2024/06/04 by Ryosuke Nakahama, Nakahama, Ryosuke
Mathematics · #17C30 #22E45 #33C67 #43A85 #Algebraic and Geometric Analysis #FOS: Mathematics #Holomorphic and Operator Theory #Meromorphic and Entire Functions #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.2406.01905
openalex publication_date 2024/06/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let (G,G1) be a symmetric pair of holomorphic type, and we consider a pair of Hermitian symmetric spaces D1=G1/K1⊂ D=G/K, realized as bounded symmetric domains in complex vector spaces \mathfrakp+1⊂\mathfrakp+ respectively. Then the universal covering group \widetildeG of G acts unitarily on the weighted Bergman space Hλ(D)\subsetO(D) on D. Its restriction to the subgroup \widetildeG1 decomposes discretely and multiplicity-freely, and its branching law is given explicitly by Hua--Kostant--Schmid--Kobayashi's formula in terms of the K1-decomposition of the space P(\mathfrakp+2) of polynomials on the orthogonal complement \mathfrakp+2 of \mathfrakp+1 in \mathfrakp+. The object of this article is to construct explicitly \widetildeG1-intertwining operators (symmetry breaking operators) Hλ(D)|_\widetildeG1\toHε1λ(D1,Pk(\mathfrakp+2)) from holomorphic discrete series representations of \widetildeG to those of \widetildeG1, which are unique up to constant multiple for sufficiently large λ. These operators are given by differential operators whose symbols are computed as the inner products of polynomials on \mathfrakp+2. In this article, we treat the case \mathfrakp+,\mathfrakp+2 are both simple of tube type and rank\mathfrakp+=rank\mathfrakp+2. When rank\mathfrakp+=3, we treat all partitions k, and when rank\mathfrakp+ is general, we treat partitions of the form k=(k,…,k,k-l).