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
frahm@math.au.dk
[Abstract-pdf]
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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