We study the prime ideal spaces of the quantized function algebras
R
q
[
G
]
R_{q}[G]
, for
G
G
a semisimple Lie group and
q
q
an indeterminate. Our method is to examine the structure of algebras satisfying a set of seven hypotheses, and then to demonstrate, using work of Joseph, Hodges and Levasseur, that the algebras
R
q
[
G
]
R_{q}[G]
satisfy this list of assumptions. Rings satisfying the assumptions are shown to satisfy normal separation, and therefore Jategaonkar’s strong second layer condition. For such rings much representation-theoretic information is carried by the graph of links of the prime spectrum, and so we proceed to a detailed study of the prime links of algebras satisfying the list of assumptions. Homogeneity is a key feature – it is proved that the clique of any prime ideal coincides with its orbit under a finite rank free abelian group of automorphisms. Bounds on the ranks of these groups are obtained in the case of
R
q
[
G
]
R_{q}[G]
. In the final section the results are specialized to the case
G
=
S
L
n
(
C
)
G= SL_{n}(\mathbb {C})
, where detailed calculations can be used to illustrate the general results. As a preliminary set of examples we show also that the multiparameter quantum coordinate rings of affine
n
n
-space satisfy our axiom scheme when the group generated by the parameters is torsionfree.