Conformal symmetry breaking operators for differential forms on spheres / Toshiyuki Kobayashi, Toshihisa Kubo, Michael Pevzner.

By:

Kobayashi, Toshiyuki, 1962- [author.]

Contributor(s):

Material type: Text

TextSeries: Lecture notes in mathematics (Springer-Verlag) ; 2170.Publisher: Singapore : Springer, 2016Description: 1 online resource (ix, 192 pages)Content type:

text

Media type:

computer

Carrier type:

online resource

ISBN:

9789811026577
9811026572
9811026564
9789811026560

Subject(s): Additional physical formats: Printed edition:: No titleDDC classification:

515/.7242 23

LOC classification:

QA329.4

Online resources:

Click here to access online

Contents:

6.6 Step 4 -- Part II: Explicit Formulæ for MIJ6.7 Step 5: Deduction from MIJ=0 to Lr(g0,g1, g2) = 0; Chapter 7 F-system for Symmetry Breaking Operators (j = i-2, i+1 case); 7.1 Proof of Theorem 7.1; Chapter 8 Basic Operators in Differential Geometry and Conformal Covariance; 8.1 Twisted Pull-Back of Differential Forms by Conformal Transformations; 8.2 Hodge Star Operator Under Conformal Transformations; 8.3 Normal Derivatives Under Conformal Transformations; 8.4 Basic Operators on Ei(Rn); 8.5 Transformation Rules Involving the Hodge Star Operator and Restxn=0.

8.6 Symbol Maps for Differential Operators Acting on FormsChapter 9 Identities of Scalar-Valued Differential Operators Dul; 9.1 Homogeneous Polynomial Inflation Ia; 9.2 Identities Among Juhl's Conformally Covariant Differential Operators; 9.3 Proof of Proposition 1.4; 9.4 Two Expressions of Di₂!-1u, a; Chapter 10 Construction of Differential Symmetry Breaking Operators; 10.1 Proof of Theorem 2.9 in the Case j=i-1; 10.2 Proof of Theorem 2.9 in the Case j=i+1; 10.3 Application of the Duality Theorem for Symmetry Breaking Operators; 10.4 Proof of Theorem 2.9 in the Case j=i

Summary: This work is the first systematic study of all possible conformally covariant differential operators transforming differential forms on a Riemannian manifold X into those on a submanifold Y with focus on the model space (X, Y) = (Sn, Sn-1). The authors give a complete classification of all such conformally covariant differential operators, and find their explicit formulæ in the flat coordinates in terms of basic operators in differential geometry and classical hypergeometric polynomials. Resulting families of operators are natural generalizations of the Rankin-Cohen brackets for modular forms and Juhl's operators from conformal holography. The matrix-valued factorization identities among all possible combinations of conformally covariant differential operators are also established. The main machinery of the proof relies on the "F-method" recently introduced and developed by the authors. It is a general method to construct intertwining operators between C∞-induced representations or to find singular vectors of Verma modules in the context of branching rules, as solutions to differential equations on the Fourier transform side. The book gives a new extension of the F-method to the matrix-valued case in the general setting, which could be applied to other problems as well. This book offers a self-contained introduction to the analysis of symmetry breaking operators for infinite-dimensional representations of reductive Lie groups. This feature will be helpful for active scientists and accessible to graduate students and young researchers in differential geometry, representation theory, and theoretical physics.

Holdings ( 1 )
Title notes ( 5 )

Holdings
Item type	Current library	Collection	Call number	Status	Date due	Barcode	Item holds
eBook	e-Library	eBook LN Mathematic		Available

Total holds: 0

Includes bibliographical references and index.

Online resource; title from PDF title page (SpringerLink, viewed October 19, 2016).

This work is the first systematic study of all possible conformally covariant differential operators transforming differential forms on a Riemannian manifold X into those on a submanifold Y with focus on the model space (X, Y) = (Sn, Sn-1). The authors give a complete classification of all such conformally covariant differential operators, and find their explicit formulæ in the flat coordinates in terms of basic operators in differential geometry and classical hypergeometric polynomials. Resulting families of operators are natural generalizations of the Rankin-Cohen brackets for modular forms and Juhl's operators from conformal holography. The matrix-valued factorization identities among all possible combinations of conformally covariant differential operators are also established. The main machinery of the proof relies on the "F-method" recently introduced and developed by the authors. It is a general method to construct intertwining operators between C∞-induced representations or to find singular vectors of Verma modules in the context of branching rules, as solutions to differential equations on the Fourier transform side. The book gives a new extension of the F-method to the matrix-valued case in the general setting, which could be applied to other problems as well. This book offers a self-contained introduction to the analysis of symmetry breaking operators for infinite-dimensional representations of reductive Lie groups. This feature will be helpful for active scientists and accessible to graduate students and young researchers in differential geometry, representation theory, and theoretical physics.

880-01 6.6 Step 4 -- Part II: Explicit Formulæ for MIJ6.7 Step 5: Deduction from MIJ=0 to Lr(g0,g1, g2) = 0; Chapter 7 F-system for Symmetry Breaking Operators (j = i-2, i+1 case); 7.1 Proof of Theorem 7.1; Chapter 8 Basic Operators in Differential Geometry and Conformal Covariance; 8.1 Twisted Pull-Back of Differential Forms by Conformal Transformations; 8.2 Hodge Star Operator Under Conformal Transformations; 8.3 Normal Derivatives Under Conformal Transformations; 8.4 Basic Operators on Ei(Rn); 8.5 Transformation Rules Involving the Hodge Star Operator and Restxn=0.