By Greene R., Yau S.-T. (eds.)

ISBN-10: 0821814958

ISBN-13: 9780821814956

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**Additional resources for Differential Geometry: Geometry in Mathematical Physics and Related Topics, Part 2**

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For each type we check the conditions for M to be a faithful nilpotent module of dimension m < 12. This means certain equations in the ai. The crucial equation is 3a2 + a8 = O. On the other hand, we construct a faithful module of dimension 22 for all filiform Lie algebras of dimension 10. Remark 1. Let G be the connected simply connected Lie group with filiform nilpotent Lie algebra as in Proposition 8. Then G does not admit an etale affine representation. There is the question whether the Lie groups corresponding to the other classes (see Proposition 7) do admit such representations.

D. FRIED, W. W. HIRSCH, Affine manifolds with nilpotent holonomy, Comment. Math. Helv. 56 (1981), 487-523. Y. B. HAKIMJANOV, Variete des lois d'algebres de Lie nilpotentes Geometriae Dedicata 40 (1991), 269-295. J. R. Acad. Sc. Paris 272 (1971), 1088-109l. J. JANTZEN, First cohomology groups for classical Lie algebras, Prog. Math. 95 (1991),289-315. H. KIM, Complete left-invariant affine structures on nilpotent Lie groups, J. Differential Geometry 24 (1986), 373-394. J. MILNOR, On fundamental groups of complete affinely flat manifolds, Advances in Math.

66 (1957), pp. 203-248. [4J T. BRATTEN, Realizing representations on generalized flag manifolds, Preprint, (1995). to appear in Compositio Math. [5J J. CHANG, Special K-types, tempered characters and the BeilinsonBernstein realization, Duke Math. , 56 (1988), pp. 345-383. [6J HARISH-CHANDRA, Representations of semisimple Lie groups VI, Amer. , 78 (1956), pp. 564-628. [7J - - , Harmonic analysis on real reductive groups I, J. Func. , 19 (1975), pp. 104-204. [8J H. HECHT AND J. , 79 (1990), pp.

### Differential Geometry: Geometry in Mathematical Physics and Related Topics, Part 2 by Greene R., Yau S.-T. (eds.)

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