Branching theorems for compact symmetric spaces

Abstract:

A compact symmetric space, for purposes of this article, is a quotient $G/K$, where $G$ is a compact connected Lie group and $K$ is the identity component of the subgroup of fixed points of an involution. A branching theorem describes how an irreducible representation decomposes upon restriction to a subgroup. The article deals with branching theorems for the passage from $G$ to $K_{2}\times K_{1}$, where $G/(K_{2}\times K_{1})$ is any of $U(n+m)/(U(n)\times U(m))$, $SO(n+m)/(SO(n)\times SO(m))$, or $Sp(n+m)/(Sp(n)\times Sp(m))$, with $n\leq m$. For each of these compact symmetric spaces, one associates another compact symmetric space $G’/K_{2}$ with the following property: To each irreducible representation $(\sigma ,V)$ of $G$ whose space $V^{K_{1}}$ of $K_{1}$-fixed vectors is nonzero, there corresponds a canonical irreducible representation $(\sigma ’,V’)$ of $G’$ such that the representations $(\sigma |_{K_{2}},V^{K_{1}})$ and $(\sigma ’,V’)$ are equivalent. For the situations under study, $G’/K_{2}$ is equal respectively to $(U(n)\times U(n))/\text {diag}(U(n))$, $U(n)/SO(n)$, and $U(2n)/Sp(n)$, independently of $m$. Hints of the kind of “duality” that is suggested by this result date back to a 1974 paper by S. Gelbart.