Solving the index problem for (curved) Bernstein-Gelfand-Gelfand sequences
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916282638532608 |
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| author | Goffeng, Magnus |
| author_facet | Goffeng, Magnus |
| contents | We study the index theory of curved Bernstein-Gelfand-Gelfand (BGG) sequences in parabolic geometry and their role in $K$-homology and noncommutative geometry. The BGG-sequences fit into $K$-homology, and we solve their index problem. We provide a condition for when the BGG-complex on the flat parabolic geometry $G/P$ of a semisimple Lie group $G$ fits into $G$-equivariant $K$-homology by means of Heisenberg calculus. For higher rank Lie groups, we prove a no-go theorem showing that the approach fails. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_07033 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Solving the index problem for (curved) Bernstein-Gelfand-Gelfand sequences Goffeng, Magnus K-Theory and Homology Differential Geometry Operator Algebras Representation Theory We study the index theory of curved Bernstein-Gelfand-Gelfand (BGG) sequences in parabolic geometry and their role in $K$-homology and noncommutative geometry. The BGG-sequences fit into $K$-homology, and we solve their index problem. We provide a condition for when the BGG-complex on the flat parabolic geometry $G/P$ of a semisimple Lie group $G$ fits into $G$-equivariant $K$-homology by means of Heisenberg calculus. For higher rank Lie groups, we prove a no-go theorem showing that the approach fails. |
| title | Solving the index problem for (curved) Bernstein-Gelfand-Gelfand sequences |
| topic | K-Theory and Homology Differential Geometry Operator Algebras Representation Theory |
| url | https://arxiv.org/abs/2406.07033 |