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Transverse parabolic structures and transverse BGG sequences

Clément Cren

TL;DR

This work develops a framework to extend Bernstein–Gelfand–Gelfand type complexes to foliated manifolds with transverse parabolic geometry by leveraging filtered calculus and transversal Rockland theory. It introduces the transversal de Rham complex on tractor bundles, proves a transversally graded Rockland property under regularity, and constructs transverse BGG operators that form a graded Rockland sequence. When the transverse de Rham sequence is a complex, the associated BGG sequence computes the same cohomology, connecting leaf-space geometry to curved BGG theory. The approach provides an index-theoretic toolkit for leaf spaces of foliations with transverse parabolic geometries and links to broader representation-theoretic and K-homology perspectives.

Abstract

Manifolds endowed with a parabolic geometry in the sense of Cartan come with natural sequences of differential operators and their analysis provide the so called (curved) BGG sequence of {\v C}ap, Slov{á}k and Sou{\v c}ek. The sequences involved do not form an elliptic complex in the sense of Atiyah but enjoy similar properties. The proper framework to study these operators is the filtered calculus associated to the natural filtration of the tangent bundle induced by the parabolic geometry. Such analysis was carried over by Dave and Haller in a very general setting. In this article we use their methods associated with the transversal index theory for filtered manifolds developped by the author in a previous paper to derive curved BGG sequences for foliated manifolds with transverse parabolic geometry.

Transverse parabolic structures and transverse BGG sequences

TL;DR

This work develops a framework to extend Bernstein–Gelfand–Gelfand type complexes to foliated manifolds with transverse parabolic geometry by leveraging filtered calculus and transversal Rockland theory. It introduces the transversal de Rham complex on tractor bundles, proves a transversally graded Rockland property under regularity, and constructs transverse BGG operators that form a graded Rockland sequence. When the transverse de Rham sequence is a complex, the associated BGG sequence computes the same cohomology, connecting leaf-space geometry to curved BGG theory. The approach provides an index-theoretic toolkit for leaf spaces of foliations with transverse parabolic geometries and links to broader representation-theoretic and K-homology perspectives.

Abstract

Manifolds endowed with a parabolic geometry in the sense of Cartan come with natural sequences of differential operators and their analysis provide the so called (curved) BGG sequence of {\v C}ap, Slov{á}k and Sou{\v c}ek. The sequences involved do not form an elliptic complex in the sense of Atiyah but enjoy similar properties. The proper framework to study these operators is the filtered calculus associated to the natural filtration of the tangent bundle induced by the parabolic geometry. Such analysis was carried over by Dave and Haller in a very general setting. In this article we use their methods associated with the transversal index theory for filtered manifolds developped by the author in a previous paper to derive curved BGG sequences for foliated manifolds with transverse parabolic geometry.
Paper Structure (11 sections, 20 theorems, 78 equations)

This paper contains 11 sections, 20 theorems, 78 equations.

Table of Contents

  1. Introduction
  2. Kostant's Hodge theory and the origins of the BGG machinery
  3. The geometric setting
  4. The transverse complex
  5. The holonomy groupoid of a foliation
  6. Invariant differential operators for the holonomy action
  7. Pseudodifferential calculus on filtered manifolds
  8. Graded order in the filtered calculus
  9. Graded (transversal) Rockland sequences
  10. Graded transverse Rockland property of the transverse complex
  11. Transverse BGG operators for transverse parabolic geometries

Key Result

Lemma 2.1

Let $k\geq 0$ and $x \in \Lambda^k\left(\faktor{\mathfrak{g}}{\mathfrak{p}}\right)^*\otimes \mathbb{E}$. If $\partial \partial^*x = 0$ then $\partial^*x = 0$. If $\partial^*\partial x = 0$ then $\partial x = 0$.

Theorems & Definitions (51)

  • Lemma 2.1
  • proof
  • Theorem 2.2: Kostant Kostant
  • proof
  • Definition 3.1
  • Example 3.2
  • Proposition 3.4
  • proof
  • Corollary 3.4.1
  • proof
  • ...and 41 more