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Conformal symmetry breaking differential operators on differential forms
About this Title
Matthias Fischmann, Andreas Juhl and Petr Somberg
Publication: Memoirs of the American Mathematical Society
Publication Year:
2020; Volume 268, Number 1304
ISBNs: 978-1-4704-4324-5 (print); 978-1-4704-6339-7 (online)
DOI: https://doi.org/10.1090/memo/1304
Published electronically: February 18, 2021
Keywords: Symmetry breaking operators,
Branson-Gover operators,
$Q$-curvature operators,
residue families,
generalized Verma modules,
branching problems,
singular vectors,
Jacobi and Gegenbauer polynomials
Table of Contents
Chapters
- 1. Introduction
- 2. Preliminaries
- 3. Singular vectors
- 4. Conformal symmetry breaking differential operators on differential forms
- 5. Geometric formulas for conformal symmetry breaking operators
- 6. Factorization identities for conformal symmetry breaking operators
- Appendix: Gegenbauer and Jacobi polynomials
Abstract
We study conformal symmetry breaking differential operators which map differential forms on $\mathbb {R} ^n$ to differential forms on a codimension one subspace $\mathbb {R} ^{n-1}$. These operators are equivariant with respect to the conformal Lie algebra of the subspace $\mathbb {R} ^{n-1}$. They correspond to homomorphisms of generalized Verma modules for $\mathfrak {g}o (n,1)$ into generalized Verma modules for $\mathfrak {g}o (n+1,1)$ 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 $\mathfrak {g}o (n+1,1)$ to $\mathfrak {g}o (n,1)$. As consequences, we derive closed formulas for all conformal symmetry breaking differential operators in terms of the first-order operators $d$, $\delta$, $\bar {d }$ and $\bar {\delta }$ 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 $\lambda$. Their values at special values of $\lambda$ appear as factors in two systems of factorization identities which involve the Branson-Gover operators of the Euclidean metrics on $\mathbb {R} ^n$ and $\mathbb {R} ^{n-1}$ and the operators $d$, $\delta$, $\bar {d }$ and $\bar {\delta }$ as factors, respectively. Moreover, they naturally recover the gauge companion and $Q$-curvature operators of the Euclidean metric on the subspace $\mathbb {R} ^{n-1}$, respectively.- Erwann Aubry and Colin Guillarmou, Conformal harmonic forms, Branson-Gover operators and Dirichlet problem at infinity, J. Eur. Math. Soc. (JEMS) 13 (2011), no. 4, 911â957. MR 2800480, DOI 10.4171/JEMS/271
- H. Bateman and A. Erdelyi, Higher transcendental functions. 1, 1953.
- Thomas P. Branson, Sharp inequalities, the functional determinant, and the complementary series, Trans. Amer. Math. Soc. 347 (1995), no. 10, 3671â3742. MR 1316845, DOI 10.1090/S0002-9947-1995-1316845-2
- Thomas Branson and A. Rod Gover, Conformally invariant operators, differential forms, cohomology and a generalisation of $Q$-curvature, Comm. Partial Differential Equations 30 (2005), no. 10-12, 1611â1669. MR 2182307, DOI 10.1080/03605300500299943
- Jean-Louis Clerc, Another approach to Juhlâs conformally covariant differential operators from $S^n$ to $S^{n-1}$, SIGMA Symmetry Integrability Geom. Methods Appl. 13 (2017), Paper No. 026, 18. MR 3635957, DOI 10.3842/SIGMA.2017.026
- Michael Eastwood and Jan SlovĂĄk, Semiholonomic Verma modules, J. Algebra 197 (1997), no. 2, 424â448. MR 1483772, DOI 10.1006/jabr.1997.7136
- Charles Fefferman and C. Robin Graham, The ambient metric, Annals of Mathematics Studies, vol. 178, Princeton University Press, Princeton, NJ, 2012. MR 2858236
- Charles Fefferman and C. Robin Graham, Juhlâs formulae for GJMS operators and $Q$-curvatures, J. Amer. Math. Soc. 26 (2013), no. 4, 1191â1207. MR 3073887, DOI 10.1090/S0894-0347-2013-00765-1
- M. Fischmann, A. Juhl and P. Somberg, Residue family operators on differential forms and Branson-Gover operators. (in preparation).
