We develop a conceptual framework for the stability of branching multiplicities arising in the restriction of finite- and infinite-dimensional representations of real reductive Lie groups. Focusing on pairs whose complexifications are of type (gl(n+1),gl(n)) and (o(n+1),o(n)), we show that branching multiplicities are locally constant on explicitly described convex regions in the joint parameter space of infinitesimal characters.The boundaries of these convex regions are given by hyperplanes defined as the zero loci of explicit rational functions in the representation parameters of G and G', arising from a universal scalar identity. We call these hyperplanes fences, and the resulting convex regions are described by interleaving patterns, generalizing classical interlacing conditions.
As a consequence, multiplicities can change only upon crossing a fence. This framework provides a uniform and conceptual account of a range of classical and recent phenomena, including Weyl’s branching law, multiplicity jumps for Verma modules, the Gan-Gross-Prasad conjecture at the real place, and the occurrence of sporadic symmetry breaking operators.
Detailed proofs are given for type A in [3, 17] and for the orthogonal case in [10].
© Toshiyuki Kobayashi