An extension of Steinberg’s theorem to biquotient pairs of subgroups

Abstract 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 upper H 1 H 1 $H_1$ and upper H 2 H 2 $H_2$ , we study the tensor product of the associated representation rings normal upper R upper H 1 R H 1 $\mathrm{R}H_1$ and normal upper R upper H 2 R H 2 $\mathrm{R}H_2$ over the representation ring normal upper R upper G R G $\mathrm{R}G$ , and prove a vanishing result for the associated higher Tor groups. This result can be viewed as a natural extension of the Theorem of Steinberg that asserts that the representation rings of maximal rank subgroups of G are free over normal upper R upper G R G $\mathrm{R}G$ . It may 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.