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.


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


Zbl 1493.22014
Full Text: DOI Link


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