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Unitarizability of certain series of representations. (English) Zbl 0561.22010
Reading this paper requires a significant amount of preparation. Even the expert will benefit by having at hand [V2] Representations of real reductive Lie groups [D. Vogan (1981; Zbl 0469.22012)] and [S-V] [B. Speh and D. Vogan, Acta Math. 145, 227-299 (1980; Zbl 0457.22011)]. [V2] is needed as a general reference for notation, for several definitions, and for proofs of some of the results. [S-V] is needed because a conjecture is raised there, which is the main theorem in the present paper.
In order to state this theorem, even in a rough way, we need some notation. Let G be a real reductive Lie group, K a maximal compact subgroup, and \(\theta\) the corresponding Cartan involution. Let \({\mathcal G}\) be the complexification of the Lie algebra of G and let \({\mathcal Q}\) be a \(\theta\)-stable parabolic subalgebra whose complex conjugate \(\bar {\mathcal Q}\) is opposite \({\mathcal Q}\). Then \(\ell ={\mathcal Q}\cap \bar {\mathcal Q}\) is a Levi factor and \({\mathcal Q}=\ell +{\mathcal U}\) where \({\mathcal U}\) is the nilradical of \({\mathcal Q}\). Define L to be the normalizer of \({\mathcal Q}\) in G. Fix a Cartan subalgebra \({\mathcal H}\subset \ell\), a weight \(\lambda\in {\mathcal H}^*\) and define \(\pi ({\mathcal U})=\sum \alpha,\) summing over the set \(\Delta\) (\({\mathcal G},{\mathcal H})\) of roots.
In [S-V] a correspondence is established between irreducible (\(\ell,L\cap K)\)-modules Y with infinitesimal character \(\lambda\)-\(\rho\) (\({\mathcal U})\) and (\({\mathcal G},K)\)-modules \({\mathcal R}Y\) of infinitesimal character \(\lambda\). The theorem deals with the unitarizability of Y and \({\mathcal R}Y\). Theorem (a) If Y is unitarizable and \(Re\ll \alpha,\lambda \gg \geq 0\) for all \(\alpha \in \Delta ({\mathcal U},{\mathcal H})\) then \({\mathcal R}Y\) is unitarizable. (b) if \({\mathcal R}Y\) is unitarizable and \(Re\ll \alpha,\lambda \gg \geq 0\) and \(\ll \alpha,\lambda \gg \neq 0\) for all \(\alpha \in \Delta ({\mathcal U},{\mathcal H})\) then Y is unitarizable.
The paper is nicely written and has an informative introduction.
Reviewer: Th.Farmer

22E46 Semisimple Lie groups and their representations
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