Comparison of Homologies and Automatic Extensions of Invariant Distributions
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 χ 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...
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| Published in | Acta mathematica scientia Vol. 43; no. 4; pp. 1561 - 1570 |
|---|---|
| Main Author | |
| Format | Journal Article |
| Language | English |
| Published |
Singapore
Springer Nature Singapore
01.07.2023
School of Sciences,Jiangnan University,Wuxi 214122,China |
| Subjects | |
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| 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
χ
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
)
⊗
χ
)
→
H
k
(
g
,
S
(
X
)
⊗
χ
)
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 -
χ
) Schwartz distributions. |
|---|---|
| AbstractList | 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
χ
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
)
⊗
χ
)
→
H
k
(
g
,
S
(
X
)
⊗
χ
)
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 -
χ
) Schwartz distributions. 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 x 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 Hk(g,S(U)(⊕)x)→ Hk(g,S(X)(⊕)x)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-x)Schwartz distributions. |
| Author | Chen, Yangyang |
| AuthorAffiliation | School of Sciences,Jiangnan University,Wuxi 214122,China |
| AuthorAffiliation_xml | – name: School of Sciences,Jiangnan University,Wuxi 214122,China |
| Author_xml | – sequence: 1 givenname: Yangyang surname: Chen fullname: Chen, Yangyang email: chenyy@amss.ac.cn organization: School of Sciences, Jiangnan University |
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| Cites_doi | 10.1007/BFb0078571 10.1007/s11856-010-0042-9 10.1007/s00209-016-1699-5 10.4153/CJM-1989-019-5 10.1007/s00208-016-1444-8 10.1007/s00229-011-0437-x 10.1016/j.jfa.2020.108817 10.1007/s11401-015-0915-7 10.1353/ajm.2013.0000 10.1093/imrn/rnm155 10.24033/asens.1628 10.1007/978-3-662-03718-8 10.1515/9781400883936 10.1007/s00209-016-1629-6 |
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| Copyright | Innovation Academy for Precision Measurement Science and Technology, Chinese Academy of Sciences 2023 Copyright © Wanfang Data Co. Ltd. All Rights Reserved. |
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| Keywords | Schwartz distributions Hausdorffness automatic extensions Schwartz functions 46T30 22E20 Lie algebra homology Schwartz distribu-tions |
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| References | Aizenbud A, Gourevitch D. Schwartz functions on Nash manifolds. Int Math Res Not, 2008, 2008 (5): Art rnm155 KnappAVoganDCohomological Induction and Unitary Representations1995PrincetonPrinceton University Press10.1515/97814008839360863.22011 ChenYSunBSchwartz homologies of representations of almost linear Nash groupsJournal of Functional Analysis20212807108817421101810.1016/j.jfa.2020.1088171478.22008 Borel A, Wallach N. Continuous cohomology, discrete subgroups, and representations of reductive groups. Annals of Mathematics Studies, 1980: No 94 CasselmanWHechtHMiličićDDoranRVaradarajanVBruhat filtrations and Whittaker vectors for real groupsProceedings of Symposia in Pure Mathematics2000Providence, RIAmer Math Soc151191 ShiotaMNash Manifolds1987BerlinSpringer-Verlag10.1007/BFb00785710629.58002 ShiotaMNash Functions and Manifolds//Broglia F. Lectures in Real Geometry1996Berlinde Gruyter AizenbudAGourevitchDSmooth transfer of Kloostermann integrals (the Archimedean case)American Journal of Mathematics2013135143182302296110.1353/ajm.2013.00001329.11049 