Representation of the quantum plane, its quantum double, and harmonic analysis on $$GL_q^+(2,\mathbb{R })$$

Abstract

We give complete detail of the description of the GNS representation of the quantum plane \(\mathcal{A }\) and its dual \({\widehat{\mathcal{A }}}\) 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 \(\mathbf{W}_m\), we give the GNS description of the quantum double \(\mathcal{D }(\mathcal{A })\) which is equivalent to \(GL_q^+(2,\mathbb{R })\). Furthermore, we study the fundamental corepresentation \(T^{\lambda ,t}\) 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 \(\mathbf{W}_m\) and prove that the space of \(L^2\) functions on the quantum double decomposes into the continuous series representation of \(U_{q\widetilde{q}}(\mathfrak{gl }(2,\mathbb{R }))\) with the quantum dilogarithm \(|S_b(Q+2i\alpha )|^2\) as the Plancherel measure. Finally, we describe certain representation theoretic meaning of integral transforms involving the quantum dilogarithm function.