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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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

Author: Richard Penney
Journal: Trans. Amer. Math. Soc. 261 (1980), 33-51
MSC: Primary 22E25; Secondary 17B56
MathSciNet review: 576862
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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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