In the special algebro-geometric case, where Y is the dual unit circle bundle of a positive line bundle on a complex projective manifold, H(Y) is the orthogonal direct sum of its isotypical components [H.sub.k](Y), k = 0,1,2, ..., under the [S.sup.1]-action; correspondingly, [PI] = [[direct sum].sub.k][[PI].sub.k], where [[PI].sub.k] is the orthogonal projector onto [H.sub.k](Y); since [H.sub.k](Y) is finite-dimensional, [[PI].sub.k] is a smoothing operator; therefore, its Schwartz kernel is a [C.sup.[infinity]] function on Y x Y.
let [H.sub.k]([X.sub.r]) [subset] H([X.sub.r]) be the finite-dimensional kth isotypical component of H([X.sub.r]) with respect to the standard [S.sup.1]-action.
The homomorphism [alpha] induces the structure of a G representation to H denoted by [H.sub.[alpha]] and we have a canonical decomposition of [H.sub.[alpha]] in isotypical components
where Irrep(G) denotes the isomorphism classes of irreducible representations of G, V is a representative of its isomorphism class of irreducible representation and [H.sup.V.sub.[alpha]] is the isotypical subspace associated to V.
We say that the homomorphism is stable if all the isotypical components of [H.sub.[alpha]] are either infinite dimensional or zero dimensional.
Take the isotypical decomposition of H [congruent to] [[direct sum].sub.W[member of]C]s [H.sup.W.sub.[alpha]] defined by [alpha].
Note that the homomorphism [PHI](Z) only disagrees with a on the subspace [mathematical expression not reproducible], that the isotypical subspace of the homomorphism [PHI](Z) associated to V is precisely ev(V [cross product] Z), i.e.
which implies that [sigma]([PHI](Z)) induces a G-equivariant unitary isomorphism between the isotypical components
is injective since for abelian groups the isotypical spaces determine the homomorphism.
Take the isotypical part corresponding to the irreducible representations in S ([??])
The homomorphisms [mathematical expression not reproducible], induce decompositions of H by isotypical components [mathematical expression not reproducible].