arXiv subjects

Gestur Olafsson

Publications and source records attributed to Gestur Olafsson.

At least 37 records · Page 2Linked to original sources

arXiv2019

Symmetric spaces with dissecting involutions

An involutive diffeomorphism $σ$ of a connected smooth manifold $M$ is called dissecting if the complement of its fixed point set is not connected. Dissecting involutions on a complete Riemannian manifold are closely related to constructive quantum field theory through the work of Dimock and Jaffe/Ritter on the construction of reflection positive Hilbert spaces. In this article we classify all pairs $(M,σ)$, where $M$ is an irreducible symmetric space, not necessarily Riemannian, and $σ$ is a dissecting involutive automorphism. In particular, we show that the only irreducible $1$-connected Riemannian symmetric spaces are $S^n$ and $H^n$ with dissecting isometric involutions whose fixed point spaces are $S^{n-1}$ and $H^{n-1}$, respectively.

math.DG↗

arXiv2019

Reflection positivity on spheres

In this article we specialize a construction of a reflection positive Hilbert space due to Dimock and Jaffe--Ritter to the sphere $\mathbb{S}^n$. We determine the resulting Osterwalder--Schrader Hilbert space, a construction that can be viewed as the step from euclidean to relativistic quantum field theory. We show that this process gives rise to an irreducible unitary spherical representation of the orthochronous Lorentz group $G^c = \mathrm{O}_{1,n}(\mathbb{R})^{\uparrow}$ and that the representations thus obtained are the irreducible unitary spherical representations of this group. A key tool is a certain complex domain $Ξ$, known as the crown of the hyperboloid, containing a half-sphere $\mathbb{S}^n_+$ and the hyperboloid $\mathbb{H}^n$ as totally real submanifolds. This domain provides a bridge between those two manifolds when we study unitary representations of $G^c$ in spaces of holomorphic functions on $Ξ$. We connect this analysis with the boundary components which are the de Sitter space and the Lorentz cone of future pointing light like vectors.

math.FA↗

arXiv2019

Toeplitz operators on the domain $\{Z\in M_{2\times2}(\mathbb{C}) \mid Z Z^* < I\}$ with $\mathrm{U}(2)\times\mathbb{T}^2$-invariant symbols

Let $D$ be the irreducible bounded symmetric domain of $2\times2$ complex matrices that satisfy $ZZ^* < I_2$. The biholomorphism group of $D$ is realized by $\mathrm{U}(2,2)$ with isotropy at the origin given by $\mathrm{U}(2)\times\mathrm{U}(2)$. Denote by $\mathbb{T}^2$ the subgroup of diagonal matrices in $\mathrm{U}(2)$. We prove that the set of $\mathrm{U}(2)\times\mathbb{T}^2$-invariant essentially bounded symbols yield Toeplitz operators that generate commutative $C^*$-algebras on all weighted Bergman spaces over $D$. Using tools from representation theory, we also provide an integral formula for the spectra of these Toeplitz operators.

math.FA↗

arXiv2019

Coorbits for projective representations with an application to Bergman spaces

Recently representation theory has been used to provide atomic decompositions for a large collection of classical Banach spaces. In this paper we extend the techniques to also include projective representations. As our main application we obtain atomic decompositions of Bergman spaces on the unit ball through the holomorphic discrete series for the group of isometries of the ball.

math.FA↗

arXiv2018

Reflection Positivity---A Representation Theoretic Perspective

Refection Positivity is a central theme at the crossroads of Lie group representations, euclidean and abstract harmonic analysis, constructive quantum field theory, and stochastic processes. This book provides the first presentation of the representation theoretic aspects of Refection Positivity and discusses its connections to those different fields on a level suitable for doctoral students and researchers in related fields.

