Representations of Quantum Lorentz Group on Gelfand Spaces

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 representations. We distinguish a class of Gelfand spaces carrying unitary representations of QLG. ∗Partially supported by Polish KBN grant No P301 020 07 and 2PO3A 030 14