Symmetry breaking operators for real reductive groups of rank one

Frahm, Jan; Weiske, Clemens

doi:10.1016/j.jfa.2020.108568

Cited by 8 publications

(6 citation statements)

References 27 publications

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“…Such operators are not only of interest when acting on functions but also when acting on sections of homogeneous vector bundles on spheres as well as in contexts where the conformal group of the sphere is replaced by other groups. We refer the interested reader to [KS15], [KS18], [KKP16], [FJS21], [FW20]. In the curved case, their definition uses a Poincaré-Einstein metric on X in the sense of Fefferman and Graham [FG12].…”

Section: Introductionmentioning

confidence: 99%

Residue families, singular Yamabe problems and extrinsic conformal Laplacians

Juhl¹,

Ørsted²

2021

Preprint

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Let (X, g) be a compact manifold with boundary M n and σ a defining function of M . To these data, we associate conformally covariant polynomial oneparameter families of differential operators C ∞ (X) → C ∞ (M ). They arise through a residue construction which generalizes an earlier construction in the framework of Poincaré-Einstein metrics and are referred to as residue families. The main ingredient of the definition of these families are eigenfunctions of the Laplacian of the singular metric σ −2 g. We prove that, if σ is an approximate solution of a singular Yamabe problem, i.e., if σ −2 g has constant scalar curvature −n(n+1) up to a sufficiently small remainder, these families can be written as compositions of certain degenerate Laplacians (Laplace-Robin operators). This shows that the notions of extrinsic conformal Laplacians and extrinsic Q-curvature introduced in recent works by Gover and Waldron naturally can be rephrased in terms of residue families. This new spectral theoretical perspective allows easy new proofs of results of Gover and Waldron. Moreover, it enables us to relate the extrinsic conformal Laplacians and the critical extrinsic Q-curvature to the scattering operator of the asymptotically hyperbolic metric σ −2 g extending work of Graham and Zworski. The relation to the scattering operators implies that the exterior conformal Laplacians are self-adjoint. We also derive new local holographic formulas for all extrinsic Q-curvatures (critical and sub-critical ones) in terms of renormalized volume coefficients, the scalar curvature of the background metric and the asymptotic expansions of eigenfunctions of the Laplacian of the singular metric σ −2 g. These results naturally extend earlier results in the Poincaré-Einstein case. Furthermore, we prove a new formula for the singular Yamabe obstruction B n . The simple structure of these formulas shows the benefit of the systematic usage of so-called adapted coordinates. We use the latter formula for B n to derive explicit expressions for the obstructions in low-order cases (confirming earlier results). Finally, we relate the obstruction B n to the supercritical Q-curvature Q n+1 .

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Section: Introductionmentioning

confidence: 99%

Residue families, singular Yamabe problems and extrinsic conformal Laplacians

Juhl¹,

Ørsted²

2021

Preprint

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“…In recent years, rapid progress in "Stage C" for the branching problems has been made in the construction and classification problems of symmetry breaking operators [23,36,81,97,99] in the "good framework" suggested by Theorem 6.7 or more strongly by Theorem 6.8. Some of them interact with parabolic geometry such as conformal geometry and also with the theory of automorphic forms.…”

Section: 1mentioning

confidence: 99%

Recent advances in branching problems of representations

Kobayashi¹

2021

Preprint

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How does an irreducible representation of a group behave when restricted to a subgroup? This is part of branching problems, which are one of the fundamental problems in representation theory, and also interact naturally with other fields of mathematics.This expository paper is an up-to-date account on some new directions in representation theory highlighting the branching problems for real reductive groups and their related topics ranging from global analysis of manifolds via group actions to the theory of discontinuous groups beyond the classical Riemannian setting.This article is an outgrowth of the invited lecture that the author delivered at the commemorative event for the 70th anniversary of the re-establishment of the Mathematical Society of Japan, and originally appeared in Japanese in Sugaku 71 (2019).

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“…F ↓ τ ρ j is called a symmetry breaking operator and F ↑ τ ρ j is called a holographic operator, according to the terminology introduced in [24,25] and [26] respectively. Such a problem is proposed by Kobayashi from the viewpoint of the representation theory (see [20]), and studied from various viewpoints by, e.g., [2,3,4,10,11,12,13,16,19,21,22,23,24,25,26,27,28,29,32,33,34,35,36,37,38,39] for holomorphic discrete series, principal series, and complementary series representations. Especially, it is proved by [19,24] that in the holomorphic setting, symmetry breaking operators F ↓ τ ρ j : O τ (D, V )| G 1 → O ρ j (D 1 , W j ) are always given by differential operators, that is, F ↓ τ ρ j are given by the form…”

Section: Introductionmentioning

confidence: 99%

Computation of Weighted Bergman Inner Products on Bounded Symmetric Domains and Restriction to Subgroups

Nakahama¹

2022

SIGMA

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Let (G, G 1 ) be a symmetric pair of holomorphic type, and we consider a pair of Hermitian symmetric spaces D 1 = G 1 /K 1 ⊂ D = G/K, realized as bounded symmetric domains in complex vector spaces p + 1 ⊂ p + respectively. Then the universal covering group G of G acts unitarily on the weighted Bergman space H λ (D) ⊂ O(D) on D. Its restriction to the subgroup G 1 decomposes discretely and multiplicity-freely, and its branching law is given explicitly by Hua-Kostant-Schmid-Kobayashi's formula in terms of the K 1decomposition of the space P(p + 2 ) of polynomials on the orthogonal complement p + 2 of p + 1 in p + . The object of this article is to compute explicitly the inner product f (x 2 ), eFor example, when p + , p + 2 are of tube type and f (x 2 ) = det(x 2 ) k , we compute this inner product explicitly by introducing a multivariate generalization of Gauss' hypergeometric polynomials 2 F 1 . Also, as an application, we construct explicitly G 1 -intertwining operators (symmetry breaking operators) H λ (D)| G1 → H µ (D 1 ) from holomorphic discrete series representations of G to those of G 1 , which are unique up to constant multiple for sufficiently large λ.

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Symmetry breaking operators for real reductive groups of rank one

Abstract: For a pair of real reductive groups G ′ ⊂ G we consider the space Hom G ′ (π| G ′ , τ ) of intertwining operators between spherical principal series representations π of G and τ of G ′ , also called symmetry breaking operators.

Cited by 8 publications

References 27 publications

Residue families, singular Yamabe problems and extrinsic conformal Laplacians

Residue families, singular Yamabe problems and extrinsic conformal Laplacians

Recent advances in branching problems of representations

Computation of Weighted Bergman Inner Products on Bounded Symmetric Domains and Restriction to Subgroups

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