Complète réductibilité  [ Complete reducibility ]
Séminaire Bourbaki : volume 2003/2004, exposés 924-937, Astérisque, no. 299 (2005), Talk no. 932, p. 195-217

The notion of complete reducibility of a linear representation Γ𝐆𝐋 n can be defined in terms of the action of Γ on the Tits building of 𝐆𝐋 n . An analogous definition can be given for any reductive group. We shall see how this translates in topological terms, and what applications can be obtained.

La notion de complète réductibilité d’une représentation linéaire Γ𝐆𝐋 n peut se définir en termes de l’action de Γ sur l’immeuble de Tits de 𝐆𝐋 n . Cela suggère une notion analogue pour tous les immeubles sphériques, et donc aussi pour tous les groupes réductifs. On verra comment cette notion se traduit en termes topologiques et quelles applications on peut en tirer.

Classification:  20-xx,  57M07
Keywords: reductive groups, spherical buildings, complete reducibility
@incollection{SB_2003-2004__46__195_0,
     author = {Serre, Jean-Pierre},
     title = {Compl\`ete r\'eductibilit\'e},
     booktitle = {S\'eminaire Bourbaki : volume 2003/2004, expos\'es 924-937},
     author = {Collectif},
     series = {Ast\'erisque},
     publisher = {Association des amis de Nicolas Bourbaki, Soci\'et\'e math\'ematique de France},
     address = {Paris},
     number = {299},
     year = {2005},
     note = {talk:932},
     pages = {195-217},
     language = {fr},
     url = {http://www.numdam.org/item/SB_2003-2004__46__195_0}
}
Serre, Jean-Pierre. Complète réductibilité, in Séminaire Bourbaki : volume 2003/2004, exposés 924-937, Astérisque, no. 299 (2005), Talk no. 932, pp. 195-217. http://www.numdam.org/item/SB_2003-2004__46__195_0/

[AS86] M. Aschbacher. Chevalley groups of type G 2 as the group of a trilinear form. J. Algebra, 109 :193-259, 1986. | MR 898346 | Zbl 0618.20030

[BMR04] M. Bate, B. M. S. Martin, and G. Röhrle. A geometric approach to complete reducibility. preprint, Univ. Birmingham 2004 ; à paraître dans Invent. math. | MR 2178661 | Zbl 1092.20038

[BT65] A. Borel and J. Tits. Groupes réductifs. Publ. Math. Inst. Hautes Études Sci., 27 :55-150, 1965. | Numdam | MR 207712 | Zbl 0145.17402

[BT71] A. Borel and J. Tits. Éléments unipotents et sous-groupes paraboliques de groupes réductifs I. Invent. math., 12 :95-104, 1971. | MR 294349 | Zbl 0238.20055

[Br89] K. Brown. Buildings. Springer-Verlag, 1989. | MR 969123 | Zbl 0922.20034

[Ch55] C. Chevalley. Théorie des Groupes de Lie, volume III. Hermann, Paris, 1955. | MR 68552 | Zbl 0186.33104

[DG70] M. Demazure and A. Grothendieck. Structure des schémas en groupes réductifs (SGA 3 III), volume 153 of Lect. Notes in Math. Springer-Verlag, 1970. | Zbl 0212.52810

[Dy52] E. B. Dynkin. Sous-groupes maximaux des groupes classiques. Trudy Moskov. Mat. Obshch., 1 :39-116 (en russe), 1952. trad. anglaise : Selected Papers, American Mathematical Society, 2000, p. 37-170.

[Gu99] R. M. Guralnick. Small representations are completely reducible. J. Algebra, 220 :531-541, 1999. | MR 1717357 | Zbl 0941.20001

[IMP03] S. Ilangovan, V. B. Mehta, and A. J. Parameswaran. Semistability and semisimplicity in representations of low height in positive characteristic. In V. Lakshmibai et al., editors, A Tribute to C.S. Seshadri. Hindustani Book Ag., New Delhi, 2003. | MR 2017588 | Zbl 1067.20061

[JLPW95] C. Jansen, K. Lux, R. Parker, and R. Wilson. An atlas of Brauer characters. LMS Monographs. Clarendon Press, Oxford, 1995. | MR 1367961 | Zbl 0831.20001

[Ja97] J. C. Jantzen. Low dimensional representations of reductive groups are semisimple. In Algebraic Groups and Lie Groups, pages 255-266. Cambridge Univ. Press, Cambridge, 1997. | MR 1635685 | Zbl 0877.20029

