Jochen Brüning - Microlocal and Global Analysis...
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Jochen Brüning - Microlocal and Global Analysis, Interactions with Geometry
Some remarks on equivariant spectral theory.
We consider a compact G-manifold, M, a G-vector bundle, E, over M, and a G-equivariant self-adjoint
elliptic operator, A, of order one acting on the smooth sections of E. We restrict the operator to the -isotypical
subspace of sections and show that its -function is meromorphic in the whole complex plane, but possibly with poles
of higher order; 0 is not a pole, though. This result is derived from the equivariant heat expansion, that in this case
contains nonlocal coecient and logarithmic terms. This approach leads to a proof of the equivariant APS-Theorem.
This is joint work with Ken Richardson.