Differential symmetry breaking operators: I. General theory and F-method

Abstract

Abstract. We prove a one-to-one correspondence between differential symmetry breaking operators for equivariant vector bundles over two homogeneous spaces and certain homomorphisms for representations of two Lie algebras, in connection with branching problems of the restriction of representations.We develop a new method (F-method) based on the algebraic Fourier transform for generalized Verma modules, which characterizes differential symmetry breaking operators by means of certain systems of partial differential equations.In contrast to the setting of real flag varieties, continuous symmetry breaking operators of Hermitian symmetric spaces are proved to be differential operators in the holomorphic setting. In this case symmetry breaking operators are characterized by differential equations of second order via the F-method.

Conformal Symmetry Breaking Operators for Anti-de Sitter Spaces

“…Thus the proof of Claim 5.1 is completed. Therefore Theorem A and Theorem B (1) follow from [15,I,Thm. 2.9].…”

“…(1) i = j = 0, ε = +, q = 0: [6], see also [3,10,14] for different approaches. The main machinery of finding symmetry breaking operators in various geometric gettings in [12], [14], and [15,II] is the "algebraic Fourier transform of generalized Verma modules" (F-method [9]), see [15, I] for a detailed exposition of the F-method.…”

“…(2) duality theorem between differential SBOs and homomorphisms for generalized Verma modules that encode branching laws [15, I, Thm. 2.9]; (3) automatic continuity theorem of differential SBOs in the Hermitian symmetric setting [15,I,Thm. 5.3].…”

“…Automatic continuity theorem is known for holomorphic differential SBOs in the Hermitian symmetric settings [15,I,Thm. 5.3].…”

“…Some of such operators are given as differential operators (e.g. [3,6,12,14,15]), and others are integral operators and their analytic continuation (e.g. [16]).…”