Character formul{\ae} and GKRS multiplets in equivariant K-theory

Let $G$ be a compact Lie group, $H$ a closed subgroup of maximal rank and $X$ a topological $G$-space. We obtain a variety of results concerning the structure of the $H$-equivariant K-ring $K_H^{\ast }(X)$ viewed as a module over the $G$-equivariant K-ring $K_G^{\ast }(X)$. One result is that the module has a nonsingular bilinear pairing; another is that the module contains multiplets which are analogous to the Gross-Kostant-Ramond-Sternberg multiplets of representation theory.