Carnot流形上亚椭圆算子的指标理论
Index theory of hypoelliptic operators on Carnot manifolds
AI总结：
本文研究Carnot流形上Heisenberg伪微分算子的Fredholm指标，利用算子K理论与Baum-Douglas几何K同调建立对偶方法，并给出Toeplitz及Δ_H+γT型算子的显式指标解。
摘要：
我们研究Carnot流形上亚椭圆算子的指标理论——这类流形的向量场李代数配备了由切丛子丛诱导的滤过。一个在van Erp-Yuncken演算中为椭圆的Heisenberg伪微分算子是亚椭圆且Fredholm的。在某些几何条件下，我们借助算子K理论计算其Fredholm指标。这些结果将Baum-van Erp（《Acta Mathematica》，2014年）针对余定向接触流形的工作，推广为在Carnot流形上以几何方式求解该指标问题的方法论。在假设Carnot流形是正则的（即在所有纤维上具有同构的密切李代数）且容许平坦余伴随轨道的条件下，我们详尽发展了源自Baum-van Erp工作的方法论。在此情形下，我们建立了K理论对偶，以Baum-Douglas式的几何K同调来计算Fredholm指标。该对偶涉及一个由平坦轨道表示构成的Hilbert空间丛。我们在几何K同调中计算了Toeplitz算子以及形如“Δ_H+γT”的算子的指标问题的显式解，分别将Boutet de Monvel和Baum-van Erp的结果从余定向接触流形推广到正则多接触流形。
英文摘要：
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.
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