  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.
