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Record Nr. |
UNINA9910959539903321 |
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Autore |
Fischmann Matthias |
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Titolo |
Conformal Symmetry Breaking Differential Operators on Differential Forms |
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Pubbl/distr/stampa |
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Providence : , : American Mathematical Society, , 2021 |
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©2020 |
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ISBN |
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Edizione |
[1st ed.] |
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Descrizione fisica |
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1 online resource (124 pages) |
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Collana |
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Memoirs of the American Mathematical Society, , 0065-9266 ; ; Number 1304 |
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Classificazione |
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22E4635J3053A3022E4733C45 |
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Altri autori (Persone) |
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Disciplina |
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Soggetti |
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Differential operators |
Conformal geometry |
Symmetry (Mathematics) |
Topological groups, Lie groups {For transformation groups, see 54H15, 57Sxx, 58-XX. For abstract harmonic analysis, see 43-XX} -- Lie groups {For the topology of Lie groups and homogeneous spaces, see |
Partial differential equations -- Elliptic equations and systems [See also 58J10, 58J20] -- Higher-order elliptic equations [See also 31A30, 31B30] |
Differential geometry {For differential topology, see 57Rxx. For foundational questions of differentiable manifolds, see 58Axx} -- Classical differential geometry -- Conformal differential geometry |
Special functions (33-XX deals with the properties of functions as functions) {For orthogonal functions, see 42Cxx; for aspects of combinatorics see 05Axx; for number-theoretic aspects see 11-XX; for |
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Lingua di pubblicazione |
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Formato |
Materiale a stampa |
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Livello bibliografico |
Monografia |
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Nota di bibliografia |
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Includes bibliographical references. |
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Nota di contenuto |
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Cover -- Title page -- Chapter 1. Introduction -- Chapter 2. Preliminaries -- 2.1. The -method -- 2.2. Notation and induced representations -- 2.3. A branching problem -- Chapter 3. Singular vectors -- 3.1. The \gol′-equivariance -- 3.2. Families of singular vectors of the first type -- 3.3. Families of singular vectors of the second type -- 3.4. Singular vectors of the third type -- 3.5. Singular |
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vectors of the fourth type -- 3.6. Middle degree cases -- Chapter 4. Conformal symmetry breaking differential operators on differential forms -- 4.1. Families of the first type -- 4.2. Families of the second type -- 4.3. Hodge conjugation -- 4.4. Operators of the third type -- 4.5. Operators of the fourth type -- 4.6. Operators on middle degree forms -- 4.7. Proof of Theorem 3 -- 4.8. Examples -- Chapter 5. Geometric formulas for conformal symmetry breaking operators -- 5.1. Preparations -- 5.2. Even-order families of the first and second type -- 5.3. Odd-order families of the first and second type -- 5.4. Operators of the third and fourth type -- Chapter 6. Factorization identities for conformal symmetry breaking operators -- 6.1. Branson-Gover, gauge companion and -curvature operators -- 6.2. Main factorizations -- 6.3. Supplementary factorizations -- 6.4. Applications -- Appendix: Gegenbauer and Jacobi polynomials -- Bibliography -- Back Cover. |
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Sommario/riassunto |
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"We study conformal symmetry breaking differential operators which map differential forms on Rn to differential forms on a codimension one subspace Rn-1. These operators are equivariant with respect to the conformal Lie algebra of the subspace Rn-1. They correspond to homomorphisms of generalized Verma modules for so(n, 1) into generalized Verma modules for so(n+1, 1) both being induced from fundamental form representations of a parabolic subalgebra. We apply the F-method to derive explicit formulas for such homomorphisms. In particular, we find explicit formulas for the generators of the intertwining operators of the related branching problems restricting generalized Verma modules for so(n+1, 1) to so(n, 1). As consequences, we derive closed formulas for all conformal symmetry breaking differential operators in terms of the first-order operators d, δ, d and δ and certain hypergeometric polynomials. A dominant role in these studies is played by two infinite sequences of symmetry breaking differential operators which depend on a complex parameter λ. Their values at special values of λ appear as factors in two systems of factorization identities which involve the Branson-Gover operators of the Euclidean metrics on Rn and Rn-1 and the operators d, δ, d and δ as factors, respectively. Moreover, they naturally recover the gauge companion and Q-curvature operators of the Euclidean metric on the subspace Rn-1, respectively"-- |
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