Representation of the Quantum Plane, its Quantum Double and Harmonic Analysis on $GL_q^+(2,R)$

We give complete detail of the description of the GNS representation of the quantum plane $\cA$ and its dual $\hat{\cA}$ as a von-Neumann algebra. In particular we obtain a rather surprising result that the multiplicative unitary $W$ is manageable in this quantum semigroup context. We study the quantum double group construction introduced by Woronowicz, and using Baaj and Vaes' construction of the multiplicative unitary $\bW_m$, we give the GNS description of the quantum double $\cD(\cA)$ which is equivalent to $GL_q^+(2,\R)$. Furthermore we study the fundamental corepresentation $T^{\l,t}$ and its matrix coefficients, and show that it can be expressed by the $b$-Hypergeometric function. We also study the regular corepresentation and representation induced by $\bW_m$, and prove that the space of $L^2$ functions on the quantum double decomposes into the continuous series representation of $U_q(\gl(2,\R))$ with the quantum dilogarithm $|S_b(Q+2i\a)|^2$ as the Plancherel measure. Finally we describe certain representation theoretic meaning of integral transforms involving the quantum dilogarithm function.