Analytic continuation of Flensted-Jensen representations

LetX=G/H be an affine symmetric space of Hermitian type. For modules in the analytic continuation of the scalar holomorphic discrete series forG, we show existence and uniqueness (that is, a multiplicity one result) of imbeddings into functions onX. The corresponding intertwining operators are analyzed using our previous methods for discrete series. In a slightly less explicit way, we also give the analogous results for the continuation of the general discrete series.     

What is Q-Curvature?

Branson’s Q-curvature is now recognized as a fundamental quantity in conformal geometry. We outline its construction and present its basic properties.     

A model for an interacting quantum field

We present here a close nonlinear analog to the free quantum field of Bose statistics, in which the linear one-particle space is replaced by a nonlinear infinite-dimensional Hermitian symmetric space D, and the quantum field is constructed as a Hilbert space of holomorphic functions on D.     

A note on the conformal quasi-invariance of the Laplacian on a pseudo-Riemannian manifold

We consider the Laplacian on a pseudo-Riemannian manifold with constant scalar curvature (e.g. Euclidian space with an arbitrary signed inner product or its conformal compactification and coverings of this) and show that for this minus a constant we have quasi-invariance with respect to an action of the conformal group on functions.     

Real double flag varieties for the symplectic group

In this paper we study a key example of a Hermitian symmetric space and a natural associated double flag variety, namely for the real symplectic group G and the symmetric subgroup L, the Levi part of the Siegel parabolic $P_S$. We give a detailed treatment of the case of the maximal parabolic subgroups Q of L corresponding to Grassmannians and the product variety of $G/P_S$ and $L/Q$; in particular we classify the L-orbits here, and find natural explicit integral transforms between degenerate principal series of L and G.     

Analysis on the minimal representation of O(p,q) I. Realization via conformal geometry

This is the first in a series of papers devoted to an analogue of the metaplectic representation, namely the minimal unitary representation of an indefinite orthogonal group; this representation corresponds to the minimal nilpotent coadjoint orbit in the philosophy of Kirillov–Kostant. We begin by applying methods from conformal geometry of pseudo-Riemannian manifolds to a general construction of an infinite-dimensional representation of the conformal group on the solution space of the Yamabe equation. By functoriality of the constructions, we obtain different models of the unitary representation, as well as giving new proofs of unitarity and irreducibility. The results in this paper play a basic role in the subsequent papers, where we give explicit branching formulae, and prove unitarization in the various models.     

Dihydroazulene-Azobenzene-Dihydroazulene Triad Photoswitches

Photoswitch triads comprising two dihydroazulene (DHA) units in conjugation with a central trans-azobenzene (AZB) unit were prepared in stepwise protocols starting from meta- and para-disubstituted azobenzenes. The para-connected triad had significantly altered optical properties and lacked the photoactivity of the separate photochromes. In contrast, for the meta-connected triad, all three photochromes could be photoisomerized to generate an isomer with two vinylheptafulvene (VHF) units and a cis-azobenzene unit. Ultrafast spectroscopy of the photoisomerizations revealed a fast DHA-to-VHF photoisomerization and a slower trans-to-cis AZB photoisomerization. This meta triad underwent thermal VHF-to-DHA back-conversion with a similar rate of all VHFs, independent of the identity of the neighboring units, and in parallel thermal cis-to-trans AZB conversion. The experimental observations were supported by computation (excitation spectra and orbital analysis of the transitions).     

Causal compactification and Hardy spaces

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

     

Analysis on the minimal representation of O(p,q): III. Ultrahyperbolic equations on

For the group O(p,q) we give a new construction of its minimal unitary representation via Euclidean Fourier analysis. This is an extension of the q=2 case, where the representation is the mass zero, spin zero representation realized in a Hilbert space of solutions to the wave equation. The group O(p,q) acts as the Möbius group of conformal transformations on , and preserves a space of solutions of the ultrahyperbolic Laplace equation on . We construct in an intrinsic and natural way a Hilbert space of solutions so that O(p,q) becomes a continuous irreducible unitary representation in this Hilbert space. We also prove that this representation is unitarily equivalent to the representation on L²(C), where C is the conical subvariety of the nilradical of a maximal parabolic subalgebra obtained by intersecting with the minimal nilpotent orbit in the Lie algebra of O(p,q).     

Conformally invariant differential equations and projective geometry

We combine harmonic analysis on certain pseudo-Riemannian symmetric spaces with results on conformally invariant linear and non-linear differential equations. This gives in many cases part of the decomposition of certain representations of the conformal group of a manifold when restricted to the isometry group.