  In the search for hypercomplex analytic functions on the half-plane, we
review the construction of induced representations of the group G=SL(2,R).
Firstly we note that G-action on the homogeneous space G/H, where H is any
one-dimensional subgroup of SL(2,R), is a linear-fractional transformation on
hypercomplex numbers. Thus we investigate various hypercomplex characters of
subgroups H. The correspondence between the structure of the group SL(2,R) and
hypercomplex numbers can be illustrated in many other situations as well. We
give examples of induced representations of SL(2,R) on spaces of hypercomplex
valued functions, which are unitary in some sense. Raising/lowering operators
for various subgroup prompt hypercomplex coefficients as well.
  The paper contains both English and Russian versions.
  Keywords: induced representation, unitary representations, SL(2,R),
semisimple Lie group, complex numbers, dual numbers, double numbers, Moebius
transformations, split-complex numbers, parabolic numbers, hyperbolic numbers,
raising/lowering operators, creation/annihilation operators
