Realization of square-integrable representations of unimodular Lie groups on 𝐿²-cohomology spaces

Rosenberg, Jonathan

doi:10.1090/S0002-9947-1980-0576861-4

Realization of square-integrable representations of unimodular Lie groups on $L^{2}$-cohomology spaces
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Trans. Amer. Math. Soc. 261 (1980), 1-32 Request permission

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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