Expedities in de representatietheorie van SL(2,ℝ) en de lorentzgroep. Van topologische tensorproducten tot conforme correlatiefuncties

Correlation functions form the key to the bootstrap programme for constructing and classifying conformal field theories. Their forms are partially dictated by the symmetries that inhere in conformal invariance, yet the four-point correlation functions, whilst subject to these requirements, are difficult to describe. Chapter 1 expounds upon the properties of correlation functions (in euclidean signature) using the Lorentz group G = SO (n ,1) e . The mathematical setup involves the subspace of invariants of the tensor product of certain G -representations. Hilbert spaces and their tensor product fail singularly in defining this subspace, though, as Chapter 2 shows. This problem is mitigated using the nuclear and (dually) Fréchet spaces of smooth and distributional vector, leading to Casselman–Wallach globalisations . Focussing on SL(2, R ) and its discrete series, functional-analytic and representation-theoretic techniques are employed to show three main results. The first pertains to appropriate versions of Schur’s Lemma for continuous intertwining operators between Casselman–Wallach globalisations. The other two show that a fourfold nuclear tensor product of distribution vectors in discrete series does have nonzero invariants, which are described using infinite series of intertwiners with coefficients of moderate growth . The third chapter studies harmonic analysis on the semisimple symmetric space G/MA , where G = SO( n ,1) e and MA is the Levi factor of a standard parabola. Vector-valued spherical functions on G relative to MA × MA are realised via spin-graph functions . These act as ‘test functions’ for four-point correlation functions and carry the action of a large class of right-invariant differential operators. This action is intimately related to Racah algebras. Show more