Acta mathematica scientia,Series B ›› 2023, Vol. 43 ›› Issue (4): 1561-1570.doi: 10.1007/s10473-023-0407-x

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COMPARISON OF HOMOLOGIES AND AUTOMATIC EXTENSIONS OF INVARIANT DISTRIBUTIONS

Yangyang CHEN   

  1. School of Sciences, Jiangnan University, Wuxi 214122, China
  • Received:2022-03-18 Revised:2022-07-12 Published:2023-08-08
  • About author:Yangyang CHEN,E-mail: chenyy@amss.ac.cn
  • Supported by:
    *This work was supported by the Fundamental Research

Abstract: Let G be a reductive Nash group, acting on a Nash manifold X. Let Z be a G -stable closed Nash submanifold of X and denote by $U$ the complement of Z in X. Let $\chi$ be a character of G and denote by g the complexified Lie algebra of G. We give a sufficient condition for the natural linear map $H_{k}(g, S(U)\otimes\chi)\rightarrow H_{k}(g, S(X)\otimes\chi)$ between the Lie algebra homologies of Schwartz functions to be an isomorphism. For k=0, by considering the dual, we obtain the automatic extensions of $g$-invariant (twisted by -$\chi$) Schwartz distributions.

Key words: Schwartz functions, Lie algebra homology, Hausdorffness, Schwartz distributions, automatic extensions

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