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Brion, M.

Représentations exceptionnelles des groupes semi-simples. (French) Zbl 0588.22010

Ann. Sci. Éc. Norm. Supér. (4) 18, 345-387 (1985).
In this paper the rational representations \(\phi\) : \(G\to GL(V)\) are studied, where V is a finite dimensional vector space and G is a semisimple group over \({\mathbb{C}}\) satisfying the following condition: the algebra \({\mathbb{C}}[V]^ U\) of U-invariant functions on V, where U is a maximal unipotent subgroup of G, is free. In this case V and the corresponding representation are called exceptional.
In the first part of the paper the closure of G-orbits in V are characterized. The second part of the paper is devoted to the classification of exceptional representations of simple groups. In the third part it is established that all singularities of the variety \(\overline{Gx}\), \(x\in V\), lie in its boundary.
[The results of this paper were announced in C. R. Acad. Sci., Paris, Sér. I 296, 5-6 (1983; Zbl 0538.14007)].
Reviewer: G.A.Soifer

Cited in 1 Review
Cited in 27 Documents

MSC:

22E46 Semisimple Lie groups and their representations
14L24 Geometric invariant theory
20G05 Representation theory for linear algebraic groups
14J17 Singularities of surfaces or higher-dimensional varieties

Keywords:

rational representations; semisimple group; exceptional representations; simple groups; singularities

Citations:

Zbl 0538.14007
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\textit{M. Brion}, Ann. Sci. Éc. Norm. Supér. (4) 18, 345--387 (1985; Zbl 0588.22010)
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