论文标题
卡诺歧管上的低细胞运算符索引理论
Index theory of hypoelliptic operators on Carnot manifolds
论文作者
论文摘要
我们研究了carnot歧管上的低纤维化算子的指数理论 - 歧管的歧管，其载体代数的代数配备了由切线束的子捆绑中引起的过滤。 Heisenberg pseudodivferential oberator，van Erp Yuncken的计算中的椭圆形，是低纤维化和弗雷德霍尔姆的。在某些几何条件下，我们通过运营商$ k $ - 理论来计算其弗雷德姆指数。这些结果将Baum-Van ERP（Acta Mathematica'2014）的共同接触歧管的工作扩展到了在Carnot歧管上几何解决此指数问题的方法。 在假设Carnot歧管是规则的，即在所有纤维中具有同构示意剂，并接受平坦的共同聚合轨道，从Baum-van ERP的工作中得出的方法是详细开发的。在这种情况下，我们通过几何$ k $ - baum-douglas来计算弗雷姆指数的理论二元性。二元性涉及一个扁平轨道表示的希尔伯特太空束。 Toeplitz操作员和“ $Δ_H+γt$”的操作员的索引问题的明确解决方案是在几何$ k $ - 人类学中计算的，分别从共同导向的触点词do botet de Monvel和Baum-van Erp的结果扩展到了普通的PolyContact cormolds。
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.
