On the Direct Integral Decomposition in Branching Laws for Real Reductive Groups

J. Frahm

Journal of Lie theory, Tome 32 (2022) no. 1, pp. 191-196

Résumé

The restriction of an irreducible unitary representation $\pi$ of a real reductive group $G$ to a reductive subgroup $H$ decomposes into a direct integral of irreducible unitary representations $\tau$ of $H$ with multiplicities $m(\pi,\tau)\in\mathbb{N}\cup\{\infty\}$. We show that on the smooth vectors of $\pi$, the direct integral is pointwise defined. This implies that $m(\pi,\tau)$ is bounded above by the dimension of the space Hom$_H(\pi^\infty|_H,\tau^\infty)$ of intertwining operators between the smooth vectors, also called \emph{symmetry breaking operators}, and provides a precise relation between these two concepts of multiplicity.

Détail

Classification : 22E45, 22E46
Mots-clés : Real reductive groups, unitary representations, branching laws, direct integral, pointwise defined, smooth vectors, symmetry breaking operators