Gestur Ólafsson

The image of the heat kernel transform on Riemannian symmetric spacesof the noncompact type

This overview is essentially the short article [13] written jointly with Olafsson and Stanton. The only dierence lies in the addition of a new section in the beginning where some classical examples (real line and circle) are discussed.

The holomorphic discrete series for affine symmetric spaces, I

Abstract Let X = G H be an affine symmetric space. We study part of the discrete spectrum of L2(X) for X of Hermitian type, a notion we define in analogy with the group case. In particular we find intertwining operators from the scalar holomorphic discrete series of G, which is automatically of Hermitian type, into L2(X). The multiplicity of this series is then shown to be one by a uniqueness result for the intertwining operators. Finally, we investigate the complexification X C of X and show that the discrete series in question admits holomorphic continuation into a certain domain in X C .

Laplace and Segal–Bargmann transforms on Hermitian symmetric spaces and orthogonal polynomials

Let D=G/K be a complex bounded symmetric domain of tube type in a complex Jordan algebra V and let DR=J∩D⊂D be its real form in a formally real Euclidean Jordan algebra J⊂V; DR=H/L is a bounded realization of the symmetric cone in J. We consider representations of H that are gotten by the generalized Segal–Bargmann transform from a unitary G-space of holomorphic functions on D to an L2-space on DR. We prove that in the unbounded realization the inverse of the unitary part of the restriction map is actually the Laplace transform. We find the extension to D of the spherical functions on DR and find their expansion in terms of the L-spherical polynomials on D, which are Jack symmetric polynomials. We prove that the coefficients are orthogonal polynomials in an L2-space, the measure being the Harish–Chandra Plancherel measure multiplied by the symbol of the Berezin transform. We prove the difference equation and recurrence relation for those polynomials by considering the action of the Lie algebra and the Cayley transform on the polynomials on D. Finally, we use the Laplace transform to study generalized Laguerre functions on symmetric cones.

The Continuous Wavelet Transform and Symmetric Spaces

The continuous wavelet transform has become a widely used tool in applied science during the last decade. In this article we discuss some generalizations coming from actions of closed subgroups H of GL(n,R) acting on Rn. If Rn has finitely many open orbits under the transposed action of H such that the union has full measure, then L2(Rn) decomposes into finitely many irreducible representations, L2(Rn)≃V1⊕⋅⋅⋅⊕Vk under the action of the semidirect product H×sRn. It is well known, that the space Vj contains an admissible vector if and only if the stabilizer in Ht of every point in Vj is compact. In this article we discuss the case where the stabilizer of a generic point in Rn is not compact, but a symmetric subgroup, a case that has not previously been discussed in the literature. In particular we show, that the wavelet transform can always be inverted in this case.

Laguerre Polynomials, Restriction Principle, and Holomorphic Representations of SL(2, R)

The restriction principle is used to implement a realization of the holomorphic representations of SL(2,R) on L2 (R+,tα dt) by way of the standard upper half plane realization. The resulting unitary equivalence establishes a correspondence between functions that transform according to the character Ψ↦ e−i(2n+α+1)Ψ; under rotations and the Laguerre polynomials. The standard recursion relations amongst Laguerre polynomials are derived from the action of the Lie algebra.

Unitary Representations of Lie Groups with Reflection Symmetry

We consider the following class of unitary representationsπof some (real) Lie groupGwhich has a matched pair of symmetries described as follows: (i) SupposeGhas a period-2 automorphismτ, and that the Hilbert spaceH(π) carries a unitary operatorJsuch thatJπ=(π∘τ)J(i.e.,selfsimilarity). (ii) An added symmetry is implied ifH(π) further contains a closed subspaceK0having a certainorder-covarianceproperty, and satisfying theK0-restricted positivity : ⦠v | Jv⦔⩾0, ∀v∈K0, where ⦠· | ·⦔ is the inner product inH(π). From (i)–(ii), we get an induced dual representation of an associated dual groupGc. All three properties, selfsimilarity, order-covariance, and positivity, are satisfied in a natural context whenGis semisimple and hermitean; but whenGis the (ax+b)-group, or the Heisenberg group, positivity is incompatible with the other two axioms for the infinite-dimensional irreducible representations. We describe a class ofG, containing the latter two, which admits a classification of the possible spacesK0⊂H(π) satisfying the axioms of selfsimilarity and order-covariance.

Symmetric spaces of hermitian type

Let M =G/H be a semisimple symmetric space,τ the corresponding involution and D =G/K the Riemannian symmetric space. Then we show that the followingare equivalent: M is of Hermitian type; τ induces a conjugation on D; thereexists an open regular H-invariant cone Ω in q =h[bottom] such that k ∩ Ω ≠ 0. We relate the spaces of Hermitian type to the regular and parahermitian symmetric spaces, analyze the fine structure of D under τ and construct an equivariant Cayley transform. We collect also some results on the classification of invariant cones in q. Finally we point out some applications in representations theory.

Equipartition of energy for waves in symmetric space

Abstract Let X = G K be an odd-dimensional semisimple Riemannian symmetric space of the noncompact type, and suppose that all Cartan subgroups of G are conjugate. Let u be a real-valued classical solution of the modified wave equation utt = (Δ + k) u on R × X, the Cauchy data of which are supported in a closed metric ball of radius a at time t = 0. Here t is the coordinate on R , Δ is the (nonpositive definite) Laplace-Beltrami operator on X, and k is a positive constant depending on the root structure of the Lie algebra of G. We show that the (t-independent) energy functional of u is eventually (for ¦t¦ ⩾ a ) partitioned into equal potential and kinetic parts; specifically, half the integrals over X of ut2, and ¦du¦ 2 − ku 2 respectively, where d is the exterior derivative in X. The proof uses Helgasons Paley-Wiener theorem for X, the classical Paley-Wiener theorem, and properties of Harish-Chandras c function.

Causal compactification and Hardy spaces

Let M = G/H be a irreducible symmetric space of Cayley type. Then M is diffeomorphic to an open and dense G-orbit in the Shilov boundary of G/K x G/K. This compactification of M is causal and can be used to give answers to questions in harmonic analysis on M. In particular we relate the Hardy space of M to the classical Hardy space on the bounded symmetric domain G/K x G/K. This gives a new formula for the Cauchy-Szego kernel for M.

Coorbit Description and Atomic Decomposition of Besov Spaces

Function spaces are central topics in analysis. Often those spaces and related analysis involves symmetries in form of an action of a Lie group. Coorbit theory as introduced by Feichtinger and Gröchenig [6–8] and then later extended in Christensen and Ólafsson [3] gives a unified method to construct Banach spaces of functions based on representations of Lie groups. In this article, we identify the homogeneous Besov spaces on stratified Lie groups introduced by Führ and Mayeli [12] as coorbit spaces in the sense of [3] and use this to derive atomic decompositions for the Besov spaces.