Mackey showed that for a compact Lie group K, the pair (K, C 0 (K)) has a unique nontrivial irreducible covariant pair of representations. We study the relevance of this result to the unitary equivalence of quantizations for an infinite-dimensional family of K × K invariant polarizations on T * K. The Kähler polarizations in the family are generated by (complex) time-τ Hamiltonian flows applied to the (Schrödinger) vertical real polarization. The unitary equivalence of the corresponding quantizations of T * K is then studied by considering covariant pairs of representations of K defined by geometric prequantization and of representations of C 0 (K) defined via Heisenberg time-(−τ ) evolution followed by time-(+τ ) geometric-quantization-induced evolution. We show that in the semiclassical and large imaginary time limits, the unitary transform whose existence is guaranteed by Mackey's theorem can be approximated by composition of the time-(+τ ) geometric-quantization-induced evolution with the time-(−τ ) evolution associated with the momentum space [17] quantization of the Hamiltonian function generating the flow. In the case of quadratic Hamiltonians, this asymptotic result is exact and unitary equivalence between quantizations is achieved by identifying the Heisenberg imaginary time evolution with heat operator evolution, in accordance with the coherent state transform of Hall.Geometric quantization has proven to be a very rich approach to the general mathematical problem of the quantization of a symplectic manifold (M, ω). In order to half-form quantize (M, ω), one needs to choose a polarization P, that is an involutive Lagrangian distribution in the complexified tangent bundle T M ⊗ C. The half-form quantization H P of (M, ω, P) is then the L 2 -closure of the space of square-integrable smooth sections of the quantum bundle L ⊗ √ K which are covariantly constant along P, where √ K is a choice of square root of the canonical bundle n P * of P and L is a Hermitian line bundle with compatible connection with curvature −iω. (Of course, in general there are topological obstructions to the existence of √ K and L, but they will play no role in this paper.) A major, perhaps even the fundamental, issue in geometric quantization is the dependence of quantization on the choice of P. In favorable cases, given two polarizations P, P ′ of (M, ω), one would like to have a natural unitary isomorphism between the corresponding Hilbert spaces of quantum states H P , H P ′ , at least up to a projective ambiguity. Moreover, this unitary isomorphism should intertwine actions of sufficiently big algebras of observables on H P and H P ′ .In the paradigmatic case when (M, ω) is a symplectic vector space and when one considers translation invariant polarizations, such an isomorphism is guaranteed to exist by the Stone-Von Neumann theorem, which gives uniqueness of the irreducible unitary representation of the Heisenberg group [1,28,18]. The linear observables, which generate translations and which, together with the constants, span th...