Linear maps preserving orbits
Annales de l'Institut Fourier, Volume 62 (2012) no. 2, p. 667-706

Let HGL(V) be a connected complex reductive group where V is a finite-dimensional complex vector space. Let vV and let G={g GL (V)gHv=Hv}. Following Raïs we say that the orbit Hv is characteristic for H if the identity component of G is H. If H is semisimple, we say that Hv is semi-characteristic for H if the identity component of G is an extension of H by a torus. We classify the H-orbits which are not (semi)-characteristic in many cases.

Soit HGL(V) un groupe complexe réductif connexe où V est un espace vectoriel complexe de dimension finie. Soient vV et G={g GL (V)gHv=Hv}. D’aprés Raïs nous disons que l’orbite Hv est caractéristique pour H si la composante connexe de l’identité de G est H. Si H est semi-simple, nous disons que Hv est semi-caractéristique pour H si la composante connexe de l’identité de G est une extension de H par un tore. Nous classifions les orbites de H qui ne sont pas (semi)-caractéristiques dans plusieurs cas.

DOI : https://doi.org/10.5802/aif.2691
Classification:  20G20,  22E46
Keywords: Characteristic orbits, linear preserver problems
@article{AIF_2012__62_2_667_0,
     author = {Schwarz, Gerald W.},
     title = {Linear maps preserving orbits},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {62},
     number = {2},
     year = {2012},
     pages = {667-706},
     doi = {10.5802/aif.2691},
     mrnumber = {2985513},
     zbl = {1255.14040},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2012__62_2_667_0}
}
Schwarz, Gerald W. Linear maps preserving orbits. Annales de l'Institut Fourier, Volume 62 (2012) no. 2, pp. 667-706. doi : 10.5802/aif.2691. http://www.numdam.org/item/AIF_2012__62_2_667_0/

[1] Doković, Dragomir Ž.; Li, Chi-Kwong Overgroups of some classical linear groups with applications to linear preserver problems, Linear Algebra Appl., Tome 197/198 (1994), pp. 31-61 (Second Conference of the International Linear Algebra Society (ILAS) (Lisbon, 1992)) | Article | MR 1275607 | Zbl 0793.15018

[2] Doković, Dragomir Ž.; Platonov, Vladimir P. Algebraic groups and linear preserver problems, C. R. Acad. Sci. Paris Sér. I Math., Tome 317 (1993) no. 10, pp. 925-930 | MR 1249362 | Zbl 0811.20044

[3] Dynkin, E. B. Maximal subgroups of the classical groups, Trudy Moskov. Mat. Obšč., Tome 1 (1952), pp. 39-166 | MR 49903 | Zbl 0048.01601

[4] Gorbatsevich, V. V.; Onishchik, A. L. Lie transformation groups [see MR0950861 (89m:22010)], Lie groups and Lie algebras, I, Springer, Berlin (Encyclopaedia Math. Sci.) Tome 20 (1993), pp. 95-235 | MR 1306739 | Zbl 0781.22004

[5] Guralnick, Robert M. Invertible preservers and algebraic groups, Proceedings of the 3rd ILAS Conference (Pensacola, FL, 1993), Tome 212/213 (1994), pp. 249-257 | Article | MR 1306980 | Zbl 0814.15002

[6] Guralnick, Robert M. Invertible preservers and algebraic groups. II. Preservers of similarity invariants and overgroups of PSL n (F), Linear and Multilinear Algebra, Tome 43 (1997) no. 1-3, pp. 221-255 | Article | MR 1613065 | Zbl 0889.20026

[7] Guralnick, Robert M.; Li, Chi-Kwong Invertible preservers and algebraic groups. III. Preservers of unitary similarity (congruence) invariants and overgroups of some unitary subgroups, Linear and Multilinear Algebra, Tome 43 (1997) no. 1-3, pp. 257-282 | Article | MR 1613069 | Zbl 0889.20027

[8] Helgason, Sigurdur Differential geometry, Lie groups, and symmetric spaces, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, Pure and Applied Mathematics, Tome 80 (1978) | MR 514561 | Zbl 0451.53038

