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 Mat. Sb. (N.S.), 1988, Volume 135(177), Number 3, Pages 385–402 (Mi msb1711)

Closed orbits of Borel subgroups

V. L. Popov

Abstract: The author considers an algebraic action of a connected reductive algebraic group $G$ defined over an algebraically closed field $k$ on an affine irreducible algebraic variety $X$, and studies the question of when the action of a Borel subgroup $B$ of $G$ on $X$ is stable, i.e., the $B$-orbit of any point belonging to some nonempty open subset of $X$ is closed in $X$. A criterion for stability is obtained: Suppose that $\operatorname{char}k=0$. In order that the action of $B$ on $X$ be stable it is necessary, and, if $G$ is semisimple and the group of divisor classes $\mathrm{Cl}X$ is periodic, also sufficient that $X$ contain a point with a finite $G$-stabilizer. For an action $G:V$ defined by a linear representation $G\to GL(V)$ the cases when $B:V$ is not stable and either $G$ is simple or $G$ is semisimple and the action $G:V$ is irreducible are listed. A general criterion for an orbit of a connected solvable group acting on an affine variety to be closed is also obtained, and it is used to obtain a simple sufficient condition for an orbit of such a group, acting linearly, to be closed.
Bibliography: 30 titles.

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English version:
Mathematics of the USSR-Sbornik, 1989, 63:2, 375–392

Bibliographic databases:

UDC: 512
MSC: Primary 14L30, 20G05; Secondary 22E45, 14D25

Citation: V. L. Popov, “Closed orbits of Borel subgroups”, Mat. Sb. (N.S.), 135(177):3 (1988), 385–402; Math. USSR-Sb., 63:2 (1989), 375–392

Citation in format AMSBIB
\Bibitem{Pop88} \by V.~L.~Popov \paper Closed orbits of Borel subgroups \jour Mat. Sb. (N.S.) \yr 1988 \vol 135(177) \issue 3 \pages 385--402 \mathnet{http://mi.mathnet.ru/msb1711} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=937648} \zmath{https://zbmath.org/?q=an:0713.20036} \transl \jour Math. USSR-Sb. \yr 1989 \vol 63 \issue 2 \pages 375--392 \crossref{https://doi.org/10.1070/SM1989v063n02ABEH003280} 

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. A. A. Premet, “The theorem on restriction of invariants, and nilpotent elements in $W_n$”, Math. USSR-Sb., 73:1 (1992), 135–159
2. V. L. Popov, “On the Closedness of Some Orbits of Algebraic Groups”, Funct. Anal. Appl., 31:4 (1997), 286–289
3. N. Vavilov, “Weight elements of Chevalley groups”, St. Petersburg Math. J., 20:1 (2009), 23–57
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