Shift operators, residue families and degenerate Laplacians

TL;DR

This paper introduces shift operators in conformal geometry, linking them to residue families and holographic formulas for Q-curvatures, providing new proofs and unifying concepts across representation theory and geometry.

Contribution

It presents shift operators as a new tool in conformal geometry, offering an alternative description of residue families and new holographic formulas for Q-curvatures.

Findings

Shift operators relate to symmetry breaking differential operators.

New holographic formulas for Q-curvatures are derived.

Residue families are described via compositions of shift operators.

Abstract

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 $Q$-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.