Graded hypoellipticity of BGG sequences

Shantanu Dave; Stefan Haller

doi:10.1007/s10455-022-09870-0

Graded hypoellipticity of BGG sequences

Shantanu Dave, Stefan Haller (Corresponding author)

Department of Mathematics

Publications: Contribution to journal › Article › Peer Reviewed

Abstract

This article studies hypoellipticity on general filtered manifolds. We extend the Rockland
criterion to a pseudodifferential calculus on filtered manifolds, construct a parametrix and
describe its precise analytic structure. We use this result to study Rockland sequences, a notion generalizing elliptic sequences to filtered manifolds. The main application that we present is to the analysis of the Bernstein–Gelfand–Gelfand (BGG) sequences over regular parabolic geometries. We do this by generalizing the BGG machinery to more general filtered manifolds (in a non-canonical way) and show that the generalized BGG sequences are Rockland in a graded sense.

Original language	English
Pages (from-to)	721-789
Number of pages	69
Journal	Annals of Global Analysis and Geometry
Volume	62
Issue number	4
Early online date	2 Sep 2022
DOIs	https://doi.org/10.1007/s10455-022-09870-0
Publication status	Published - Nov 2022

Austrian Fields of Science 2012

101002 Analysis
101006 Differential geometry

Keywords

Filtered manifold
Pseudodifferential operator
Hypoelliptic operator
Rockland operator
Hypoelliptic sequence
Rockland sequence
BGG sequence
Rumin–Seshadri operator
Engel structure
Generic rank two distribution in dimension five
SUBELLIPTIC OPERATORS
COMPLEX
Rumin-Seshadri operator
SPACES
ALGEBRA
PSEUDODIFFERENTIAL-OPERATORS
DIFFERENTIAL FORMS
SINGER INDEX FORMULA
HEISENBERG GROUP

Access to Document

10.1007/s10455-022-09870-0Licence: CC BY 4.0

https://phaidra.univie.ac.at/o:1649452Licence: CC BY 4.0

Cite this

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title = "Graded hypoellipticity of BGG sequences",

abstract = "This article studies hypoellipticity on general filtered manifolds. We extend the Rocklandcriterion to a pseudodifferential calculus on filtered manifolds, construct a parametrix anddescribe its precise analytic structure. We use this result to study Rockland sequences, a notion generalizing elliptic sequences to filtered manifolds. The main application that we present is to the analysis of the Bernstein–Gelfand–Gelfand (BGG) sequences over regular parabolic geometries. We do this by generalizing the BGG machinery to more general filtered manifolds (in a non-canonical way) and show that the generalized BGG sequences are Rockland in a graded sense.",

keywords = "Filtered manifold, Pseudodifferential operator, Hypoelliptic operator, Rockland operator, Hypoelliptic sequence, Rockland sequence, BGG sequence, Rumin–Seshadri operator, Engel structure, Generic rank two distribution in dimension five, SUBELLIPTIC OPERATORS, COMPLEX, Rumin-Seshadri operator, SPACES, ALGEBRA, PSEUDODIFFERENTIAL-OPERATORS, DIFFERENTIAL FORMS, SINGER INDEX FORMULA, HEISENBERG GROUP",

author = "Shantanu Dave and Stefan Haller",

note = "Publisher Copyright: {\textcopyright} 2022, The Author(s). Publisher Copyright: {\textcopyright} 2022, The Author(s).",

year = "2022",

month = nov,

doi = "10.1007/s10455-022-09870-0",

language = "English",

volume = "62",

pages = "721--789",

journal = "Annals of Global Analysis and Geometry",

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TY - JOUR

T1 - Graded hypoellipticity of BGG sequences

AU - Dave, Shantanu

AU - Haller, Stefan

PY - 2022/11

Y1 - 2022/11

N2 - This article studies hypoellipticity on general filtered manifolds. We extend the Rocklandcriterion to a pseudodifferential calculus on filtered manifolds, construct a parametrix anddescribe its precise analytic structure. We use this result to study Rockland sequences, a notion generalizing elliptic sequences to filtered manifolds. The main application that we present is to the analysis of the Bernstein–Gelfand–Gelfand (BGG) sequences over regular parabolic geometries. We do this by generalizing the BGG machinery to more general filtered manifolds (in a non-canonical way) and show that the generalized BGG sequences are Rockland in a graded sense.

AB - This article studies hypoellipticity on general filtered manifolds. We extend the Rocklandcriterion to a pseudodifferential calculus on filtered manifolds, construct a parametrix anddescribe its precise analytic structure. We use this result to study Rockland sequences, a notion generalizing elliptic sequences to filtered manifolds. The main application that we present is to the analysis of the Bernstein–Gelfand–Gelfand (BGG) sequences over regular parabolic geometries. We do this by generalizing the BGG machinery to more general filtered manifolds (in a non-canonical way) and show that the generalized BGG sequences are Rockland in a graded sense.

KW - Filtered manifold

KW - Pseudodifferential operator

KW - Hypoelliptic operator

KW - Rockland operator

KW - Hypoelliptic sequence

KW - Rockland sequence

KW - BGG sequence

KW - Rumin–Seshadri operator

KW - Engel structure

KW - Generic rank two distribution in dimension five

KW - SUBELLIPTIC OPERATORS

KW - COMPLEX

KW - Rumin-Seshadri operator

KW - SPACES

KW - ALGEBRA

KW - PSEUDODIFFERENTIAL-OPERATORS

KW - DIFFERENTIAL FORMS

KW - SINGER INDEX FORMULA

KW - HEISENBERG GROUP

UR - https://arxiv.org/abs/2107.02062

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DO - 10.1007/s10455-022-09870-0

M3 - Article

VL - 62

SP - 721

EP - 789

JO - Annals of Global Analysis and Geometry

JF - Annals of Global Analysis and Geometry

SN - 0232-704X

IS - 4

ER -

Graded hypoellipticity of BGG sequences

Abstract

Austrian Fields of Science 2012

Keywords

Access to Document

Other files and links

Isoperimetr. Struktur v Anfangs

Analytic torsion of filtered manifolds

Singularity Theorems and Comparison Geometry

Regularized determinants of the Rumin complex on nilmanifolds with (2,3,5) geometry

Analytic torsion of the Rumin complex on filtered 5-manifolds with growth vector (2,3,5)

Analytic torsion of (2,3,5) geometries

Cite this

Graded hypoellipticity of BGG sequences

Abstract

Austrian Fields of Science 2012

Keywords

Access to Document

Other files and links

Projects

Isoperimetr. Struktur v Anfangs

Analytic torsion of filtered manifolds

Singularity Theorems and Comparison Geometry

Activities

Regularized determinants of the Rumin complex on nilmanifolds with (2,3,5) geometry

Analytic torsion of the Rumin complex on filtered 5-manifolds with growth vector (2,3,5)

Analytic torsion of (2,3,5) geometries

Cite this