- A. Rod Gover, Conformal de Rham Hodge theory and operators generalising the $Q$-curvature, Rend. Circ. Mat. Palermo (2) Suppl. 75 (2005), 109â137. MR 2152358
- A. Rod Gover, Emanuele Latini, and Andrew Waldron, Poincaré-Einstein holography for forms via conformal geometry in the bulk, Mem. Amer. Math. Soc. 235 (2015), no. 1106, vi+95. MR 3338300, DOI 10.1090/memo/1106
- A. Rod Gover and Andrew Waldron, Boundary calculus for conformally compact manifolds, Indiana Univ. Math. J. 63 (2014), no. 1, 119â163. MR 3218267, DOI 10.1512/iumj.2014.63.5057
- C. Robin Graham, Ralph Jenne, Lionel J. Mason, and George A. J. Sparling, Conformally invariant powers of the Laplacian. I. Existence, J. London Math. Soc. (2) 46 (1992), no. 3, 557â565. MR 1190438, DOI 10.1112/jlms/s2-46.3.557
- C. Robin Graham and Andreas Juhl, Holographic formula for $Q$-curvature, Adv. Math. 216 (2007), no. 2, 841â853. MR 2351380, DOI 10.1016/j.aim.2007.05.021
- Akira Ikeda and Yoshiharu Taniguchi, Spectra and eigenforms of the Laplacian on $S^{n}$ and $P^{n}(\textbf {C})$, Osaka Math. J. 15 (1978), no. 3, 515â546. MR 510492
- Andreas Juhl, Cohomological theory of dynamical zeta functions, Progress in Mathematics, vol. 194, BirkhÀuser Verlag, Basel, 2001. MR 1808296
- Andreas Juhl, Families of conformally covariant differential operators, $Q$-curvature and holography, Progress in Mathematics, vol. 275, BirkhÀuser Verlag, Basel, 2009. MR 2521913
- Andreas Juhl, Explicit formulas for GJMS-operators and $Q$-curvatures, Geom. Funct. Anal. 23 (2013), no. 4, 1278â1370. MR 3077914, DOI 10.1007/s00039-013-0232-9
- Toshiyuki Kobayashi, Restrictions of generalized Verma modules to symmetric pairs, Transform. Groups 17 (2012), no. 2, 523â546. MR 2921076, DOI 10.1007/s00031-012-9180-y
- Toshiyuki Kobayashi, F-method for symmetry breaking operators, Differential Geom. Appl. 33 (2014), no. suppl., 272â289. MR 3159963, DOI 10.1016/j.difgeo.2013.10.003
- Toshiyuki Kobayashi, Toshihisa Kubo, and Michael Pevzner, Vector-valued covariant differential operators for the Möbius transformation, Lie theory and its applications in physics, Springer Proc. Math. Stat., vol. 111, Springer, Tokyo, 2014, pp. 67â85. MR 3619943, DOI 10.1007/978-4-431-55285-7_{6}
- Toshiyuki Kobayashi, Toshihisa Kubo, and Michael Pevzner, Conformal symmetry breaking operators for differential forms on spheres, Lecture Notes in Mathematics, vol. 2170, Springer, Singapore, 2016. MR 3559958
- Toshiyuki Kobayashi, Bent Ărsted, Petr Somberg, and VladimĂr SouÄek, Branching laws for Verma modules and applications in parabolic geometry. I, Adv. Math. 285 (2015), 1796â1852. MR 3406542, DOI 10.1016/j.aim.2015.08.020
- Toshiyuki Kobayashi and Michael Pevzner, Differential symmetry breaking operators: I. General theory and F-method, Selecta Math. (N.S.) 22 (2016), no. 2, 801â845. MR 3477336, DOI 10.1007/s00029-015-0207-9
- Toshiyuki Kobayashi and Birgit Speh, Symmetry breaking for representations of rank one orthogonal groups, Mem. Amer. Math. Soc. 238 (2015), no. 1126, v+110. MR 3400768, DOI 10.1090/memo/1126
- P. Somberg, Algebraic analysis on generalized Verma modules and differential invariants in parabolic geometries, Habilitation, Prague (2013). electronically available at https://is.cuni.cz/habilitace/download?variableId=1422541&forceFilename=Somberg_Habilitation_Thesis.pdf I