  How far can the elementary description of centralizers of parabolic
subalgebras of Hecke algebras of finite real reflection groups be generalized
to the complex reflection group case? In this paper we begin to answer this
question by establishing results in two directions. First, under conditions
closely analogous to those existing for the real case, we give explicit
relations between coefficients in an element centralizing a generator. Second,
we introduce a tool for dealing with a major challenge of the complex case --
the ``instability'' of certain double cosets -- through the definition and use
of a double coset graph. We use these results to find integral bases for the
centralizers of generators as well as the centres of the Hecke algebras of
types $G_4$ and $G(4,1,2)$.
  Keywords: complex reflection group; Hecke algebra; centre; centralizer;
modular; double coset.