[9] Hochschild, G. The structure of Lie groups, Holden-Day Inc., San Francisco (1965) | MR 207883 | Zbl 0131.02702

[10] Jacobson, N. A note on automorphisms of Lie algebras, Pacific J. Math., Tome 12 (1962), pp. 303-315 | MR 148716 | Zbl 0109.26201

[11] Jacobson, Nathan Lie algebras, Interscience Publishers (a division of John Wiley & Sons), New York-London, Interscience Tracts in Pure and Applied Mathematics, No. 10 (1962) | MR 143793 | Zbl 0121.27504

[12] Jacobson, Nathan Lie algebras, Dover Publications Inc., New York (1979) (Republication of the 1962 original) | MR 559927

[13] Li, Chi-Kwong; Pierce, Stephen Linear preserver problems, Amer. Math. Monthly, Tome 108 (2001) no. 7, pp. 591-605 | Article | MR 1862098

[14] Luna, Domingo Slices étales, Sur les groupes algébriques, Soc. Math. France, Paris (1973), p. 81-105. Bull. Soc. Math. France, Paris, Mémoire 33 | Numdam | MR 318167 | Zbl 0286.14014

[15] Luna, Domingo Adhérences d’orbite et invariants, Invent. Math., Tome 29 (1975) no. 3, pp. 231-238 | Article | MR 376704 | Zbl 0315.14018

[16] Oniščik, Arkadi L. Inclusion relations between transitive compact transformation groups, Trudy Moskov. Mat. Obšč., Tome 11 (1962), pp. 199-242 | MR 153779 | Zbl 0192.12601

[17] Oniščik, Arkadi L. Decompositions of reductive Lie groups, Mat. Sb. (N.S.), Tome 80 (122) (1969), pp. 553-599 | MR 277660 | Zbl 0222.22011

[18] Oniščik, Arkadi L. Topology of transitive transformation groups, Johann Ambrosius Barth Verlag GmbH, Leipzig (1994) | MR 1266842 | Zbl 0796.57001

[19] Platonov, Vladimir P.; Doković, Dragomir Ž. Linear preserver problems and algebraic groups, Math. Ann., Tome 303 (1995) no. 1, pp. 165-184 | Article | MR 1348361 | Zbl 0836.20065

[20] Raïs, Mustapha Notes sur la notion d’invariant caractéristique (2007) (arxiv.org/abs/0707.0782v1)

[21] Schwarz, Gerald W. Algebraic quotients of compact group actions, J. Algebra, Tome 244 (2001) no. 2, pp. 365-378 | Article | MR 1857750

[22] Stanley, Richard P. Invariants of finite groups and their applications to combinatorics, Bull. Amer. Math. Soc. (N.S.), Tome 1 (1979) no. 3, pp. 475-511 | Article | MR 526968 | Zbl 0497.20002

[23] Tefera, Akalu What is a Wilf-Zeilberger pair?, Notices Amer. Math. Soc., Tome 57 (2010) no. 4, p. 508-509 | MR 2647850

[24] Van Leeuwen, M. A. A. LiE, a software package for Lie group computations, Euromath Bull., Tome 1 (1994) no. 2, pp. 83-94 | MR 1283465 | Zbl 0807.17001

[25] Van Leeuwen, M. A. A.; Cohen, A. M.; Lisser, B. A package for Lie group computations, Computer Algebra Nederland, Amsterdam (1992)

[26] Vinberg, È. B.; Popov, V. L. A certain class of quasihomogeneous affine varieties, Izv. Akad. Nauk SSSR Ser. Mat., Tome 36 (1972), pp. 749-764 | MR 313260 | Zbl 0248.14014

[27] Weyl, Hermann The Classical Groups. Their Invariants and Representations, Princeton University Press, Princeton, N.J. (1939) | MR 1488158 | Zbl 1024.20502

[28] Wilf, Herbert S.; Zeilberger, Doron Rational functions certify combinatorial identities, J. Amer. Math. Soc., Tome 3 (1990) no. 1, pp. 147-158 | Article | MR 1007910 | Zbl 0695.05004