Harmonic Analysis on the quantum Lorentz group

This works begins with a review of complexification and realification of Hopf algebras. We emphasize the notion of multiplier Hopf algebras for the description of different classes of functions (compact supported, bounded, unbounded) on complex quantum groups and the construction of the associated left and right Haar measure. Using a continuation of $6j$ symbols of $SU_q (2)$ with complex spins, we give a new description of the unitary representations of $SL_q (2,\CC)_{\RR}$ and find explicit expressions for the characters of $SL_q (2,\CC)_{\RR}$. We give the precise form of the representation of the $R$ matrix in the tensor product of two infinite dimensional unitary representations of $SL_q (2,\CC)_{\RR}$ and compute the Clebsch-Gordan coefficients of the tensor product of a finite dimensional representation with a finite or infinite dimensional irreducible representation. The major theorem of this article is the Plancherel formula for the Quantum Lorentz Group, which has a purely combinatorial proof. We have worked out the case of $SL_q (2,\CC)_{\RR}$ for $q$ real. It will be clear to the reader that the structures we have found are quite general and are likely to be generalized to quantization of other complex groups. The generalization of these constructions to the case where $q$ is an arbitrary complex number requires a more careful study which will be developed elsewhere.