Construction of Intertwining Operators between Holomorphic Discrete Series Representations

Abstract

In this paper, we explicitly construct G₁-intertwining operators between holomorphic discrete series representations H of a Lie group G and those H₁ of a subgroup G₁⊂G when (G, G₁) is a symmetric pair of holomorphic type. More precisely, we construct G₁-intertwining projection operators from H onto H₁ as differential operators, in the case (G, G₁)=(G₀×G₀, ΔG₀) and both H, H₁ are of scalar type, and also construct G₁-intertwining embedding operators from H₁ into H as infinite-order differential operators, in the case G is simple, H is of scalar type, and H₁ is multiplicity-free under a maximal compact subgroup K₁⊂K. In the actual computation, we make use of series expansions of integral kernels and the result of Faraut-Korányi (1990) or the author's previous result (2016) on norm computation. As an application, we observe the behavior of residues of the intertwining operators, which define the maps from some subquotient modules, when the parameters are at poles.