We present a method to calculate intertwining operators between the underlying
Harish-Chandra modules of degenerate principal series representations of a reductive Lie group
and a reductive
subgroup
, and
between their composition factors. Our method describes the restriction of these operators to
the
-isotypic
components,
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
