Lie cohomology of representations of nilpotent Lie groups and holomorphically induced representations

Penney, Richard

doi:10.1090/S0002-9947-1980-0576862-6

Lie cohomology of representations of nilpotent Lie groups and holomorphically induced representations
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Trans. Amer. Math. Soc. 261 (1980), 33-51 Request permission

Abstract:

Let U be a locally injective, Moore-Wolf square integrable representation of a nilpotent Lie group N. Let $(\mathcal {H}, \lambda )$ be a complex, maximal subordinate pair corresponding to U and let ${\mathcal {H}_0} = \ker \lambda \cap \mathcal {H}$. The space ${C^\infty }(U)$ of differentiable vectors for U is an ${\mathcal {H}_0}$ module. In this work we compute the Lie algebra cohomology ${H^p}({\mathcal {H}_0}, {C^\infty }(U))$ of this Lie module. We show that the cohomology is zero for all but one value of p and that for this specific value the cohomology is one dimensional. These results, when combined with earlier results of ours, yield the existence and irreducibility of holomorphically induced representations for arbitrary (nonpositive), totally complex polarizations.

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A vanishing theorem for

cohomology in the nilpotent case

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