Branching theorems for compact symmetric spaces (Q2761186)

The purpose of this paper is to prove some branching theorems for compact symmetric spaces, where ''compact symmetric space'' denotes a homogeneous space \(G/K\) with \(G\) a compact connected real Lie group and \(K\) the identity component of the subgroup of fixed points of an involution of \(G\). Branching theorems tell how an irreducible representation of \(G\) decomposes when restricted to \(K\) and, in mathematics, they are studied particularly as tools for decomposing induced representations by means of Frobenius reciprocity. NEWLINE\[NEWLINE\begin{aligned} &G = U(n+m) \quad (n \leq m), \qquad K_2 \times K_1 = U(n) \times U(m),\\ &G'/K_2 = (U(n) \times U(n))/\text{diag }U(n),\end{aligned}\tag{1}NEWLINE\]NEWLINE NEWLINE\[NEWLINE\begin{aligned} &G = SO(n+m) \quad (n \leq m), \qquad K_2 \times K_1 = SO(n) \times SO(m),\\ & G'/K_2 = U(n)/SO(n),\end{aligned} \tag{2}NEWLINE\]NEWLINE NEWLINE\[NEWLINE\begin{aligned} & G = Sp(n+m) \quad (n \leq m), \qquad K_2 \times K_1 = Sp(n) \times Sp(m),\\ &G'/K_2 = U(2n)/Sp(n),\end{aligned}\tag{3}NEWLINE\]NEWLINE where \(K = K_2 \times K_1\) and \(K_1\) is larger than \(K_2\). The author shows that, in all three cases, there is another compact symmetric space \(G'/K_2\) (given above) with the following property: for each irreducible (finite-dimensional complex) representation \((\sigma,V)\) of \(G\) whose space \(V^{K_1}\) of \(K_1\)-fixed vectors is nonzero, there is a canonical irreducible representation \((\sigma',V')\) of \(G'\) such that the representations \((\sigma|_{K_2},V^{K_1})\) and \((\sigma'|_{K_2},V')\) are equivalent. The proof constructs a linear mapping of a certain vector subspace \(V'\) of \(V\) into \(V^{K_1}\), equivariant with respect to \(K_2\), and shows that this mapping is bijective. The stated result may be viewed as a generalization of a theorem of \textit{S. Helgason} [Adv. Math. 5, 1-154 (1970; Zbl 0209.25403)] that gives, in the case of a compact symmetric space \(G/K\), the multiplicity of the trivial representation in the restriction to \(K\) of an irreducible representation of \(G\).