Abstract
This paper is a continuation and a completion of the work of the first and the third author on the Jordan decomposition. We extend the Jordan decomposition of blocks: we show that blocks of finite groups of Lie type in nondescribing characteristic are Morita equivalent to blocks of subgroups associated to isolated elements of the dual group — this is the modular version of a fundamental result of Lusztig, and the best approximation of the character-theoretic Jordan decomposition that can be obtained via Deligne-Lusztig varieties. The key new result is the invariance of the part of the cohomology in a given modular series of Deligne-Lusztig varieties associated to a given Levi subgroup, under certain variations of parabolic subgroups.\looseness=-1
We also bring in local block theory methods: we show that the equivalence arises from a splendid Rickard equivalence. Even in the setting of the original work of the first and the third author, the finer homotopy equivalence was unknown. As a consequence, the equivalences preserve defect groups and categories of subpairs. We finally determine when Deligne-Lusztig induced representations of tori generate the derived category of representations. An additional new feature is an extension of the results to disconnected reductive algebraic groups, which is required to handle local subgroups.
-
[qi]
C. Bonnafé, "Quasi-isolated elements in reductive groups," Comm. Algebra, vol. 33, iss. 7, pp. 2315-2337, 2005.
@ARTICLE{qi, mrkey = {2153225},
number = {7},
issn = {0092-7872},
author = {Bonnaf{é},
C{é}dric},
mrclass = {20G15},
doi = {10.1081/AGB-200063602},
journal = {Comm. Algebra},
zblnumber = {1096.20037},
volume = {33},
mrnumber = {2153225},
fjournal = {Communications in Algebra},
mrreviewer = {Evgenii L. Bashkirov},
coden = {COALDM},
title = {Quasi-isolated elements in reductive groups},
year = {2005},
pages = {2315--2337},
} -
[asterisque] C. Bonnafé, Sur les Caractères des Groupes Réductifs Finis à Centre Non Connexe: Applications aux Groupes Spéciaux Linéaires et Unitaires, Paris: Math. Soc. France, 2006, vol. 306.
@book{asterisque, mrkey = {2274998},
issn = {0303-1179},
author = {Bonnaf{é},
C{é}dric},
mrclass = {20G40 (20G05)},
series = {Astérisque},
ADDRESS={Paris},
PUBLISHER={Math. Soc. France},
volume = {306},
mrnumber = {2274998},
mrreviewer = {Robert B{é}dard},
title = {Sur les Caractères des Groupes Réductifs Finis à Centre Non Connexe: Applications aux Groupes Spéciaux Linéaires et Unitaires},
year = {2006},
pages = {vi+165},
zblnumber = {1157.20022},
} -
[bm]
C. Bonnafé and J. Michel, "Computational proof of the Mackey formula for $q>2$," J. Algebra, vol. 327, pp. 506-526, 2011.
@ARTICLE{bm, mrkey = {2746047},
issn = {0021-8693},
author = {Bonnaf{é},
C{é}dric and Michel, Jean},
mrclass = {20C33 (20G05 20G40)},
doi = {10.1016/j.jalgebra.2010.10.030},
journal = {J. Algebra},
zblnumber = {1231.20041},
volume = {327},
mrnumber = {2746047},
fjournal = {Journal of Algebra},
mrreviewer = {Simon M. Goodwin},
coden = {JALGA4},
title = {Computational proof of the {M}ackey formula for {$q>2$}},
year = {2011},
pages = {506--526},
} -
[BR]
C. Bonnafé and R. Rouquier, "Catégories dérivées et variétés de Deligne-Lusztig," Publ. Math. Inst. Hautes Études Sci., vol. 97, pp. 1-59, 2003.
@ARTICLE{BR, mrkey = {2010739},
volume = {97},
issn = {0073-8301},
author = {Bonnaf{é},
C{é}dric and Rouquier, Rapha{ë}l},
mrclass = {20G05 (18E30 20G40)},
doi = {10.1007/s10240-003-0013-3},
journal = {Publ. Math. Inst. Hautes Études Sci.},
zblnumber = {1054.20024},
mrnumber = {2010739},
fjournal = {Publications Mathématiques. Institut de Hautes Études Scientifiques},
mrreviewer = {Jean Michel},
title = {Catégories dérivées et variétés de {D}eligne-{L}usztig},
year = {2003},
pages = {1--59},
} -
[BR2]
