Аннотация:
In this talk I shall explain how cyclic cohomology and K-theory can be used in order to define and investigate primary and secondary invariants of G-invariant Dirac operators on a cocompact G-proper manifold, with G a connected real reductive Lie group .
I will first treat the case of cyclic cocycles associated to elements in the differentiable cohomology of $G$; I will then move to delocalized cyclic cocycles.
I will explain the challenges in defining the delocalized eta invariant associated to the orbital integral defined by a semisimple element g in G and in showing that such an invariant enters in an Atiyah-Patodi-Singer index theorem for cocompact G-proper manifolds. I will then move to a higher version of these results, using the higher delocalized cyclic cocycles defined by Song and Tang.
This talk is based on two articles with Hessel Posthuma and a recent article with Hessel Posthuma, Yanli Song and Xiang Tang.
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