  The non-equivariant topology of Stiefel manifolds has been studied
extensively, culminating in a result of Miller demonstrating that a Stiefel
manifold splits stably to a wedge of Thom spaces over Grassmannians.
Equivariantly, one can consider spaces of isometries between representations as
an analogue to Stiefel manifolds. This concept, however, yields a different
theory to the non-equivariant case. We first construct a variation on the
theory of the functional calculus before studying the homotopy-theoretic
properties of this theory. This allows us to construct the main result; a
natural tower of G-spectra running down from equivariant isometries which
manifests the pieces of the non-equivariant splitting in the form of the
homotopy cofibres of the tower. Furthermore, we detail extra topological
properties and special cases of this theory, developing explicit expressions
covering the tower's geometric and topological structure. We conclude with two
detailed conjectures which provide an avenue for future study. Firstly we
explore how our theory interacts with the splitting of Miller, proving partial
results linking in our work with Miller's and conjecturing even deeper
connections. Finally, we begin to calculate the equivariant K-theory of the
tower, conjecturing and providing evidence towards the idea that the rich
topological structure will be mirrored on the K-theory level by a similarly
deep algebraic structure.
