The Wigner–Weyl isomorphism for quantum mechanics on a compact simple Lie
group G is developed in detail. Several features are shown to arise which have no
counterparts in the familiar Cartesian case. Notable among these is the notion of a
semiquantized phase space, a structure on which the Weyl symbols of operators
turn out to be naturally defined and, figuratively speaking, located midway between
the classical phase space T*G and the Hilbert space of square integrable functions
on G. General expressions for the star product for Weyl symbols are presented and
explicitly worked out for the angle-angular momentum case.