HongJSunBGeneralized semi-invariant distributions on p-adic spacesMath Ann201736717271776362323610.1007/s00208-016-1444-81362.22012 AizenbudAGourevitchDThe de Rham theorem and Shapiro lemma for Schwartz functions on Nash manifoldsIsrael J Math2010177155188268441710.1007/s11856-010-0042-91208.58002 AizenbudAGourevitchDKrötzBLiuGErratum to: Hausdorffness for Lie algebra homology of Schwartz spaces and applications to the comparison conjectureMath Zeit201628399399410.1007/s00209-016-1699-51404.22028 TrèvesFTopological Vector Spaces, Distributions and Kernels1967New YorkAcademic Press0171.10402 SunBAlmost linear Nash groupsChina Ann Math Ser B201536355400334116410.1007/s11401-015-0915-71322.22009 BochnakJCosteMRoyM FReal Algebraic Geometry1998BerlinSpringer10.1007/978-3-662-03718-80912.14023 CasselmanWCanonical extensions of Harish-Chandra modules to representations of GCanad J Math198941385438101346210.4153/CJM-1989-019-50702.22016 Taylor J L. Notes on locally convex topological vector spaces. Lecture Notes, University of Utah, 1995 du ClouxFSur les représentations différentiables des groupes de Lie algébriquesAnn Sci Ecole Norm Sup1991243257318110099210.24033/asens.16280748.22006 SunBZhuC BA general form of Gelfand-Kazhdan criterionManu Math2011136185197282040110.1007/s00229-011-0437-x1229.22014 AizenbudAGourevitchDKrötzBLiuGHausdorffness for Lie algebra homology of Schwartz spaces and applications to the comparison conjectureMath Zeit2016283979992351999010.1007/s00209-016-1629-61347.22010 407_CR1 W Casselman (407_CR9) 2000 B Sun (407_CR17) 2011; 136 A Aizenbud (407_CR4) 2016; 283 A Knapp (407_CR13) 1995 407_CR18 B Sun (407_CR16) 2015; 36 Y Chen (407_CR10) 2021; 280 A Aizenbud (407_CR3) 2010; 177 W Casselman (407_CR8) 1989; 41 M Shiota (407_CR15) 1996 A Aizenbud (407_CR5) 2016; 283 M Shiota (407_CR14) 1987 J Hong (407_CR12) 2017; 367 J Bochnak (407_CR6) 1998 407_CR7 F Trèves (407_CR19) 1967 A Aizenbud (407_CR2) 2013; 135 F du Cloux (407_CR11) 1991; 24 |
| References_xml | – reference: du ClouxFSur les représentations différentiables des groupes de Lie algébriquesAnn Sci Ecole Norm Sup1991243257318110099210.24033/asens.16280748.22006 – reference: ShiotaMNash Functions and Manifolds//Broglia F. Lectures in Real Geometry1996Berlinde Gruyter – reference: AizenbudAGourevitchDKrötzBLiuGErratum to: Hausdorffness for Lie algebra homology of Schwartz spaces and applications to the comparison conjectureMath Zeit201628399399410.1007/s00209-016-1699-51404.22028 – reference: CasselmanWHechtHMiličićDDoranRVaradarajanVBruhat filtrations and Whittaker vectors for real groupsProceedings of Symposia in Pure Mathematics2000Providence, RIAmer Math Soc151191 – reference: BochnakJCosteMRoyM FReal Algebraic Geometry1998BerlinSpringer10.1007/978-3-662-03718-80912.14023 – reference: ShiotaMNash Manifolds1987BerlinSpringer-Verlag10.1007/BFb00785710629.58002 – reference: KnappAVoganDCohomological Induction and Unitary Representations1995PrincetonPrinceton University Press10.1515/97814008839360863.22011 – reference: TrèvesFTopological Vector Spaces, Distributions and Kernels1967New YorkAcademic Press0171.10402 – reference: AizenbudAGourevitchDSmooth transfer of Kloostermann integrals (the Archimedean case)American Journal of Mathematics2013135143182302296110.1353/ajm.2013.00001329.11049 – reference: AizenbudAGourevitchDThe de Rham theorem and Shapiro lemma for Schwartz functions on Nash manifoldsIsrael J Math2010177155188268441710.1007/s11856-010-0042-91208.58002 – reference: SunBZhuC BA general form of Gelfand-Kazhdan criterionManu