Higher Orbital Integrals, Cyclic Cocycles and Noncommutative Geometry
Let G be a linear real reductive Lie group. Orbital integrals define traces on the group algebra of G. We introduce a construction of higher orbital integrals in the direction of higher cyclic cocycles on the Harish-Chandra Schwartz algebra of G. We analyze these higher orbital integrals via Fourier...
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| Format: | Article |
| Language: | English |
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Cambridge University Press
2025-01-01
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| Series: | Forum of Mathematics, Sigma |
| Online Access: | https://www.cambridge.org/core/product/identifier/S2050509424001154/type/journal_article |
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| author | Yanli Song Xiang Tang |
| author_facet | Yanli Song Xiang Tang |
| author_sort | Yanli Song |
| collection | DOAJ |
| description | Let G be a linear real reductive Lie group. Orbital integrals define traces on the group algebra of G. We introduce a construction of higher orbital integrals in the direction of higher cyclic cocycles on the Harish-Chandra Schwartz algebra of G. We analyze these higher orbital integrals via Fourier transform by expressing them as integrals on the tempered dual of G. We obtain explicit formulas for the pairing between the higher orbital integrals and the K-theory of the reduced group
$C^{*}$
-algebra, and we discuss their application to K-theory. |
| format | Article |
| id | doaj-art-dda7bd1227c84892bd103906dff162e5 |
| institution | Kabale University |
| issn | 2050-5094 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | Cambridge University Press |
| record_format | Article |
| series | Forum of Mathematics, Sigma |
| spelling | doaj-art-dda7bd1227c84892bd103906dff162e52025-02-10T12:03:37ZengCambridge University PressForum of Mathematics, Sigma2050-50942025-01-011310.1017/fms.2024.115Higher Orbital Integrals, Cyclic Cocycles and Noncommutative GeometryYanli Song0https://orcid.org/0000-0003-4506-2937Xiang Tang1Department of Mathematics and Statistics, Washington University in St. Louis, St. Louis, MO, 63130, U.S.A.Department of Mathematics and Statistics, Washington University in St. Louis, St. Louis, MO, 63130, U.S.A.; E-mail:Let G be a linear real reductive Lie group. Orbital integrals define traces on the group algebra of G. We introduce a construction of higher orbital integrals in the direction of higher cyclic cocycles on the Harish-Chandra Schwartz algebra of G. We analyze these higher orbital integrals via Fourier transform by expressing them as integrals on the tempered dual of G. We obtain explicit formulas for the pairing between the higher orbital integrals and the K-theory of the reduced group $C^{*}$ -algebra, and we discuss their application to K-theory.https://www.cambridge.org/core/product/identifier/S2050509424001154/type/journal_article |
| spellingShingle | Yanli Song Xiang Tang Higher Orbital Integrals, Cyclic Cocycles and Noncommutative Geometry Forum of Mathematics, Sigma |
| title | Higher Orbital Integrals, Cyclic Cocycles and Noncommutative Geometry |
| title_full | Higher Orbital Integrals, Cyclic Cocycles and Noncommutative Geometry |
| title_fullStr | Higher Orbital Integrals, Cyclic Cocycles and Noncommutative Geometry |
| title_full_unstemmed | Higher Orbital Integrals, Cyclic Cocycles and Noncommutative Geometry |
| title_short | Higher Orbital Integrals, Cyclic Cocycles and Noncommutative Geometry |
| title_sort | higher orbital integrals cyclic cocycles and noncommutative geometry |
| url | https://www.cambridge.org/core/product/identifier/S2050509424001154/type/journal_article |
| work_keys_str_mv | AT yanlisong higherorbitalintegralscycliccocyclesandnoncommutativegeometry AT xiangtang higherorbitalintegralscycliccocyclesandnoncommutativegeometry |