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

Jan Möllers, Bent Ørsted

Published 2014-04-04, updated 2016-03-17Version 2

We present a method to calculate intertwining operators between the underlying Harish-Chandra modules of degenerate principal series representations of a semisimple Lie group $G$ and a semisimple subgroup $G'$, and between their composition factors. Our method decribes the restriction of these operators to the $K'$-isotypic components, $K'\subseteq G'$ 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--Speh, gives new proofs of their main results and extends them to unitary groups. Applications of our classification for orthogonal groups include the construction of discrete components in the restriction of certain unitary representations, a Funk--Hecke type formula and the computation of the spectrum of Juhl's conformally invariant differential operators.