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Transactions of the American Mathematical Society

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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
DOI: https://doi.org/10.1090/S0002-9947-1980-0576862-6
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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  • [1] J. Camora, Représentations du groupe de Heisenberg dans les espaces de $ (0,\,q)$ formes, Math. Ann. 205 (1973), 89-112. MR 0342643 (49:7389)
  • [2] L. Corwin, F. Greenleaf and R. Penney, A general character formula for irreducible projections on $ {L^2}$ of a nilmanifold, Math. Ann. 225 (1977), 21-32. MR 0425021 (54:12979)
  • [3] G. Hochschild and J. P. Serre, Cohomology of Lie algebras, Ann. of Math. (2) 57 (1953), 591-603. MR 0054581 (14:943c)
  • [4] A. Kirillov, Unitary representations of nilpotent Lie groups, Russian Math. Surveys 17 (1962), 53-104. MR 0142001 (25:5396)
  • [5] C. Moore and J. Wolf, Square integrable representations of nilpotent groups, Trans. Amer. Math. Soc. 185 (1973), 445-462. MR 0338267 (49:3033)
  • [6] H. Moscovici and A. Verona, Harmonically induced representations of nilpotent Lie groups, Invent. Math. 48 (1978), 61-64. MR 508089 (80a:22011)
  • [7] H. Moscovici, A vanishing theorem for $ {L^2}$-cohomology in the nilpotent case, (Conference on Non-Commutative Harmonic Analysis, Marseille-Luminy, June, 1978), Springer, Berlin and New York.
  • [8] R. Penney, Harmonically induced representations on nilpotent Lie groups and automorphic forms on nilmanijolds, Trans. Amer. Math. Soc. 260 (1980), 123-145. MR 570782 (81h:22008)
  • [9] N. S. Poulsen, On $ {C^\infty }$-vectors and intertwining bilinear forms for representations of Lie groups, J. Functional Analysis 9 (1972), 87-120. MR 0310137 (46:9239)
  • [10] I. Satake, Unitary representations of semi-direct product of Lie groups on $ \bar \partial $-cohomology spaces, Math. Ann. 190 (1971), 177-202. MR 0296213 (45:5274)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1980-0576862-6
Article copyright: © Copyright 1980 American Mathematical Society

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