  Let $\G$ be a locally compact group satisfying some technical requirements
and $\wG$ its unitary dual. Using the theory of twisted crossed product
$C^*$-algebras, we develop a twisted global quantization for symbols defined on
$\G\times\wG$ and taking operator values. The emphasis is on the
representation-theoretic aspect. For nilpotent Lie groups, the connection is
made with a scalar quantization of the cotangent bundle $T^*(\G)$ and with a
Quantum Mechanical theory of observables in the presence of variable magnetic
fields.
