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Formes harmoniques et cohomologie relative des algèbres de Lie. (Harmonic forms and relative cohomology of Lie algebras). (French) Zbl 0621.58013

In this interesting paper the author develops further her ideas on the relative cohomologies of certain Lie algebras published earlier in a paper with an identical title [J. Reine Angew. Math. 344, 71-86 (1983; Zbl 0512.58006)]. The main result of the present work is the following: Let G be a unimodular Lie group, K an Ad-compact subgroup and \({\mathfrak g}^ a \)totally complex K-invariant subalgebra of the complex Lie algebra of G. Then for any traceable representation of G the two spaces of \(C^{\infty}\)-vector-valued and distribution-vector-valued (\({\mathfrak g},K)\)-relative homologies are the same and are finite dimensional. The result implies Poincaré duality and its meaning is explained in the context of the orbit method.
Reviewer: C.S.Sharma

MSC:

58E30 Variational principles in infinite-dimensional spaces
17B55 Homological methods in Lie (super)algebras
17B56 Cohomology of Lie (super)algebras
35J50 Variational methods for elliptic systems

Citations:

Zbl 0512.58006
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References:

[1] BOREL (A.) and WALLACH (N.) . - Continuous cohomology , discrete subgroups and representations of reductive groups, Study n^\circ 94, Princeton, 1980 . MR 83c:22018 | Zbl 0443.22010 · Zbl 0443.22010
[2] CONNES (A.) and MOSCOVICI (H.) . - The L2-index theorem for homogeneous spaces of Lie groups , Ann. of Math., vol. 115, 1982 , p. 291-330. MR 84f:58108 | Zbl 0515.58031 · Zbl 0515.58031 · doi:10.2307/1971393
[3] DIXMIER (J.) . - Les C*-algèbres et leurs représentations , Gauthier-Villars, Paris, 1964 . MR 30 #1404 | Zbl 0152.32902 · Zbl 0152.32902
[4] DUFLO (M.) . - Construction de représentations unitaires d’un groupe de Lie , dans Harmonic Analysis and Group Representations, Liguoni, Napoli, 1982 . MR 87b:22028 · Zbl 0522.22011
[5] GOODMAN (R.) . - One parameter-groups generated by operators in an enveloping algebra , J. Funct. Anal., vol. 6, 1970 , p. 218-236. MR 42 #3229 | Zbl 0203.44202 · Zbl 0203.44202 · doi:10.1016/0022-1236(70)90059-5
[6] HELGASON (S.) . - Differential geometry and symmetric spaces , Academic Press, New York, 1962 . MR 26 #2986 | Zbl 0111.18101 · Zbl 0111.18101
[7] HERSANT (A.) . - Formes harmoniques et cohomologie relative des algèbres de Lie , J. Reine Angew. Math., vol. 344, 1983 , p. 71-86. MR 86a:22030 | Zbl 0512.58006 · Zbl 0512.58006 · doi:10.1515/crll.1983.344.71
[8] KOORNWINDER (T. H.) . - Invariant differential operators on non-reductive homogeneous spaces , Preprint, Mathematisch Centrum, Amsterdam, 1981 . arXiv | MR 82g:43011 | Zbl 0454.22006 · Zbl 0454.22006
[9] KOSZUL (J. L.) . - Homologie et cohomologie des algèbres de Lie , Bull. SMF, vol. 78, 1950 , p. 65-127. Numdam | MR 12,120g | Zbl 0039.02901 · Zbl 0039.02901
[10] PENNEY (R.) . - Harmonically induced representations on nilpotent Lie groups and automorphic forms on nilmanifolds , Trans. Amer. Math. Soc., vol. 260, 1980 , p. 123-145. MR 81h:22008 | Zbl 0439.22012 · Zbl 0439.22012 · doi:10.2307/1999879
[11] N. S. POULSEN . - On C;-vectors and intertwining bilinear forms for representations of Lie groups , J. Funct. Anal., vol. 9, 1972 , p. 87-120. MR 46 #9239 | Zbl 0237.22013 · Zbl 0237.22013 · doi:10.1016/0022-1236(72)90016-X
[12] ROSENBERG (J.) . - Realization of square-integrable representations of unimodular Lie groups on L2-cohomology spaces , Trans. Amer. Math. Soc., vol. 261, 1980 , p. 1-32. MR 81k:22012 | Zbl 0446.22010 · Zbl 0446.22010 · doi:10.2307/1998315
[13] ROSENBERG (J.) and VERGNE (M.) . - Harmonically induced representations of solvable Lie groups , Preprint I.H.E.S., 1984 . · Zbl 0602.22008
[14] SCHMID (W.) . - On a conjecture of Langlands , Ann. of Math., vol. 93, 1971 , p. 1-42. MR 44 #4149 | Zbl 0291.43013 · Zbl 0291.43013 · doi:10.2307/1970751
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