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Zbl 0678.17012
Mathieu, Olivier
Construction d'un groupe de Kac-Moody et applications. (Construction of a Kac-Moody group and applications). (French)
[J] Compos. Math. 69, No.1, 37-60 (1989). ISSN 0010-437X; ISSN 1570-5846/e

Let A be a generalized Cartan matrix. The author constructs a so called ind-group scheme G associated to A. There is also a notion of Borel subgroup scheme B, and the quotient G/B exists with $G\to G/B$ locally trivial. He proves that the ``Schubert variety'' $S\sb{w\lambda}$ associated to a Weyl group element w and an integral dominant weight $\lambda$ is normal (and projectively normal with respect to its natural embedding). Finally he proves the PRV-conjecture (for symmetrizable Kac- Moody algebras over a field of characteristic 0). An important ingredient in the proofs is the techniques of Frobenius splitting in characteristic p. \par {\it P. Polo} proved the PRV-conjecture for finite dimensional Lie algebras of type A [Variétés de Schubert et excellentes filtrations, Astérisque 173-174, 281-311 (1989; Zbl 0733.20021)] while {\it S. Kumar} handled the general case of semisimple Lie algebras [Invent. Math. 93, 117-130 (1988; Zbl 0668.17008)].
[H.H.Andersen]

MSC 2000:: *17B67 Kac-Moody algebras
22E65 Infinite-dimensional Lie groups
14L15 Group schemes

Keywords: projective normality; generalized Cartan matrix; ind-group scheme; Borel subgroup scheme; Schubert variety; PRV-conjecture; symmetrizable Kac- Moody algebras; Frobenius splitting

Citations: Zbl 0666.17006; Zbl 0668.17008; Zbl 0733.20021

Cited in: Zbl 1229.22013 Zbl 0668.17008

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