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Index theory for Heisenberg elliptic and transversally Heisenberg elliptic operators from $KK$-theoretic viewpoint

Minjie Tian

TL;DR

This work develops a KK-theoretic framework for index theory of Heisenberg elliptic and transversally Heisenberg elliptic operators on filtered manifolds. It extends the classical elliptic theory by integrating the osculating groupoid $Gr(H)$, the tangent-Clifford symbolic data, and the Connes–Thom isomorphism into a transversally enriched setting, culminating in explicit KK-product formulas for indices. A key contribution is the definition and analysis of transversally H-elliptic operators, including the construction of leafwise Dirac operators, the essential self-adjointness of associated cosymbols, and the tangent-Clifford symbol calculus that yields the main index theorem via KK-theory. The paper also provides a Fourier-transform-based method to verify compactness conditions for multipliers in nilpotent group C*-algebras and discusses concrete examples, establishing a robust foundation for hypoelliptic operator theory on complex filtered manifolds.

Abstract

This research comprehensively describes the basic theory of transversally Heisenberg elliptic operators, and investigates the index theory of Heisenberg elliptic and transversally Heisenberg elliptic operators from the perspective of $KK$-theory, applying Kasparov's methodology. Moreover, the analysis methodically examines specific conditions, with a focus on the Fourier transform of the nilpotent group $C^{\ast}$-algebra. We demonstrate enhanced methods for analyzing the hypoellipticity of operators, presenting a robust framework for defining and understanding transversal Heisenberg ellipticity in a $KK$-theoretic context. This work provides a solid foundation for future research into the properties of hypoelliptic differential operators in complicated manifolds.

Index theory for Heisenberg elliptic and transversally Heisenberg elliptic operators from $KK$-theoretic viewpoint

TL;DR

This work develops a KK-theoretic framework for index theory of Heisenberg elliptic and transversally Heisenberg elliptic operators on filtered manifolds. It extends the classical elliptic theory by integrating the osculating groupoid , the tangent-Clifford symbolic data, and the Connes–Thom isomorphism into a transversally enriched setting, culminating in explicit KK-product formulas for indices. A key contribution is the definition and analysis of transversally H-elliptic operators, including the construction of leafwise Dirac operators, the essential self-adjointness of associated cosymbols, and the tangent-Clifford symbol calculus that yields the main index theorem via KK-theory. The paper also provides a Fourier-transform-based method to verify compactness conditions for multipliers in nilpotent group C*-algebras and discusses concrete examples, establishing a robust foundation for hypoelliptic operator theory on complex filtered manifolds.

Abstract

This research comprehensively describes the basic theory of transversally Heisenberg elliptic operators, and investigates the index theory of Heisenberg elliptic and transversally Heisenberg elliptic operators from the perspective of -theory, applying Kasparov's methodology. Moreover, the analysis methodically examines specific conditions, with a focus on the Fourier transform of the nilpotent group -algebra. We demonstrate enhanced methods for analyzing the hypoellipticity of operators, presenting a robust framework for defining and understanding transversal Heisenberg ellipticity in a -theoretic context. This work provides a solid foundation for future research into the properties of hypoelliptic differential operators in complicated manifolds.
Paper Structure (36 sections, 52 theorems, 144 equations)

This paper contains 36 sections, 52 theorems, 144 equations.

Table of Contents

  1. Preliminaries for H-elliptic theory
  2. Filtered manifold and the osculating groupoid
  3. Kirillov's orbit method
  4. Groupoid $C^{\ast}$-algebra
  5. Heisenberg calculus for differential operators and H-elliptic operators
  6. Hilbert modules and $KK$-theory
  7. Heisenberg calculus for pseudo-differential operators
  8. Index map for H-elliptic operators
  9. Groupoid approach and Connes-Thom's isomorphism
  10. Elliptic operators and transversally elliptic operators
  11. Elliptic operators from $KK$-theoretic viewpoint
  12. Transversally elliptic operators from $KK$-theoretic viewpoint
  13. H-elliptic index theory
  14. The groupoid approach for H-elliptic operators
  15. Mohsen's Index theorem for H-elliptic operators from $KK$-theoretic viewpoint
  16. ...and 21 more sections

Key Result

Theorem 2

Let $M$ be a complete filtered Riemannian manifold and $G$ be a locally compact Lie group isometrically and properly acting on $M$ and preserving the filtered structure. Let $P$ be a $G$-invariant transversally H-elliptic operator on $M$ of order $0$, then we have in which $[\mathcal{D}_{M,{\Gamma}}^{cl'}]\in K^0(C^{\ast}(G,C^{\ast}(Gr(H))\hat{\otimes}_{C_0(M)}Cl_{\Gamma}(M)))$ indicates the Clif

Theorems & Definitions (155)

  • Definition 1
  • Theorem 2
  • Theorem 3
  • Definition 1.1.1
  • Lemma 1.1.2
  • proof
  • Definition 1.1.3
  • Remark 1.1.4
  • Definition 1.2.1
  • Theorem 1.2.2: kirillov2004lectures
  • ...and 145 more