Induced representations of the group SL2(R) and hypercomplex numbers

We review the construction of induced representations of the group G = SL2(R). Firstly we note that G-action on the homogeneous space G/H, where H is any onedimensional subgroup of SL2(R), is a linear-fractional transformation on hypercomplex numbers. This observation can be extended to further correspondences between structural components of SL2(R) and hypercomplex systems. Thus we investigate various hypercomplex characters of subgroups H. In particular we give examples of induced representations of SL2(R) on spaces of hypercomplex valued functions, which are unitary in some sense.