Residue formula for regular symmetry breaking operators

TL;DR

This paper derives an explicit residue formula for conformally covariant operators, enabling a new construction of symmetry breaking operators and analyzing their zeros in principal series representations.

Contribution

It introduces a residue formula for meromorphic continuations of conformally covariant operators and constructs symmetry breaking operators between differential forms on spheres and hyperplanes.

Findings

Explicit residue formula for conformally covariant operators.

New construction of symmetry breaking operators on spheres.

Determination of zeros of matrix-valued operators in principal series.

Abstract

We prove an explicit residue formula for a meromorphic continuation of conformally covariant integral operators between differential forms on ${\bf R}^n$ and on its hyperplane. The results provide a simple and new construction of the conformally covariant differential symmetry breaking operators between differential forms on the sphere and those on its totally geodesic hypersurface that were introduced in [Kobayashi-Kubo-Pevzner, Lect. Notes Math. (2016)]. Moreover, we determine the zeros of the matrix-valued regular symmetry breaking operators between principal series representations of $O(n+1,1)$ and $O(n,1)$.