By Toshiyuki Kobayashi, Toshihisa Kubo, Michael Pevzner

This paintings is the 1st systematic learn of all attainable conformally covariant differential operators remodeling differential kinds on a Riemannian manifold *X* into these on a submanifold *Y* with concentrate on the version area (*X*, *Y*) = (*S ^{n}*,

*S*

^{n}^{-1}).

The authors provide an entire type of all such conformally covariant differential operators, and locate their specific formulæ within the flat coordinates by way of uncomplicated operators in differential geometry and classical hypergeometric polynomials. ensuing households of operators are average generalizations of the Rankin–Cohen brackets for modular types and Juhl's operators from conformal holography. The matrix-valued factorization identities between all attainable combos of conformally covariant differential operators also are established.

The major equipment of the facts will depend on the "F-method" lately brought and built by means of the authors. it's a basic approach to build intertwining operators among *C*^{∞}-induced representations or to discover singular vectors of Verma modules within the context of branching principles, as options to differential equations at the Fourier rework part. The e-book provides a brand new extension of the F-method to the matrix-valued case within the common surroundings, that may be utilized to different difficulties as well.

This booklet bargains a self-contained creation to the research of symmetry breaking operators for infinite-dimensional representations of reductive Lie teams. this option could be valuable for lively scientists and available to graduate scholars and younger researchers in differential geometry, illustration conception, and theoretical physics.

**Read Online or Download Conformal Symmetry Breaking Operators for Differential Forms on Spheres PDF**

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**Extra info for Conformal Symmetry Breaking Operators for Differential Forms on Spheres**

**Sample text**

C) |i − j| = k = 1. 6, nonzero (k) O(N)-homomorphisms Hi→ j were constructed in Sect. 3. 6) implies the following proposition. 6. Then, we have HomO(N) i (CN ), j (k) (CN ) ⊗ H k (CN ) = CHi→ j . 8. 3)), any O(N)-homomorphism from i (CN ) to j (CN ) ⊗ Pol(CN ) can be written as a linear (k) combination of QN Hi→ j ( ∈ N, k ∈ {0, 1, 2}). 6. For this, we observe that i (CN ) may be thought of as a U(N)-module, whereas H k (CN ) is just an O(N)-module. Then our strategy is to use the branching laws with respect to a chain of subgroups 56 5 Application of Finite-Dimensional Representation Theory U(N) ×U(N) ⊃ U(N) ⊃ O(N), and the proof is divided into the following two steps.

1007/978-981-10-2657-7_4 41 4 Matrix-Valued F-method for O(n + 1, 1) 42 Sol(n+ ; σλ , τν ) = ψ ∈ HomL (V,W ⊗ Pol[ζ1 , . . , ζn ]) : dπ(σ ,λ )∗ (N1+ ) ⊗ idW ψ = 0 . 2) (2) Suppose ψ ∈ Sol(n+ ; σλ , τν ). Let D be the HomC (V,W )-valued differential operator on Rn with constant coefficients such that Symb(D) = ψ. Then the differential operator Restxn =0 ◦ D : C∞ (Rn ) ⊗V → C∞ (Rn−1 ) ⊗W G extends to a symmetry breaking operator from IndG P (σλ ) to IndP (τν ). G (3) Conversely, any G -equivariant differential operator from IndG P (σλ ) to IndP (τν ) is obtained in this manner.

Ii) For some nonzero C0 ∈ n+ , dπ(σ ,λ )∗ (C0 ) ⊗ idW ψ = 0. Proof. The implication (i)⇒(ii) is obvious. We shall prove (ii)⇒(i). We set µ := σλ∗ . Suppose ψ ∈ HomL (V, Pol(n+ ) ⊗W ) χ2ρ ( )µ( −1 ) ⊗ Ad# ( −1 (V ∨ ⊗ Pol(n+ ) ⊗W )L . This means that ) ⊗ τν ( −1 )ψ = ψ ∈L, for all where Ad# ( ) : Pol(n+ ) → Pol(n+ ) is defined by Ad# ( ) : f (·) → f (Ad( −1 )·), (see [22, p. 825]). If ψ satisfies (ii), then we have dπµ (C0 ) ⊗ idW µ( −1 )Ad# ( −1 ) ψ = 0. 7) We let the group L act on V ∨ ⊗ Pol(n− ) by πµ ( ) = µ( )Ad# ( ).