Branching laws for Verma modules and applications in parabolic geometry. I

Abstract

We initiate a new study of differential operators with symmetries and combine this with the study of branching laws for Verma modules of reductive Lie algebras. By the criterion for discretely decomposable and multiplicity-free restrictions of generalized Verma modules [T. Kobayashi, Transf. Groups (2012)], we are brought to natural settings of parabolic geometries for which there exist unique equivariant differential operators to submanifolds. Then we apply a new method (F-method) relying on the Fourier transform to find singular vectors in generalized Verma modules, which significantly simplifies and generalizes many preceding works. In certain cases, it also determines the Jordan-Hölder series of the restriction for singular parameters. The F-method yields an explicit formula of such unique operators, for example, giving an intrinsic and new proof of Juhl's conformally invariant differential operators [Juhl, Progr. Math. 2009] and its generalizations to spinor bundles. This article is the first in the series, and the next ones include their extension to curved cases together with more applications of the F-method to various settings in parabolic geometries.

Bernstein-Sato identities and conformal symmetry breaking operators

“…In the final section we highlight different origins of Bernstein-Sato operators. Furthermore, we discuss several applications related to conformal symmetry breaking differential operators [KØSS15,FJS16,KKP16]. As a consequence we shall observe that the Bernstein-Sato operators recover conformal symmetry breaking differential operators for functions, spinors and differential forms by partially new formulas.…”

“…Conformal symmetry breaking differential operators for functions. The operator P (λ), see (3.5), recovers conformal symmetry breaking differential operators [J09, KS15,KØSS15] D N (λ) : C ∞ (R n ) → C ∞ (R n−1 ), (4.5) which are given by…”

“…Our convention for the distribution kernels are as duals to what they are considered as in the standard literature mentioned above. This choice is justified by the use of Fourier transform, which is applied to these distribution kernels directly without further dualization and leads to generalizations of singular vectors studied in [KØSS15,FJS16,KKP16]. The proposal to find Bernstein-Sato operators of interest in the context of conformal symmetry breaking operators was initiated in [PS15], where certain shift operators for Gegenbauer polynomials regarded as the residues of Fourier transformed K ± λ,ν (x ′ , x n ) are studied.…”

Conformal higher-spin gravity: linearized spectrum = symmetry algebra

“…In order to derive the branching rules for on-shell Fradkin-Tseytlin modules, we first need the following lemma [114,115]: The idea of the proof (for more technical details, see e.g. [114,115]) is simple enough, when expressed in terms of the Poincaré-Birkhoff-Witt basis, so that we recall it here, before giving another proof in terms of characters.…”

Projective structure, $\widetilde{\operatorname{SL}}(3,\mathbb{R})$ and the symplectic Dirac operator

“…An approach to find precise positions of singular vectors in the representation space can be found in [5], [8], [9]. Let V denote a complex simple highest weight L-module, extended to P -module by U acting trivially.…”