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Proper actions of Lie groups and numeric invariants of Dirac operators

By Paolo Piazza

Appears in collection : Geometry and analysis on non-compact manifolds / Géométrie et analyse sur les variétés non compactes

I shall explain how 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. This involves cyclic cohomology and Ktheory. After treating the case of cyclic cocycles associated to elements in the differentiable cohomology of G I will move to delocalized cyclic cocycles, in particular, 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 consider a higher version of these results, based on the Song-Tang higher orbital integrals associated to a cuspidal parabolic subgroup P¡G with Langlands decomposition P=MAN and a semisimple element g in M. This talk is based on articles with Hessel Posthuma and with Hessel Postrhuma, Yanli Song and Xiang Tang.

Information about the video

Citation data

  • DOI 10.24350/CIRM.V.19902203
  • Cite this video Piazza Paolo (3/31/22). Proper actions of Lie groups and numeric invariants of Dirac operators. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.19902203
  • URL https://dx.doi.org/10.24350/CIRM.V.19902203

Bibliography

  • PIAZZA, Paolo et POSTHUMA, Hessel B. Higher genera for proper actions of Lie groups. Annals of K-Theory, 2019, vol. 4, no 3, p. 473-504. - https://doi.org/10.2140/akt.2019.4.473
  • PIAZZA, Paolo et POSTHUMA, Hessel B. Higher genera for proper actions of Lie groups, II: The case of manifolds with boundary. Annals of K-Theory, 2022, vol. 6, no 4, p. 713-782. - https://doi.org/10.2140/akt.2021.6.713
  • PIAZZA, Paolo, POSTHUMA, Hessel, SONG, Yanli, et al. Higher orbital integrals, rho numbers and index theory. arXiv preprint arXiv:2108.00982, 2021. - https://arxiv.org/abs/2108.00982

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