (approximation), if a sequence $(f_n)$ of functions of class $C^1$ from $\mathbb{R}^N$ to $\mathbb{R}$ converges uniformly to a function $f$ of class $C^1$, then the gradient of $f$ is a limit of gradients of $f_n$ in the sense that $\operatorname{graph}(\nabla f)\subset \liminf_{n\to +\infty} \operatorname{graph}(\nabla f_n)$; second (regularization), the functions $(f_n)$ can be chosen to be of class $C^{\infty}$ and $C^1$-converging to $f$ in the sense that $\lim_{n\to +\infty} \Vert f_n-f\Vert _{\infty}+ \Vert\nabla f_n-\nabla f\Vert _{\infty}=0$. In other words, the space of $C^{\infty}$ functions is dense in the space of $C^1$ functions endowed with the $C^1$ pseudo-norm. We first deepen the properties of Warga's counterexample (1981) for the extension of the approximation part to the Lipschitzian setting. This part cannot be extended, even if one restricts the approximation schemes to the classical convolution and the Lasry-Lions regularization. We thus make more precise various results in the literature on the convergence of subdifferentials. We then show that the regularization part can be extended to the Lipschitzian setting, namely if $f:\mathbb{R}^N \rightarrow {\mathbb{R}}$ is a locally Lipschitz function, we build a sequence of smooth functions $(f_n)_{n \in \mathbb{N}}$ such that $\displaystyle \lim_{n\to +\infty} \Vert f_n-f\Vert _{\infty}=0,$ $\displaystyle \lim_{n\to +\infty} d_{Haus}(\operatorname{graph}(\nabla f_n), \operatorname{graph}(\partial f))=0.$ In other words, the space of $C^{\infty}$ functions is dense in the space of locally Lipschitz functions endowed with an appropriate Lipschitz pseudo-distance. Up to now, Rockafellar and Wets (1998) have shown that the convolution procedure permits us to have the equality $\limsup_{n\to +\infty} \operatorname{graph}(\nabla f_n) =\operatorname{graph}(\partial f)$, which cannot provide the exactness of our result. As a consequence, we obtain a similar result on the regularization of epi-Lipschitz sets. With both functional and set parts, we improve previous results in the literature on the regularization of functions and sets.
Generic systems of co-rank one vector distributions
Howard
Jacobowitz
4521-4531
Abstract: This paper studies a generic class of sub-bundles of the complexified tangent bundle. Involutive, generic structures always exist and have Levi forms with only simple zeroes. For a compact, orientable three-manifold the Chern class of the sub-bundle is mod $2$ equivalent to the Poincaré dual of the characteristic set of the associated system of linear partial differential equations.
Martingales and character ratios
Jason
Fulman
4533-4552
Abstract: Some general connections between martingales and character ratios of finite groups are developed. As an application we sharpen the convergence rate in a central limit theorem for the character ratio of a random representation of the symmetric group on transpositions. A generalization of these results is given for Jack measure on partitions. We also give a probabilistic proof of a result of Burnside and Brauer on the decomposition of tensor products.
Spectra of quantized hyperalgebras
William
Chin;
Leonid
Krop
4553-4567
Abstract: We describe the prime and primitive spectra for quantized enveloping algebras at roots of 1 in characteristic zero in terms of the prime spectrum of the underlying enveloping algebra. Our methods come from the theory of Hopf algebra crossed products. For primitive ideals we obtain an analogue of Duflo's Theorem, which says that every primitive ideal is the annihilator of a simple highest weight module. This depends on an extension of Lusztig's tensor product theorem.
The spectrum of twisted Dirac operators on compact flat manifolds
Roberto
J.
Miatello;
Ricardo
A.
Podestá
4569-4603
Abstract: Let $M$ be an orientable compact flat Riemannian manifold endowed with a spin structure. In this paper we determine the spectrum of Dirac operators acting on smooth sections of twisted spinor bundles of $M$, and we derive a formula for the corresponding eta series. In the case of manifolds with holonomy group $\mathbb{Z}_2^k$, we give a very simple expression for the multiplicities of eigenvalues that allows us to compute explicitly the $\eta$-series, in terms of values of Hurwitz zeta functions, and the $\eta$-invariant. We give the dimension of the space of harmonic spinors and characterize all $\mathbb{Z}_2^k$-manifolds having asymmetric Dirac spectrum. Furthermore, we exhibit many examples of Dirac isospectral pairs of $\mathbb{Z}_2^k$-manifolds which do not satisfy other types of isospectrality. In one of the main examples, we construct a large family of Dirac isospectral compact flat $n$-manifolds, pairwise nonhomeomorphic to each other of the order of $a^n$.
Finite edge-transitive Cayley graphs and rotary Cayley maps
Cai
Heng
Li
4605-4635
Abstract: This paper aims to develop a theory for studying Cayley graphs, especially for those with a high degree of symmetry. The theory consists of analysing several types of basic Cayley graphs (normal, bi-normal, and core-free), and analysing several operations of Cayley graphs (core quotient, normal quotient, and imprimitive quotient). It provides methods for constructing and characterising various combinatorial objects, such as half-transitive graphs, (orientable and non-orientable) regular Cayley maps, vertex-transitive non-Cayley graphs, and permutation groups containing certain regular subgroups. In particular, a characterisation is given of locally primitive holomorph Cayley graphs, and a classification is given of rotary Cayley maps of simple groups. Also a complete classification is given of primitive permutation groups that contain a regular dihedral subgroup.
Vanishing and non-vanishing of traces of Hecke operators
Jeremy
Rouse
4637-4651
Abstract: Using a reformulation of the Eichler-Selberg trace formula, due to Frechette, Ono and Papanikolas, we consider the problem of the vanishing (resp. non-vanishing) of traces of Hecke operators on spaces of even weight cusp forms with trivial Nebentypus character. For example, we show that for a fixed operator and weight, the set of levels for which the trace vanishes is effectively computable. Also, for a fixed operator the set of weights for which the trace vanishes (for any level) is finite. These results motivate the ``generalized Lehmer conjecture'', that the trace does not vanish for even weights $2k \geq 16$ or $2k = 12$.
On the non-unitary unramified dual for classical $p$--adic groups
Goran
Muic
4653-4687
Abstract: In this paper we give a Zelevinsky type classification of unramified irreducible representations of split classical groups.