Journal of Lie Theory

no. 1
On the Direct Integral Decomposition in Branching Laws for Real Reductive Groups
Journal of Lie Theory, Volume 32 (2022) no. 1, pp. 191-196
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.
DOI: 10.5802/jolt.1226
Classification: 22E45, 22E46
Keywords: Real reductive groups, unitary representations, branching laws, direct integral, pointwise defined, smooth vectors, symmetry breaking operators 
@article{JOLT_2022_32_1_a9,
     author = {J. Frahm},
     title = {On the {Direct} {Integral} {Decomposition} in {Branching} {Laws} for {Real} {Reductive} {Groups}},
     journal = {Journal of Lie Theory},
     pages = {191--196},
     year = {2022},
     volume = {32},
     number = {1},
     doi = {10.5802/jolt.1226},
     zbl = {1493.22010},
     url = {https://jolt.centre-mersenne.org/articles/10.5802/jolt.1226/}
}
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J. Frahm. On the Direct Integral Decomposition in Branching Laws for Real Reductive Groups. Journal of Lie Theory, Volume 32 (2022) no. 1, pp. 191-196. doi: 10.5802/jolt.1226

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