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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

##### MSC:
 2.2e+47 Semisimple Lie groups and their representations
##### Keywords:
real reductive Lie group; Cartan involution; Levi factor
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