F-method for symmetry breaking operators

TL;DR

This paper introduces an extension of the F-method for analyzing symmetry breaking operators in reductive groups, providing geometric criteria and explicit formulas, with applications in conformal geometry.

Contribution

It extends the F-method to non-local operators and offers geometric criteria for the finiteness of intertwining operators, with concrete examples in conformal geometry.

Findings

Extended F-method to non-local operators

Derived geometric criteria for operator finiteness

Obtained explicit residue formulas for symmetry breaking operators

Abstract

We provide some insights in the study of branching problems of reductive groups, and a method of investigations into symmetry breaking operators. First, we give geometric criteria for finiteness property of linearly independent continuous (respectively, differential) operators that intertwine two induced representations of reductive Lie groups and their reductive subgroups. Second, we extend the F-method known for local operators to non-local operators. We then illustrate the idea by concrete examples in conformal geometry, and explain how the F-method works for detailed analysis of symmetry breaking operators, e.g., finding functional equations and explicit residue formulae of regular symmetry breaking operators with meromorphic parameters.