REPRESENTATIONS OF QUANTUM LORENTZ GROUP ON GELFAND SPACES

Abstract

A large class of representations of the quantum Lorentz group QLG (the one admitting Iwasawa decomposition) is found and described in detail. In a sense the class contains all irreducible unitary representations of QLG. Parabolic subgroup P of the group QLG is introduced. It is a smooth deformation of the subgroup of SL (2, C) consisting of the upper-triangular matrices. A description of the set of all 1-dimensional representations (the characters) of P is given. It turns out that the topological structure of this set is not the same as for the parabolic subgroup of the classical Lorentz group. The class of (in general non-unitary) representations of QLG induced by characters of its parabolic subgroup P is investigated. Representations act on spaces of smooth sections of (quantum) line boundles over the homogeneous space P\QLG (Gelfand spaces) as in the classical case. For any pair of Gelfand spaces the set of all non-zero invariant bilinear forms is described. This set is not empty only for certain pairs. We give a complete list of such pairs. Using this list we solve the problems of equivalence and irreducibility of the representation. We distinguish a class of Gelfand spaces carrying unitary representations of QLG.