Construction \`a la Ibukiyama of symmetry breaking differential operators, I

TL;DR

This paper extends Ibukiyama's method for constructing symmetry breaking differential operators from the Siegel upper half space to general Hermitian symmetric spaces of tube-type, using Euclidean Jordan algebras.

Contribution

It generalizes the construction of symmetry breaking operators to broader Hermitian symmetric spaces and provides explicit examples involving Euclidean Jordan algebras.

Findings

Extended Ibukiyama's construction to Hermitian symmetric spaces of tube-type

Derived explicit differential operators for specific restrictions

Connected the construction to Euclidean Jordan algebras

Abstract

The construction of symmetry breaking differential operators, using invariant pluri-harmonic polynomials, due to T. Ibukiyama in the context of the Siegel upper half space, is extended for scalar representations to general Hermitian symmetric spaces of tube-type. The new context is described in terms of Euclidean Jordan algebras and their representations. As an example, new and explicit differential operators are obtained for the restriction from the tube domain over the light cone to the product of two upper half-planes.