Vol. 302, No. 1, 2019

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

Jan Frahm and Bent Ørsted

Vol. 302 (2019), No. 1, 23–76
Abstract

We 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 ${G}^{\prime }$, and between their composition factors. Our method describes the restriction of these operators to the ${K}^{\prime }$-isotypic components, ${K}^{\prime }\subseteq {G}^{\prime }$ 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 we calculate these relations explicitly and use them to classify intertwining operators. We further show that in these cases automatic continuity holds; i.e., every intertwiner between the Harish-Chandra modules extends to an intertwiner between the Casselman–Wallach completions, verifying a conjecture by 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.

Keywords
symmetry-breaking operators, intertwining operators, Harish-Chandra modules, principal series, spectrum-generating operator
Mathematical Subject Classification 2010
Primary: 22E46
Secondary: 05E10, 17B15
Milestones
Received: 14 February 2017
Accepted: 29 October 2018
Published: 5 November 2019
Authors
 Jan Frahm Department of Mathematics Aarhus University Aarhus Denmark Bent Ørsted Department of Mathematics Aarhus University Aarhus Denmark