Characters on a homogeneous space

We will define the notion of cohomological triviality of
fibrations and show that for a fibration of homogeneous spaces of a
reductive linear algebraic group  is cohomologically trivial when the base
is $\mathbb{C}^*$ with kernel of the associated character $S$ is
connected. We will use the method of the proof  to obtain a classification
of quasi-reductive groups (algebraic groups whose unipotent radical is
trivial) up to isogeny and a similar result of cohomological triviality
can be obtained for homogeneous spaces of connected algebraic groups. We
will be working over the field of complex numbers and singular cohomology
with rational coefficients. This is talk is based on a joint work with
Prof. A. J. Parameswaran.