Holographic and symmetry breaking operators of holomorphic discrete series representations for real rank 3 cases (Recent Developments in Representation Theory and Related Topics)

Abstract

Let (𝐺, ([σ]𝐺)₀) be a symmetric pair of holomorphic type, and let [Hλ](𝐷) be a holomorphic discrete series representation of scalar type of 𝐺. Then the restriction [Hλ](𝐷)l([Gσ])₀ is decomposed multiplicity-freely into the Hilbert direct sum of countable holomorphic discrete series representations, and its branching law is given explicitly by the Hua-Kostant-Schmid-Kobayashi formula. Especially, there exist uniquely (up to constant multiple) the ([σ]𝐺)₀-intertwining operators (holographic operator, symmetry breaking operator) between [Hλ](𝐷)l([Gσ])₀ and each irreducible subrepresentation of ([σ]𝐺)₀. In this article, we treat the results on explicit construction of all intertwining operators for [Hλ](𝐷)l([Gσ])₀ when 𝐺 and the associated symmetric subgroup ([σ]𝐺ᶿ)₀ are both of real rank 3, of tube type and their noncompact parts are simple.