Pseudo-differential Operators, Wigner Transforms and Weyl Transforms on the Poincaré Unit Disk.

Using the affine group and the Cayley transform from the unit disk D<inline-graphic></inline-graphic> onto the upper half plane, we can turn D<inline-graphic></inline-graphic> into a group, which we call the Poincaré unit disk. With this construction, D<inline-graphic></inline-graphic> is a noncompact and nonunimodular Lie group. We characterize all infinite-dimensional, irreducible and unitary representations of D<inline-graphic></inline-graphic>. By means of these representations, the Fourier transform on D<inline-graphic></inline-graphic> is defined. The Plancherel theorem and hence the Fourier inversion formula can be given. Then pseudo-differential operators with operator-valued symbols, operator-valued Wigner transforms, and Weyl transforms on D<inline-graphic></inline-graphic> are defined. [ABSTRACT FROM AUTHOR]