math.RT↗

arXiv2017

Antiunitary representations and modular theory

Antiunitary representations of Lie groups take values in the group of unitary and antiunitary operators on a Hilbert space H. In quantum physics, antiunitary operators implement time inversion or a PCT symmetry, and in the modular theory of operator algebras they arise as modular conjugations from cyclic separating vectors of von Neumann algebras. We survey some of the key concepts at the borderline between the theory of local observables (Quantum Field Theory (QFT) in the sense of Araki--Haag--Kastler) and modular theory of operator algebras from the perspective of antiunitary group representations. Here a central point is to encode modular objects in standard subspaces V in H which in turn are in one-to-one correspondence with antiunitary representations of the multiplicative group R^x. Half-sided modular inclusions and modular intersections of standard subspaces correspond to antiunitary representations of Aff(R), and these provide the basic building blocks for a general theory started in the 90s with the ground breaking work of Borchers and Wiesbrock and developed in various directions in the QFT context. The emphasis of these notes lies on the translation between configurations of standard subspaces as they arise in the context of modular localization developed by Brunetti, Guido and Longo, and the more classical context of von Neumann algebras with cyclic separating vectors. Our main point is that configurations of standard subspaces can be studied from the perspective of antiunitary Lie group representations and the geometry of the corresponding spaces, which are often fiber bundles over ordered symmetric spaces. We expect this perspective to provide new and systematic insight into the much richer configurations of nets of local observables in QFT.

math.RT↗

arXiv2016

KMS conditions, standard real subspaces and reflection positivity on the circle group

In the present paper we continue our investigations of the representation theoretic side of reflection positivity by studying positive definite functions ψon the additive group (R,+) satisfying a suitably defined KMS condition. These functions take values in the space Bil(V) of bilinear forms on a real vector space V. As in quantum statistical mechanics, the KMS condition is defined in terms of an analytic continuation of ψto the strip { z \in C\: 0 \leq Im z \leq b} with a coupling condition ψ(ib + t) = \oline{ψ(t)} on the boundary. Our first main result consists of a characterization of these functions in terms of modular objects (Δ, J) (J an antilinear involution and Δ> 0 selfadjoint with JΔJ = Δ^{-1}) and an integral representation. Our second main result is the existence of a Bil(V)-valued positive definite function f on the group R_τ= R \rtimes {\id_\R,τ} with τ(t) = -t satisfying f(t,τ) = ψ(it) for t \in R. We thus obtain a 2b-periodic unitary one-parameter group on the GNS space H_f for which the one-parameter group on the GNS space H_ψis obtained by Osterwalder--Schrader quantization. Finally, we show that the building blocks of these representations arise from bundle-valued Sobolev spaces corresponding to the kernels 1/(λ^2 - (d^2)/(dt^2}) on the circle R/bZ of length b.

math-ph↗

arXiv2016

Reflection Positive Stochastic Processes Indexed by Lie Groups

Reflection positivity originates from one of the Osterwalder-Schrader axioms for constructive quantum field theory. It serves as a bridge between euclidean and relativistic quantum field theory. In mathematics, more specifically, in representation theory, it is related to the Cartan duality of symmetric Lie groups (Lie groups with an involution) and results in a transformation of a unitary representation of a symmetric Lie group to a unitary representation of its Cartan dual. In this article we continue our investigation of representation theoretic aspects of reflection positivity by discussing reflection positive Markov processes indexed by Lie groups, measures on path spaces, and invariant gaussian measures in spaces of distribution vectors. This provides new constructions of reflection positive unitary representations.

math-ph↗

arXiv2015

The $\ct$ transform on line bundles over compact Hermitian symmetric spaces

In a previous article the second author together with A. Pasquale determined the spectrum of the $Cos^λ$ transform on smooth functions on the Grassmann manifolds $G_{p,n+1}$. This article extends those results to line bundles over certain Grassmannians. In particular we define the $Cos^λ$ transform on smooth sections of homogeneous line bundles over$G_{p,n+1}$ and show that it is an intertwining operator between generalized ($χ$-spherical) principal series representations induced from a maximal parabolic subgroup of $\mathrm{SL} (n+1, \mathbb{K})$. Then we use the spectrum generating method to determine the $K$-spectrum of the $Cos^λ$ transform.

