In this paper, we introduce new aspects in conformal geometry of some very natural
second-order differential operators. These operators are termed shift operators. In
the flat space, they are intertwining operators which are closely related to
symmetry breaking differential operators. In the curved case, they are closely
connected with ideas of holography and the works of Fefferman–Graham,
Gover–Waldron and one of the authors. In particular, we obtain an alternative
description of the so-called residue families in conformal geometry in terms of
compositions of shift operators. This relation allows easy new proofs of some of
their basic properties. In addition, we derive new holographic formulas for
-curvatures
in even dimension. Since these turn out to be equivalent to earlier holographic
formulas, the novelty here is their conceptually very natural proof. The overall
discussion leads to a unification of constructions in representation theory and
conformal geometry.
