COMPARISON OF HOMOLOGIES AND AUTOMATIC EXTENSIONS OF INVARIANT DISTRIBUTIONS∗

doi:10.1007/s10473-023-0407-x

数学物理学报(英文版) ›› 2023, Vol. 43 ›› Issue (4): 1561-1570.doi: 10.1007/s10473-023-0407-x

COMPARISON OF HOMOLOGIES AND AUTOMATIC EXTENSIONS OF INVARIANT DISTRIBUTIONS^∗

Yangyang CHEN

School of Sciences, Jiangnan University, Wuxi 214122, China

收稿日期:2022-03-18 修回日期:2022-07-12 发布日期:2023-08-08
作者简介:Yangyang CHEN,E-mail: chenyy@amss.ac.cn
基金资助:
*This work was supported by the Fundamental Research

COMPARISON OF HOMOLOGIES AND AUTOMATIC EXTENSIONS OF INVARIANT DISTRIBUTIONS^∗

Yangyang CHEN

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

摘要/Abstract

摘要： 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.

关键词: Schwartz functions, Lie algebra homology, Hausdorffness, Schwartz distributions, automatic extensions

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

Yangyang CHEN. COMPARISON OF HOMOLOGIES AND AUTOMATIC EXTENSIONS OF INVARIANT DISTRIBUTIONS^∗[J]. 数学物理学报(英文版), 2023, 43(4): 1561-1570.

Yangyang CHEN. COMPARISON OF HOMOLOGIES AND AUTOMATIC EXTENSIONS OF INVARIANT DISTRIBUTIONS^∗[J]. Acta mathematica scientia,Series B, 2023, 43(4): 1561-1570.

参考文献

[1] Aizenbud A, Gourevitch D. Schwartz functions on Nash manifolds. Int Math Res Not, 2008, 2008(5): Art rnm155
[2] Aizenbud A, Gourevitch D. Smooth transfer of Kloostermann integrals (the Archimedean case). American Journal of Mathematics, 2013, 135: 143-182
[3] Aizenbud A, Gourevitch D. The de Rham theorem and Shapiro lemma for Schwartz functions on Nash manifolds. Israel J Math, 2010, 177: 155-188
[4] Aizenbud A, Gourevitch D, Krötz B, Liu G. Hausdorffness for Lie algebra homology of Schwartz spaces and applications to the comparison conjecture. Math Zeit, 2016, 283: 979-992
[5] Aizenbud A, Gourevitch D, Krötz B, Liu G. Erratum to: Hausdorffness for Lie algebra homology of Schwartz spaces and applications to the comparison conjecture. Math Zeit, 2016, 283: 993-994
[6] Bochnak J, Coste M, Roy M F. Real Algebraic Geometry. Berlin: Springer, 1998
[7] Borel A, Wallach N. Continuous cohomology, discrete subgroups,representations of reductive groups. Annals of Mathematics Studies, 1980: No 94
[8] Casselman W. Canonical extensions of Harish-Chandra modules to representations of $G$. Canad J Math, 1989, 41: 385-438
[9] Casselman W, Hecht H, Miliɩiɣ D. Bruhat filtrations and Whittaker vectors for real groups//Doran R, Varadarajan V. Proceedings of Symposia in Pure Mathematics. Providence, RI: Amer Math Soc, 2000: 151-191
[10] Chen Y, Sun B. Schwartz homologies of representations of almost linear Nash groups. Journal of Functional Analysis, 2021, 280(7): 108817
[11] du Cloux F. Sur les représentations différentiables des groupes de Lie algébriques. Ann Sci Ecole Norm Sup,1991, 24(3): 257-318
[12] Hong J, Sun B. Generalized semi-invariant distributions on $p$-adic spaces. Math Ann, 2017, 367: 1727-1776
[13] Knapp A, Vogan D.Cohomological Induction and Unitary Representations. Princeton: Princeton University Press, 1995
[14] Shiota M. Nash Manifolds.Berlin: Springer-Verlag, 1987
[15] Shiota M.Nash Functions and Manifolds//Broglia F. Lectures in Real Geometry. Berlin: de Gruyter, 1996
[16] Sun B. Almost linear Nash groups. China Ann Math Ser B, 2015, 36: 355-400
[17] Sun B, Zhu C B. A general form of Gelfand-Kazhdan criterion. Manu Math, 2011, 136: 185-197
[18] Taylor J L.Notes on locally convex topological vector spaces. Lecture Notes, University of Utah, 1995
[19] Trèves F. Topological Vector Spaces, Distributions and Kernels. New York: Academic Press, 1967

COMPARISON OF HOMOLOGIES AND AUTOMATIC EXTENSIONS OF INVARIANT DISTRIBUTIONS^∗

COMPARISON OF HOMOLOGIES AND AUTOMATIC EXTENSIONS OF INVARIANT DISTRIBUTIONS^∗

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 0

Metrics

本文评价

推荐阅读 10