math.RT↗

arXiv2015

Paley-Wiener Theorem for Line Bundles over Compact Symmetric Spaces and New Estimates for the Heckman-Opdam Hypergeometric Functions

Paley-Wiener type theorems describe the image of a given space of functions, often compactly supported functions, under an integral transform, usually a Fourier transform on a group or homogeneous space. Several authors have studied Paley-Wiener type theorems for Euclidean spaces, Riemannian symmetric spaces of compact or non-compact type as well as affine Riemannian symmetric spaces. In this article we prove a Paley-Wiener theorem for homogeneous line bundles over a compact symmetric space $U/K$. The Paley-Wiener theorem characterizes f with sufficiently small support in terms of holomorphic extendability and exponential growth of their Fourier transforms. An important tool is a generalization of Opdam's estimate for the hypergeometric functions for multiplicity functions that are not necessarily positive. The domain where this estimate is valid is also bigger. This is done in an appendix.

math.RT↗

arXiv2015

Conical Representations for Direct Limits of Symmetric Spaces

We extend the definition of conical representations for Riemannian symmetric spaces to a certain class of infinite-dimensional Riemannian symmetric spaces. Using an infinite-dimensional version of Weyl's Unitary Trick, there is a correspondence between smooth representations of infinite-dimensional noncompact-type Riemannian symmetric spaces and smooth representations of infinite-dimensional compact-type symmetric spaces. We classify all smooth conical representations which are unitary on the compact-type side. Finally, a new class of non-smooth unitary conical representations appears on the compact-type side which has no analogue in the finite-dimensional case. We classify these representations and show how to decompose them into direct integrals of irreducible conical representations.

math.RT↗

arXiv2015

Wave Front Sets of Reductive Lie Group Representations

If $G$ is a Lie group, $H\subset G$ is a closed subgroup, and $τ$ is a unitary representation of $H$, then the authors give a sufficient condition on $ξ\in i\mathfrak{g}^*$ to be in the wave front set of $\operatorname{Ind}_H^Gτ$. In the special case where $τ$ is the trivial representation, this result was conjectured by Howe. If $G$ is a real, reductive algebraic group and $π$ is a unitary representation of $G$ that is weakly contained in the regular representation, then the authors give a geometric description of $\operatorname{WF}(π)$ in terms of the direct integral decomposition of $π$ into irreducibles. Special cases of this result were previously obtained by Kashiwara-Vergne, Howe, and Rossmann. The authors give applications to harmonic analysis problems and branching problems.

math.RT↗

arXiv2014

Reflection positivity for the circle group

In this note we characterize those unitary one-parameter groups U^c which admit euclidean realizations in the sense that they are obtained by the analytic continuation process corresponding to reflection positivity from a unitary representation $U$ of the circle group. These are precisely the ones for which there exists an anti-unitary involution $J$ commuting with $U^c$. This provides an interesting link with the modular data arising in Tomita--Takesaki theory. Introducing the concept of a positive definite function with values in the space of sesquilinear forms, we further establish a link between KMS states and reflection positivity on the circle.

math.RT↗

arXiv2014

Reflection positive one-parameter groups and dilations

The concept of reflection positivity has its origins in the work of Osterwalder--Schrader on constructive quantum field theory. It is a fundamental tool to construct a relativistic quantum field theory as a unitary representation of the Poincare group from a non-relativistic field theory as a representation of the euclidean motion group. This is the second article in a series on the mathematical foundations of reflection positivity. We develop the theory of reflection positive one-parameter groups and the dual theory of dilations of contractive hermitian semigroups. In particular, we connect reflection positivity with the outgoing realization of unitary one-parameter groups by Lax and Phillips. We further show that our results provide effective tools to construct reflection positive representations of general symmetric Lie groups, including the ax+b-group, the Heisenberg group, the euclidean motion group and the euclidean conformal group.

