  The quantum group analogue of the normalizer of SU(1,1) in SL(2,C) is an
important and non-trivial example of a non-compact quantum group. The general
theory of locally compact quantum groups in the operator algebra setting
implies the existence of the dual quantum group. The first main goal of the
paper is to give an explicit description of the dual quantum group for this
example involving the quantized enveloping algebra U_q(su(1,1)). It turns out
that U_q(su(1,1)) does not suffice to generate the dual quantum group. The dual
quantum group is graded with respect to commutation and anticommutation with a
suitable analogue of the Casimir operator characterized by an affiliation
relation to a von Neumann algebra. This is used to obtain an explicit set of
generators. Having the dual quantum group the left regular corepresentation of
the quantum group analogue of the normalizer of SU(1,1) in SL(2,C) is
decomposed into irreducible corepresentations. Upon restricting the irreducible
corepresentations to U_q(su(1,1))-representation one finds combinations of the
positive and negative discrete series representations with the strange series
representations as well as combinations of the principal unitary series
representations. The detailed analysis of this example involves analysis of
special functions of basic hypergeometric type and, in particular, some results
on these special functions are obtained, which are stated separately.
  The paper is split into two parts; the first part gives almost all of the
statements and the results, and the statements in the first part are
independent of the second part. The second part contains the proofs of all the
statements.
