Differential symmetry breaking operators. I-Genreal theory and F-method. II-Rankin-Cohen Operators for Symmetric Pairs

Abstract

Part I. 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. Part II. Rankin--Cohen brackets are symmetry breaking operators for the tensor product of two holomorphic discrete series representations of SL(2,R). We address a general problem to find explicit formulae for such intertwining operatorsin the setting of multiplicity-free branching laws for reductive symmetric pairs. We discover explicit formulae of new differential symmetry breaking operators for all the six different complex geometries arising from semisimple symmetric pairs of split rank one, and reveal an intrinsic reason why the coefficients of orthogonal polynomials appear in these operators (Rankin--Cohen type) in the three geometries and why normal derivatives are symmetry breaking operators in the other three cases. Further, we analyze a new phenomenon that the multiplicities in the branching laws of Verma modules may jump up at singular parameters.

References in corpus (2)

• Symmetry breaking for representations of rank one orthogonal groups
• Symmetry breaking for representations of rank one orthogonal groups II

Cited by in corpus (11)

• A program for branching problems in the representation theory of real reductive groups
• Symmetry breaking for representations of rank one orthogonal groups II
• F-method for symmetry breaking operators
• F-method for constructing equivariant differential operators
• Another Approach to Juhl's Conformally Covariant Differential Operators from $S^n$ to $S^{n-1}$
• Invariant differential operators on H-type groups and discrete components in restrictions of complementary series of rank one semisimple groups
• Algebraic analysis on scalar generalized Verma modules of Heisenberg parabolic type I.: $A_n$-series
• On the homomorphisms between the generalized Verma modules arising from conformally invariant systems
• Knapp-Stein type intertwining operators for symmetric pairs
• Tensor products of complementary series of rank one Lie groups
• Branching problems and $\mathfrak{s}\mathfrak{l}(2,\mathbb{C})$-actions