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Realization of square-integrable representations of unimodular Lie groups on $ L\sp{2}$-cohomology spaces


Author: Jonathan Rosenberg
Journal: Trans. Amer. Math. Soc. 261 (1980), 1-32
MSC: Primary 22E45; Secondary 22E25
DOI: https://doi.org/10.1090/S0002-9947-1980-0576861-4
MathSciNet review: 576861
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Abstract: An analogue of the ``Langlands conjecture'' is proved for a large class of connected unimodular Lie groups having square-integrable representations (modulo their centers). For nilpotent groups, it is shown (without restrictions on the group or the polarization) that the $ {L^2}$-cohomology spaces of a homogeneous holomorphic line bundle, associated with a totally complex polarization for a flat orbit, vanish except in one degree given by the ``deviation from positivity'' of the polarization. In this degree the group acts irreducibly by a square-integrable representation, confirming a conjecture of Moscovici and Verona. Analogous results which improve on theorems of Satake are proved for extensions of a nilpotent group having square-integrable representations by a reductive group, by combining the theorem for the nilpotent case with Schmid's proof of the Langlands conjecture. Some related results on Lie algebra cohomology and the ``Harish-Chandra homomorphism'' for Lie algebras with a triangular decomposition are also given.


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DOI: https://doi.org/10.1090/S0002-9947-1980-0576861-4
Article copyright: © Copyright 1980 American Mathematical Society

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