Shifting Operators on Spherical Functions of Compact Symmetric Pairs

Abstract

Representation theory provides a unified framework for understanding many of the classical special functions appearing in theoretical physics, such as Legendre polynomials, by interpreting them as spherical functions of relevant symmetry groups. In modern applications such as conformal field theory (CFT), one needs to consider matrix spherical functions of the conformal group to decompose four-point correlation functions into conformal blocks. A recent paper by Buric and Schomerus [BS23] introduces a method for shifting between conformal blocks using the action of the conformal Lie algebra. We extend this idea to the setting of a compact symmetric pair (G, K), where shifting corresponds to tensoring the representation governing the transformation behaviour of the matrix spherical function with the complexification of the isotropy representation of K on the tangent space at the identity of the symmetric space G/K. We analyse this isotropy representation and determine, for specific symmetric pairs, seeding sets from which all matrix spherical functions can be generated by the shifting action.