Stability of Branching Multiplicities for Orthogonal Gelfand Pairs

TL;DR

This paper introduces a unified framework for understanding how branching multiplicities behave in representation theory, especially for orthogonal Gelfand pairs, using linear inequalities and geometric fences.

Contribution

It develops a universal system of inequalities and geometric fences to describe the stability and variation of branching multiplicities across different representations.

Findings

Branching multiplicities are governed by linear inequalities.

Multiplicity functions are locally constant within convex regions.

Framework applies to both finite-dimensional and smooth Fréchet representations.

Abstract

We propose a structural framework for branching multiplicities in representation theory, emphasizing their behavior under variation of infinitesimal characters. For the orthogonal reductive pairs $(G,G')$ with complexified Lie algebras $(\mathfrak{o}(n+1,\mathbb{C}), \mathfrak{o}(n,\mathbb{C}))$, we show that branching multiplicities are governed by universal systems of linear inequalities on the parameter space of reduced coherent families introduced in this paper. To describe the loci where multiplicities may change, we introduce \emph{fences}: piecewise-linear hypersurfaces that divide the parameter space into convex regions. We prove that the multiplicity function is locally constant on each such region bounded by these fences. The framework applies uniformly to finite-dimensional representations and to admissible smooth Fr\'echet representations of real reductive groups. It accounts for classical results such as the Weyl branching law and provides a unified explanation for a range of phenomena, including the Gross--Prasad conjecture, sporadic symmetry breaking operators, and fusion rules for Verma modules. These results establish a general paradigm for branching multiplicities in orthogonal Gelfand pairs.