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Zbl 0612.17005
Bürgstein, Hartmut; Hesselink, Wim H.
Algorithmic orbit classification for some Borel group actions. (English)
[J] Compos. Math. 61, 3-41 (1987). ISSN 0010-437X; ISSN 1570-5846/e

Let H be a reductive algebraic group over a base field K, let G be a Borel subgroup, and let U be the Lie algebra of the unipotent radical of G. The problem is to describe the orbit structures in the G-modules U and $U\sp*$. To attack this problem the authors have developed an algorithm which uses the root system of G and the weight space of the G-module. The procedure, which is quite technical, is presented in detail in the paper and is illustrated by several examples. \par The algorithm was implemented on a computer to obtain a complete classification of the orbits in U or $U\sp*$ when H is $A\sb n$ (n$\le 6)$, $B\sb 2$, $G\sb 2$, $B\sb 3$ or $C\sb 3$. The complete classification is also obtained for the orbits in $U\sp*$ when H is $C\sb 4$ or $A\sb 7$. In other cases, partial results are given.
[Th.Farmer]

MSC 2000:: *17B20 Simple and semisimple Lie algebras
20G05 Representation theory of linear algebraic groups
22E25 Nilpotent and solvable Lie groups

Keywords: nilpotent matrices; reductive algebraic group; Borel subgroup; algorithm; orbits

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