A hidden symmetry of a branching law

TL;DR

This paper uncovers a hidden symmetry in the branching laws for restricting certain unitary representations of orthogonal groups, simplifying the understanding of their subrepresentations and confirming a conjecture by Orsted-Speh.

Contribution

It introduces a new simple branching law for irreducible representations of $O(p,q)$ restricted to $O(p-1,q)$, revealing a hidden symmetry and confirming a prior conjecture.

Findings

Derived a simplified branching law for $O(p,q)$ representations

Confirmed the conjecture by Orsted-Speh

Revealed a hidden symmetry in the restriction process

Abstract

We consider branching laws for the restriction of some irreducible unitary representations $\Pi$ of $G=O(p,q)$ to its subgroup $H=O(p-1,q)$. In Kobayashi (arXiv:1907.07994), the irreducible subrepresentations of $O(p-1,q)$ in the restriction of the unitary $\Pi|_{O(p-1,q) }$ are determined. By considering the restriction of packets of irreducible representations we obtain another very simple branching law, which was conjectured in Orsted-Speh (arXiv:1907.07544).