Decomposition of the unitary representation of SU(1,1) on the unit disk into irreducible components

In this thesis, we decompose the representation of SU(1,1) on the unit disk into ir reducible components. We start with the decomposition over the maximal compact subgroup K, we identify the modules of eigenfunctions which are square integrable with respect to the quasi invariant measure on the unit disk. These modules rep resent the discrete series representations. Then, we use the induction in stages method to find the principal series representation. The matrix coefficient with the principal series and a K-invariant vector turns to be an important function which is called a spherical function. There is a nice function (Harish Chandra’s function) controlling the decay of the spherical function at infinity. Finally, we use a new approach to find the inversion formula which is equivalent to decomposition into irreducible representations using the geometry of cycles with dual numbers and the covariant transform.