Index theory of hypoelliptic operators on Carnot manifolds
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866911831302340608 |
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| author | Goffeng, Magnus Kuzmin, Alexey |
| author_facet | Goffeng, Magnus Kuzmin, Alexey |
| contents | We study the index theory of hypoelliptic operators on Carnot manifolds -- manifolds whose Lie algebra of vector fields is equipped with a filtration induced from sub-bundles of the tangent bundle. A Heisenberg pseudodifferential operator, elliptic in the calculus of van Erp-Yuncken, is hypoelliptic and Fredholm. Under some geometric conditions, we compute its Fredholm index by means of operator $K$-theory. These results extend the work of Baum-van Erp (Acta Mathematica '2014) for co-oriented contact manifolds to a methodology for solving this index problem geometrically on Carnot manifolds.
Under the assumption that the Carnot manifold is regular, i.e. has isomorphic osculating Lie algebras in all fibres, and admits a flat coadjoint orbit, the methodology derived from Baum-van Erp's work is developed in full detail. In this case, we develope $K$-theoretical dualities computing the Fredholm index by means of geometric $K$-homology a la Baum-Douglas. The duality involves a Hilbert space bundle of flat orbit representations. Explicit solutions to the index problem for Toeplitz operators and operators of the form "$Δ_H+γT$" are computed in geometric $K$-homology, extending results of Boutet de Monvel and Baum-van Erp, respectively, from co-oriented contact manifolds to regular polycontact manifolds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2203_04717 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Index theory of hypoelliptic operators on Carnot manifolds Goffeng, Magnus Kuzmin, Alexey Differential Geometry Analysis of PDEs K-Theory and Homology Operator Algebras Representation Theory 19K56 (primary), 19K33, 22E25, 53A40, 58B34, 58J40, 58J42 (secondary) We study the index theory of hypoelliptic operators on Carnot manifolds -- manifolds whose Lie algebra of vector fields is equipped with a filtration induced from sub-bundles of the tangent bundle. A Heisenberg pseudodifferential operator, elliptic in the calculus of van Erp-Yuncken, is hypoelliptic and Fredholm. Under some geometric conditions, we compute its Fredholm index by means of operator $K$-theory. These results extend the work of Baum-van Erp (Acta Mathematica '2014) for co-oriented contact manifolds to a methodology for solving this index problem geometrically on Carnot manifolds. Under the assumption that the Carnot manifold is regular, i.e. has isomorphic osculating Lie algebras in all fibres, and admits a flat coadjoint orbit, the methodology derived from Baum-van Erp's work is developed in full detail. In this case, we develope $K$-theoretical dualities computing the Fredholm index by means of geometric $K$-homology a la Baum-Douglas. The duality involves a Hilbert space bundle of flat orbit representations. Explicit solutions to the index problem for Toeplitz operators and operators of the form "$Δ_H+γT$" are computed in geometric $K$-homology, extending results of Boutet de Monvel and Baum-van Erp, respectively, from co-oriented contact manifolds to regular polycontact manifolds. |
| title | Index theory of hypoelliptic operators on Carnot manifolds |
| topic | Differential Geometry Analysis of PDEs K-Theory and Homology Operator Algebras Representation Theory 19K56 (primary), 19K33, 22E25, 53A40, 58B34, 58J40, 58J42 (secondary) |
| url | https://arxiv.org/abs/2203.04717 |