Intertwining Families

Part of the Progress in Mathematics book series (PM, volume 275)


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.


Explicit Formula Operator Family Factorization Identity Algebraic Theory Normal Order 
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© Birkhäuser Verlag AG 2009

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