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ORBITAL AND TRANSVERSE INDEX THEORY FOR A PROPER LIE-GROUP ACTION
- Speaker(s)
- GENNADI KASPAROV
- Affiliation
- Vanderbilt University, Nashville, USA
- Date
- May 29, 2024, 5:15 p.m.
- Link
- https://uw-edu-pl.zoom.us/j/95105055663?pwd=TTIvVkxmMndhaHpqMFUrdm8xbzlHdz09
- Information about the event
- 405 IMPAN & ZOOM
- Seminar
- North Atlantic Noncommutative Geometry Seminar
For the index theory of elliptic operators on a compact manifold X, the standard choice of a symbol algebra for (scalar) operators of negative order is C0(TX). The zero-order symbols are bounded multipliers of this algebra (i.e. elements of Cb(TX)). In the case when a compact Lie group G acts on a compact manifold X, one can consider operators which are leaf-wise (i.e. orbit-wise) elliptic. Also, there are transversely elliptic operators (defined by M. Atiyah). For each of these two classes of operators, there is a corresponding algebra of negative-order operators: Slf(X) for the leaf-wise operators and Str(X) for the transverse operators. Both are subalgebras of Cb(TX). The index group for leaf-wise operators is a certain K-theory group associated with the crossed product C*(G,X). The index group for transverse operators is the K-homology group of C*(G,X). In this talk, I will explain the K-duality between Slf(X) and Str(X), namely, how the K-homology of each of them is isomorphic to the K-theory of the other. I will also state the corresponding index theorems (leaf-wise and transverse) and give examples. The results naturally generalize to a proper Lie-group action on a complete manifold.
