  We study the derived tensor product of the representation rings of subgroups
of a given compact Lie group G. That is, given two such subgroups H_1 and H_2,
we study the tensor product of the associated representation rings R(H_1) and
R(H_2) over the representation ring RG, and prove a vanishing result for the
associated higher Tor-groups. This result can be viewed as a natural
generalization of the Theorem of Steinberg that asserts that the representation
rings of maximal rank subgroups of G are free over RG. It my also be viewed as
an analogue of a result of Singhof on the cohomology of classifying spaces. We
include an immediate application to the complex K-theory of biquotient
manifolds.
