  The phase space given by the cotangent bundle of a Lie group appears in the
context of several models for physical systems. A representation for the
quantum system in terms of non-commutative functions on the (dual) Lie algebra,
and a generalized notion of (non-commutative) Fourier transform, different from
standard harmonic analysis, has been recently developed, and found several
applications, especially in the quantum gravity literature. We show that this
algebra representation can be defined on the sole basis of a quantization map
of the classical Poisson algebra, and identify the conditions for its
existence. In particular, the corresponding non-commutative star-product
carried by this representation is obtained directly from the quantization map
via deformation quantization. We then clarify under which conditions a unitary
intertwiner between such algebra representation and the usual group
representation can be constructed giving rise to the non-commutative plane
waves and consequently, the non-commutative Fourier transform. The compact
groups U(1) and SU(2) are considered for different choices of quantization
maps, such as the symmetric and the Duflo map, and we exhibit the corresponding
star-products, algebra representations and non-commutative plane waves.