Math2011136185197282040110.1007/s00229-011-0437-x1229.22014 – reference: SunBAlmost linear Nash groupsChina Ann Math Ser B201536355400334116410.1007/s11401-015-0915-71322.22009 – reference: Aizenbud A, Gourevitch D. Schwartz functions on Nash manifolds. Int Math Res Not, 2008, 2008 (5): Art rnm155 – reference: Borel A, Wallach N. Continuous cohomology, discrete subgroups, and representations of reductive groups. Annals of Mathematics Studies, 1980: No 94 – reference: CasselmanWCanonical extensions of Harish-Chandra modules to representations of GCanad J Math198941385438101346210.4153/CJM-1989-019-50702.22016 – reference: AizenbudAGourevitchDKrötzBLiuGHausdorffness for Lie algebra homology of Schwartz spaces and applications to the comparison conjectureMath Zeit2016283979992351999010.1007/s00209-016-1629-61347.22010 – reference: HongJSunBGeneralized semi-invariant distributions on p-adic spacesMath Ann201736717271776362323610.1007/s00208-016-1444-81362.22012 – reference: ChenYSunBSchwartz homologies of representations of almost linear Nash groupsJournal of Functional Analysis20212807108817421101810.1016/j.jfa.2020.1088171478.22008 – reference: Taylor J L. Notes on locally convex topological vector spaces. Lecture Notes, University of Utah, 1995 – volume-title: Nash Manifolds year: 1987 ident: 407_CR14 doi: 10.1007/BFb0078571 – start-page: 151 volume-title: Proceedings of Symposia in Pure Mathematics year: 2000 ident: 407_CR9 – volume: 177 start-page: 155 year: 2010 ident: 407_CR3 publication-title: Israel J Math doi: 10.1007/s11856-010-0042-9 – volume: 283 start-page: 993 year: 2016 ident: 407_CR5 publication-title: Math Zeit doi: 10.1007/s00209-016-1699-5 – volume: 41 start-page: 385 year: 1989 ident: 407_CR8 publication-title: Canad J Math doi: 10.4153/CJM-1989-019-5 – volume: 367 start-page: 1727 year: 2017 ident: 407_CR12 publication-title: Math Ann doi: 10.1007/s00208-016-1444-8 – volume: 136 start-page: 185 year: 2011 ident: 407_CR17 publication-title: Manu Math doi: 10.1007/s00229-011-0437-x – volume-title: Topological Vector Spaces, Distributions and Kernels year: 1967 ident: 407_CR19 – volume: 280 start-page: 108817 issue: 7 year: 2021 ident: 407_CR10 publication-title: Journal of Functional Analysis doi: 10.1016/j.jfa.2020.108817 – volume: 36 start-page: 355 year: 2015 ident: 407_CR16 publication-title: China Ann Math Ser B doi: 10.1007/s11401-015-0915-7 – volume-title: Nash Functions and Manifolds//Broglia F. Lectures in Real Geometry year: 1996 ident: 407_CR15 – volume: 135 start-page: 143 year: 2013 ident: 407_CR2 publication-title: American Journal of Mathematics doi: 10.1353/ajm.2013.0000 – ident: 407_CR18 – ident: 407_CR1 doi: 10.1093/imrn/rnm155 – volume: 24 start-page: 257 issue: 3 year: 1991 ident: 407_CR11 publication-title: Ann Sci Ecole Norm Sup doi: 10.24033/asens.1628 – ident: 407_CR7 – volume-title: Real Algebraic Geometry year: 1998 ident: 407_CR6 doi: 10.1007/978-3-662-03718-8 – volume-title: Cohomological Induction and Unitary Representations year: 1995 ident: 407_CR13 doi: 10.1515/9781400883936 – volume: 283 start-page: 979 year: 2016 ident: 407_CR4 publication-title: Math Zeit doi: 10.1007/s00209-016-1629-6 |
| SSID | ssj0016264 |
| Score | 2.278339 |
| Snippet | 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... 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 x be... |
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| SubjectTerms | Analysis Mathematics Mathematics and Statistics |
| Title | Comparison of Homologies and Automatic Extensions of Invariant Distributions |
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