The compact picture of symmetry-breaking operators for rank-one orthogonal and unitary groups (Q2305364)

The authors present a method to calculate intertwining operators between the underlying Harish-Chandra modules of degenerate principal series representations of a reductive Lie group \(G\) and a reductive subgroup \(G_0\), and between their composition factors. For these, the authors give an alternative approach to the class of symmetry breaking operators based on the Harish-Chandra module, i.e., the K finite vectors in the representation, in analogy with the idea of spectrum-generating operators [\textit{T. Branson} et al., J. Funct. Anal. 135, No. 1, 163--205 (1996; Zbl 0841.22011)]. The method describes the restriction of these operators to the \(K_0\)-isotypic components, \(K_0 \subset G_0\) a maximal compact subgroup, and reduces the representation-theoretic problem to an infinite system of scalar equations of a combinatorial nature. For rank-one orthogonal and unitary groups and spherical principal series representations the authors calculate these relations explicitly and use them to classify intertwining operators. They show how their presentations of the intertwiners fits into the integral representations for intertwiners. They authors show that automatic continuity holds. That is, every intertwiner between class one principal series Harish-Chandra modules (resp. any of their sub quotients) for \(G\) into a similar representation for \(G_0\) extends to an intertwiner between the Casselman-Wallach completions, verifying a conjecture by Toshiyuki Kobayashi. Altogether, this establishes the compact picture of the recently studied symmetry breaking operators for orthogonal groups by Kobayashi and Speh, gives new proofs of their main results, and extends them to unitary groups. Moreover, their algebraic framework provides an alternative proof of the discrete spectrum in certain unitary representations. The approach is quite general.