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

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