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Transactions of the Moscow Mathematical Society

ISSN 1547-738X(online) ISSN 0077-1554(print)



On the orbit space of a compact linear Lie group with commutative connected component

Author: O. G. Styrt
Translated by: E. Khukhro
Original publication: Trudy Moskovskogo Matematicheskogo Obshchestva, tom 70 (2009).
Journal: Trans. Moscow Math. Soc. 2009, 171-206
MSC (2000): Primary 22E45; Secondary 20C15, 22C05, 22E15
Published electronically: December 3, 2009
MathSciNet review: 2573640
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Abstract: This paper is devoted to the study of topological quotients of compact linear Lie groups. More precisely, it investigates the question of when such a quotient is a topological or a smooth manifold.

The topological quotient of a finite linear group was studied by Mikhaĭlova in 1984. Here the connected component $ G^0$ of the original Lie group $ G$ is assumed to be a torus of positive dimension.

The main method used here is to consider the weight system corresponding to the decomposition of the representation of the commutative group $ G^0$ into irreducible representations. In §8 an arbitrary linear group is reduced to a linear group of special type, namely, one with a $ 2$-stable weight system (for the definition and properties of $ q$-stable sets of vectors, where $ q\in\mathbb{N}$, see §§1, 4). The main results for a group with a $ 2$-stable weight system are stated in the Introduction (Theorems 1.3-1.8)

and proved in §§6 and 7.

References [Enhancements On Off] (What's this?)

  • 1. M. A. Mikhaĭlova, A factor space with respect to the action of a finite group generated by pseudoreflections, Izv. Akad. Nauk SSSR Ser. Mat. 48 (1984), no. 1, 104–126 (Russian). MR 733360
  • 2. Glen E. Bredon, Introduction to compact transformation groups, Academic Press, New York-London, 1972. Pure and Applied Mathematics, Vol. 46. MR 0413144
  • 3. Jean-Pierre Serre, Représentations linéaires des groupes finis, Third revised edition, Hermann, Paris, 1978 (French). MR 543841
  • 4. G. C. Shephard and J. A. Todd, Finite unitary reflection groups, Canadian J. Math. 6 (1954), 274–304. MR 0059914
  • 5. H. F. Blichfeldt, Finite collineation groups, Univ. of Chicago Press, 1917.

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Additional Information

O. G. Styrt
Affiliation: Moscow State University, Moscow, Russia

Keywords: Compact, linear group, Lie group, orbit space, smooth manifold
Published electronically: December 3, 2009
Article copyright: © Copyright 2009 American Mathematical Society

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