  Let G be a compact Lie group. We build a tower of G-spectra over the
suspension spectrum of the space of linear isometries from one G-representation
to another. The stable cofibres of the maps running down the tower are certain
interesting Thom spaces. We conjecture that this tower provides an equivariant
extension of Miller's stable splitting of Stiefel manifolds. We provide a
cohomological obstruction to the tower producing a splitting in most cases;
however, this obstruction does not rule out a split tower in the case where the
Miller splitting is possible. We claim that in this case we have a split tower
which would then produce an equivariant version of the Miller splitting, we
prove this claim in certain special cases though the general case remains a
conjecture. To achieve these results we construct a variation of the functional
calculus with useful homotopy-theoretic properties and explore the geometric
links between certain equivariant Gysin maps and residue theory.
