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Abstract

In the present chapter, we construct two sequences of polynomial families D N c (λ): C(S n ) → C(Sn−1) and D N nc (λ) : C(ℝ n ) → C(ℝn−1) of differential intertwining operators for spherical principal series representations. These families are induced by families \( \mathcal{D}_N \)(λ) of homomorphisms of Verma modules. We show how D N nc (λ) naturally arises from the asymptotics of eigenfunctions of the Laplacian of the hyperbolic metric on the upper half-space. This is the simplest special case of the construction of residue families in Section 6.6. In Chapter 6, the coincidence of two constructions of different nature for D N nc (λ) will be interpreted as the simplest case of a relation between residue families and tractor families (holographic duality). The induction and the mutual relations between the families are discussed in Section 5.2. We prove that all families \( \mathcal{D}_N \)(λ) satisfy a system of factorization identities. These relations imply corresponding identities for the induced families of differential operators. In turn, these give rise to a recursive algorithm which allows determination of explicit formulas for the families D N nc (λ). In Section 6.11, analogs of these factorization identities for residue families will shed light on the recursive structure of Q-curvatures and GJMS-operators.

Keywords

Explicit Formula Operator Family Factorization Identity Algebraic Theory Normal Order 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Birkhäuser Verlag AG 2009

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