Vladimir KisilAMJAD SALEH MUHAMMED ALGHAMDI2022-06-042022-05-242022-06-04111103https://drepo.sdl.edu.sa/handle/20.500.14154/66410This thesis is concerned with the representation theory for the special linear Lie group SL_2(R). After the introduction and background material in the second chapter, we present the representation theory of the affine group, which is a subgroup of SL_2(R). The classification of the irreducible unitary representation of SL_2(R) was achieved by Bargmann [4]. Here, a new study of the SL_2(R) unitary representation is provided by using an inducing procedure and Gelfand method [13,VII]. Also, we investigate the representation of the group SL_2(R) on holomorphic spaces of function on the unit disc.105enThesis