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Multiplicity in restricting small representations. (English) Zbl 1508.22009

This paper is a compressed announcement of the paper [T. Kobayashi, J. Lie Theory 32, No. 1, 197–238 (2022; Zbl 1493.22014)] and detailed proofs appear there. However, ideas of the proofs and strategies are explained to a large extent in this short notice.
Let \(G\) be a noncompact real reductive Lie group and denote the category of moderate growth smooth representations of \(G\) by \(\mathcal{M}(G)\). Let \(G'\) is a reductive subgroup of \(G\) and consider \(\Pi\in\mathcal{M}(G)\) and \(\pi\in\mathcal{M}(G')\). Then \([\Pi\big|_{G'}:\pi]=\dim\mathrm{Hom}_{G'}(\Pi\big|_{G'},\pi)\) denotes the multiplicity of \(\pi\) in the restriction of \(\Pi\). Note that it is becoming infinity in the most general settings. Also denote \(m(\Pi\big|_{G'})=\sup_{\pi\in\mathrm{Irr}(G')}[\Pi\big|_{G'}:\pi]\) (this might be infinity as well).
If \(G'=H\) is a symmetric subgroup defined by an involution, there are notions of a “Borel subgroup” for \(G/H\) denoted by \(B_{G/H}\subset G_{\mathbb{C}}\) and a “minimal parabolic subgroup” \(P_{G/H} \subset G\). Then one of the results in the paper under review claims that \(\{m(\Pi\big|_{G'})\mid\Pi\in\mathrm{Irr}(G)_H\}\) is uniformly bounded if and only if \(G_{\mathbb{C}}/B_{G/H}\) is \(G'_{\mathbb{C}}\)-spherical. Here \(\mathrm{Irr}(G)_H\) denotes irreducible representations which are \(H\)-distinguished. An another result says that \(G/P_{G/H}\) is \(G'\)-real spherical if and only if \([\Pi\big|_{G'}:\pi]<\infty\) for any \(\Pi \in\mathrm{Irr}(G)_H\) and any \(\pi\in\mathrm{Irr}(G')\).
There are similar and much more precise results when the representations are limited to the class of degenerate principal series representations.

MSC:

22E46 Semisimple Lie groups and their representations
22E45 Representations of Lie and linear algebraic groups over real fields: analytic methods
53C35 Differential geometry of symmetric spaces

Citations:

Zbl 1493.22014
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References:

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