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

Abstract

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 $G^{\prime }$, 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--Speh, gives new proofs of their main results and extends them to unitary groups.

45 pages; v2: worked out all details for the case of rank one unitary groups, v3: final published version

References in corpus (3)

• A program for branching problems in the representation theory of real reductive groups
• Shintani functions, real spherical manifolds, and symmetry breaking operators
• Invariant differential operators on H-type groups and discrete components in restrictions of complementary series of rank one semisimple groups

Cited by in corpus (2)

• Knapp-Stein Type Intertwining Operators for Symmetric Pairs II. -- The Translation Principle and Intertwining Operators for Spinors
• Symmetry breaking operators for line bundles over real projective spaces