math.RT↗

arXiv2014

Integrability of unitary representations on reproducing kernel spaces

Let g be a Banach Lie algebra and τ: g ---> g an involution. Write g=h+q for the eigenspace decomposition of g with respect to τand g^c := h+iq for the dual Lie algebra. In this article we show the integrability of two types of infinitesimally unitary representations of g^c. The first class of representation is determined by a smooth positive definite kernel K on a locally convex manifold M. The kernel is assumed to satisfying a natural invariance condition with respect to an infinitesimal action β: g \to V(M) by locally integrable vector fields that is compatible with a smooth action of a connected Lie group $H$ with Lie algebra h. The second class is constructed from a positive definite kernel corresponding to a positive definite distribution K \in C^{-\infty}(M \times M) on a finite dimensional smooth manifold M which satisfies a similar invariance condition with respect to a homomorphism β: g \to V(M). As a consequence, we get a generalization of the Luscher--Mack Theorem which applies to a class of semigroups that need not have a polar decomposition. Our integrability results also apply naturally to local representations and representations arising in the context of reflection positivity.

math.RT↗

arXiv2013

The Radon transform and its dual for limits of symmetric spaces

The Radon transform and its dual are central objects in geometric analysis on Riemannian symmetric spaces of the noncompact type. In this article we study algebraic versions of those transforms on inductive limits of symmetric spaces. In particular, we show that normalized versions exists on some spaces of regular functions on the limit. We give a formula for the normalized transform using integral kernels and relate them to limits of double fibration transforms on spheres.

math.RT↗

arXiv2013

The Continuous Spectrum in Discrete Series Branching Laws

If $G$ is a reductive Lie group of Harish-Chandra class, $H$ is a symmetric subgroup, and $π$ is a discrete series representation of $G$, the authors give a condition on the pair $(G,H)$ which guarantees that the direct integral decomposition of $π|_H$ contains each irreducible representation of $H$ with finite multiplicity. In addition, if $G$ is a reductive Lie group of Harish-Chandra class, and $H\subset G$ is a closed, reductive subgroup of Harish-Chandra class, the authors show that the multiplicity function in the direct integral decomposition of $π|_H$ is constant along `continuous parameters'. In obtaining these results, the authors develop a new technique for studying multiplicities in the restriction $π|_H$ via convolution with Harish-Chandra characters. This technique has the advantage of being useful for studying the continuous spectrum as well as the discrete spectrum.

math.RT↗

arXiv2013

Reflection Positivity and Conformal Symmetry

The concept of reflection positivity has its origins in the work of Osterwalder--Schrader on constructive quantum field theory and duality between unitary representations of the euclidean motion group and the Poincare group. On the mathematical side this duality can be made precise as follows. If $\g$ is a Lie algebra with an involutive automorphism $τ$. Decompose $\g = \fh \oplus \fq = \ker(τ- \1) \oplus \ker(τ+ \1)$ into $τ$-eigenspaces and let $\g^c := \fh \oplus i \fq$. At the core of the notion of reflection positivity is the idea that this duality can sometimes be implemented on the level of unitary representations. The idea is simple on the Lie algebra level: Let $(π,\cH^0)$ be a representation of $\g$ where $π$ acts by skew-symmetric operators. Assume that there exists a unitary operator $J$ of order two such that $J πJ = π\circ τ$ and a $\g$-invariant subspace $\cK^0$ which is {\it $J$-positive}. Then complex linear extension leads to a representation of $\g^c$ on $\cK^0$ by operators which are skew-symmetric with respect to $h_J$, so that we obtain a "unitary" representation of $\g^c$ on the pre-Hilbert space $\cK_J^0 := \cK_J/{v \: \cK^0 \: h_J(v,v)=0}$. The aim of this article is twofold. First we discuss reflection positivity in an abstract setting using {\it reflection positive distributions} on the Lie group $G_τ=G\rtimes {1,τ}$ and {\it reflection positive distribution vectors} of a unitary representation of $G_τ$. Then we apply these ideas to the conformal group $\OO_{1,n+1}^+(\R)$ of the sphere $\bS^n$ as well as the the half-space picture mostly used in physics.

math.RT↗