2022/03/09 by Goffeng, Magnus, Kuzmin, Alexey · 4 citations
#19K33 #19K56 (primary) #22E25 #53A40 #58B34 #58J40 #58J42 (secondary) #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #K-Theory and Homology (math.KT) #Operator Algebras (math.OA) #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.2203.04717
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 "ΔH+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.