[Ke78] G. R. Kempf. Instability in invariant theory. Ann. of Math., 108 :299-316, 1978. | MR 506989 | Zbl 0406.14031

[LS96] M. W. Liebeck and G. M. Seitz. Reductive Subgroups of Exceptional Algebraic Groups, volume 580 of Mem. Amer. Math. Soc. American Mathematical Society, 1996. | MR 1329942 | Zbl 0851.20045

[Ma03a] B. M. S. Martin. Reductive subgroups of reductive groups in nonzero characteristic. J. Algebra, 262 :265-286, 2003. | MR 1971039 | Zbl 1103.20045

[Ma03b] B. M. S. Martin. A normal subgroup of a strongly reductive subgroup is strongly reductive. J. Algebra, 265 :669-674, 2003. | MR 1987023 | Zbl 1032.20030

[Mc98] G. J. Mcninch. Dimensional criteria for semisimplicity of representations. Proc. London Math. Soc. (3), 76 :95-149, 1998. | MR 1476899 | Zbl 0891.20032

[Mc00] G. J. Mcninch. Semisimplicity of exterior powers of semisimple representations of groups. J. Algebra, 225 :646-666, 2000. | MR 1741556 | Zbl 0971.20005

[Mo56] G. D. Mostow. Fully reducible subgroups of algebraic groups. Amer. J. Math., 78 :200-221, 1956. | MR 92928 | Zbl 0073.01603

[Mu65] D. Mumford. Geometric Invariant Theory. Springer-Verlag. 1965 ; third enlarged edit. (D. Mumford, J. Fogarty, F. Kirwan), 1994. | MR 214602 | Zbl 0797.14004

[Mue97] B. Mühlherr. Complete reducibility in projective spaces and polar spaces. preprint, Dortmund, 1997.

[No87] M. V. Nori. On subgroups of 𝐆𝐋(n,𝐅 p ). Invent. math., 88 :257-275, 1987. | MR 880952 | Zbl 0632.20030

[Ri88] R. W. Richardson. Conjugacy classes of n-tuples in Lie algebras and algebraic groups. Duke Math. J., 57 :1-35, 1988. | MR 952224 | Zbl 0685.20035

[Ron89] M. Ronan. Lectures on Buildings. Acad. Press, San Diego, 1989. | MR 1005533 | Zbl 1190.51008

[Rou78] G. Rousseau. Immeubles sphériques et théorie des invariants. C. R. Acad. Sci. Paris Sér. I Math., 286 :247-250, 1978. | MR 506257 | Zbl 0375.14013

[Sei00] G. M. Seitz. Unipotent elements, tilting modules, and saturation. Invent. math., 141 :467-502, 2000. | MR 1779618 | Zbl 1053.20043

[Se94] J-P. Serre. Sur la semi-simplicité des produits tensoriels de représentations de groupes. Invent. math., 116 :513-530, 1994. volume dédié à Armand Borel. | MR 1253203 | Zbl 0816.20014

[Se97a] J-P. Serre. Semisimplicity and tensor products of group representations : converse theorems. J. Algebra, 194 :496-520, 1997. with an Appendix by Walter Feit. | MR 1467165 | Zbl 0896.20004

[Se97b] J-P. Serre. La notion de complète réductibilité dans les immeubles sphériques et les groupes réductifs. Séminaire au Collège de France, 1997, résumé dans [Ti97], p. 93-98.

[Se98] J-P. Serre. The notion of complete reducibility in group theory. In Moursund Lectures Part II (Eugene, 1998). Notes by W.E. Duckworth, http://darkwing.uoregon.edu/~math/serre/index.html.

[So69] L. Solomon. The Steinberg character of a finite group with BN-pair. In Theory of Finite Groups, pages 213-221. Benjamin, 1969. | MR 246951 | Zbl 0216.08001

[Te95] D. M. Testerman. A 1 -type overgroups of elements of order p in semisimple algebraic groups and the associated finite groups. J. Algebra, 177 :34-76, 1995. | MR 1356359 | Zbl 0857.20025

[Ti74] J. Tits. Buildings of spherical type and finite BN-pairs, volume 386 of Lect. Notes in Math. Springer-Verlag, 1974. | MR 470099 | Zbl 0295.20047

[Ti97] J. Tits. Résumé des cours de 1996-1997. In Annuaire du Collège de France, volume 97, pages 89-102. 1997.

[TW02] J. Tits and R. M. Weiss. Moufang Polygons. Springer-Verlag, 2002. | MR 1938841 | Zbl 1010.20017