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Journal of Lie Theory 32 (2022), No. 1, 191--196
Copyright Heldermann Verlag 2022

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

Jan Frahm
Dept. of Mathematics, Aarhus University, Denmark


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.

Keywords: Real reductive groups, unitary representations, branching laws, direct integral, pointwise defined, smooth vectors, symmetry breaking operators.

MSC: 22E45; 22E46.

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