# Elliptic complexes for bundles with infinite dimensional fibers

I am looking for differential geometric treatment of bundles over finite dimensional manifolds which have infinite dimensional fibers. It would be nice to have theory for general locally convex topological vector spaces as fibers, but I'd be happy even with separable Hilbert space. More specifically - is there a version of the theory of elliptic complexes of differential operators between sections of such bundles?

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What structure group are you working with? The general linear group of a LCTVS tends to be either contractible or not a regular Lie group so one has to place some restriction thereon. –  Loop Space Nov 13 '10 at 14:55
I am interested in bundles constructed from finite sums of (infinite dimensional) irreducible representations of simple Lie groups. –  Vít Tuček Nov 14 '10 at 12:49
Check out the development of the theory in e.g. Gilkey's book and try to skip the condition that the vector bundles are finite-dimensional. I expect that symbol calculus, Sobolev embedding theorem, invertibility modulo smoothing operators work for Hilbert bundles in the same way. When it comes to the Fredholm property, then the theory suffers a serious breakdown, though. The reason is that the Rellich lemma is false for Sobolev spaces of functions with values in infinite-dimensional spaces, due to the failure of local compactness of infinite-dimensional Hilbert spaces. –  Johannes Ebert Nov 14 '10 at 14:03
There is however a shadow of the Fredholm theory for these operators in the context of $C^*$-algebras (I mention Miscenko, Fomenko), but whether this is useful for you depends very much on the specific problem you are interested in. –  Johannes Ebert Nov 14 '10 at 14:10