The Dual Quantum Group for the Quantum Group Analog of the Normalizer of SU(1, 1) in $SL(2,\mathbb C)$

The quantum group analog of the normalizer of SU(1, 1) in $SL(2,\mathbb C)$ is an important and nontrivial example of a noncompact 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 this article is to give an explicit description of the dual quantum group for this example involving the quantized enveloping algebra $U_q(\mathfrak {su}(1,1))$. It turns out that $U_q(\mathfrak {su}(1,1))$ does not suffice to generate the dual quantum group. The dual quantum group is graded with respect to commutation and anticommutation with a suitable analog 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 analog of the normalizer of SU(1, 1) in $SL(2,\mathbb C)$ is decomposed into irreducible corepresentations. Upon restricting the irreducible corepresentations to $U_q(\mathfrak {su}(1,1))$-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 the analysis of special functions of basic hypergeometric type and, in particular, some results on these special functions are obtained, which are stated separately. This article is split into two parts: the first part gives almost all of the statements and the results, and the statements of this part are independent of the second part. The second part contains the proofs of all the statements.