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Finite, tame, and wild actions of parabolic subgroups in \(\text{GL}(V)\) on certain unipotent subgroups. (English) Zbl 0968.20023
A parabolic subgroup of \(\text{GL}(V)\) is the stabilizer of a (possibly incomplete) flag of subspaces of \(V\). If \(P\) is such a parabolic subgroup, the authors study the action of \(P\) on the Lie algebra of its unipotent radical and more generally on the \(l\)-th term of the descending central series. They characterize the cases where this action has a finite number of orbits.
The proof relies on seeing the orbits as isomorphism classes of some category, which is then shown to be equivalent to a category of modules over an algebra.

MSC:
20G05 Representation theory for linear algebraic groups
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[1] Auslander, M.; Reiten, I.; Smalø, S.O., Representation theory of Artin algebras, Cambridge studies in advanced mathematics, 36, (1995), Cambridge Univ. Press Cambridge
[2] Bongartz, K., A criterion for finite representation type, Math. ann., 269, 1-12, (1984) · Zbl 0552.16012
[3] Bongartz, K.; Gabriel, P., Covering spaces in representation theory, Invent. math., 65, 331-378, (1982) · Zbl 0482.16026
[4] Brüstle, Th., Matrix-finite bimodules: an algorithm, C. R. acad. sci. Paris Sér. I, 319, 1141-1145, (1997) · Zbl 0821.16010
[5] Th. Brüstle, and, D. Guhe, xparabol, a computer program that computes the orbits of an action of P on ideals in \(p\)u and their quotients. Available via ftp: Server, ftp.uni-bielefeld.de, directory: pub/math/f-d-alg/crep.
[6] Th. Brüstle, and, L. Hille, Matrices over upper triangular bimodules and Delta-filtered modules over quasi-hereditary algebras, Colloq. Math, to appear.
[7] Crawley-Boevey, W.W., Matrix problems and Drozd’s theorem, Topics in algebra, part 1 (Warsaw, 1988), Banach center publ., 26, (1990), PWN Warsaw, p. 199-222 · Zbl 0734.16004
[8] de la Peña, J.A.; Tomé, B., Iterated tubular algebras, J. pure appl. algebra, 64, 303-314, (1990) · Zbl 0704.16006
[9] Dlab, V.; Ringel, C.M., (), 200-224
[10] Gabriel, P.; Roiter, A.V., Representations of finite-dimensional algebras, Encyclopaedia of math. sci. vol. 73, algebra VIII, (1992)
[11] Happel, D.; Ringel, C.M., The derived category of a tubular algebra, Representation theory, I (Ottawa, ont., 1984), Lecture notes in math., 1177, (1986), Springer Berlin/New York, p. 156-180
[12] Happel, D.; Vossieck, D., Minimal algebras of infinite representation type with preprojective component, Manuscripta math., 42, 221-243, (1983) · Zbl 0516.16023
[13] Hille, L.; Röhrle, G., On parabolic subgroups of classical groups with a finite number of orbits on the unipotent radical, C. R. acad. sci. Paris Sér. I, 325, 465-470, (1997) · Zbl 0922.20049
[14] Hille, L.; Röhrle, G., A classification of parabolic subgroups in classical groups with a finite number of orbits on the unipotent radical, Transform. groups, 4, 35-52, (1999) · Zbl 0924.20035
[15] Ringel, C.M., Tame algebras and integral quadratic forms, Lecture notes in mathematics, 1099, (1984), Springer-Verlag Berlin/Heidelberg/New York/Tokyo
[16] Ringel, C.M., The category of modules with good filtrations over a quasi-hereditary algebra has almost split sequences, Math. Z., 208, 209-223, (1991) · Zbl 0725.16011
[17] Unger, L., The concealed algebras of the minimal wild, hereditary algebras, Bayreuth. math. schr., 31, 145-154, (1990) · Zbl 0739.16012
[18] Vinberg, E.B., Complexity of actions of reductive groups, Funct. anal. appl., 20, (1986) · Zbl 1054.14064
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