C. Bonnafé and R. Rouquier, "Coxeter orbits and modular representations," Nagoya Math. J., vol. 183, pp. 1-34, 2006.
@ARTICLE{BR2, mrkey = {2253885},
issn = {0027-7630},
author = {Bonnaf{é},
C{é}dric and Rouquier, Rapha{ë}l},
mrclass = {20G05 (18E30 20G40)},
journal = {Nagoya Math. J.},
zblnumber = {1109.20038},
volume = {183},
mrnumber = {2253885},
fjournal = {Nagoya Mathematical Journal},
mrreviewer = {Eric C. Rowell},
coden = {NGMJA2},
title = {Coxeter orbits and modular representations},
year = {2006},
pages = {1--34},
doi = {10.1017/S0027763000009259},
} -
[Bou]
S. Bouc, "Résolutions de foncteurs de Mackey," in Group Representations: Cohomology, Group Actions and Topology, Providence, RI: Amer. Math. Soc., 1998, vol. 63, pp. 31-83.
@INCOLLECTION{Bou, mrkey = {1603131},
author = {Bouc, S.},
mrclass = {20C20 (19A22)},
series = {Proc. Sympos. Pure Math.},
address = {Providence, RI},
publisher = {Amer. Math. Soc.},
doi = {10.1090/pspum/063/1603131},
zblnumber = {0897.19001},
volume = {63},
mrnumber = {1603131},
booktitle = {Group Representations: Cohomology, Group Actions and Topology},
mrreviewer = {Jacques Th{é}venaz},
venue = {{S}eattle, {WA},
1996},
title = {Résolutions de foncteurs de {M}ackey},
pages = {31--83},
year = {1998},
} -
[broueperm]
M. Broué, "On Scott modules and $p$-permutation modules: an approach through the Brauer morphism," Proc. Amer. Math. Soc., vol. 93, iss. 3, pp. 401-408, 1985.
@ARTICLE{broueperm, mrkey = {0773988},
number = {3},
issn = {0002-9939},
author = {Brou{é},
Michel},
mrclass = {20C11 (20C05)},
doi = {10.2307/2045600},
journal = {Proc. Amer. Math. Soc.},
zblnumber = {0574.20005},
volume = {93},
mrnumber = {0773988},
fjournal = {Proceedings of the American Mathematical Society},
mrreviewer = {Peter W. Donovan},
coden = {PAMYAR},
title = {On {S}cott modules and {$p$}-permutation modules: an approach through the {B}rauer morphism},
year = {1985},
pages = {401--408},
} -
[broue]
M. Broué, "Isométries de caractères et équivalences de Morita ou dérivées," Inst. Hautes Études Sci. Publ. Math., vol. 71, pp. 45-63, 1990.
@ARTICLE{broue, mrkey = {1079643},
volume = {71},
issn = {0073-8301},
author = {Brou{é},
Michel},
mrclass = {20C20 (16D90 20C33)},
journal = {Inst. Hautes Études Sci. Publ. Math.},
zblnumber = {0727.20005},
mrnumber = {1079643},
fjournal = {Institut des Hautes Études Scientifiques. Publications Mathématiques},
mrreviewer = {Jeremy Rickard},
coden = {PMIHA6},
title = {Isométries de caractères et équivalences de {M}orita ou dérivées},
year = {1990},
pages = {45--63},
doi = {10.1007/BF02699877},
} -
[BrMi]
M. Broué and J. Michel, "Blocs et séries de Lusztig dans un groupe réductif fini," J. Reine Angew. Math., vol. 395, pp. 56-67, 1989.
@ARTICLE{BrMi, mrkey = {0983059},
issn = {0075-4102},
author = {Brou{é},
Michel and Michel, Jean},
mrclass = {20G05 (20C20)},
doi = {10.1515/crll.1989.395.56},
journal = {J. Reine Angew. Math.},
zblnumber = {0654.20048},
volume = {395},
mrnumber = {0983059},
fjournal = {Journal für die Reine und Angewandte Mathematik},
mrreviewer = {Bhama Srinivasan},
coden = {JRMAA8},
title = {Blocs et séries de {L}usztig dans un groupe réductif fini},
year = {1989},
pages = {56--67},
} -
[Ca]
M. Cabanes, "On Jordan decomposition of characters for ${ SU}(n,q)$," J. Algebra, vol. 374, pp. 216-230, 2013.
@ARTICLE{Ca, mrkey = {2998804},
issn = {0021-8693},
author = {Cabanes, Marc},
mrclass = {20G40 (20G05)},
doi = {10.1016/j.jalgebra.2012.10.015},
journal = {J. Algebra},
zblnumber = {1272.20049},
volume = {374},
mrnumber = {2998804},
fjournal = {Journal of Algebra},
mrreviewer = {Bhama Srinivasan},
coden = {JALGA4},
title = {On {J}ordan decomposition of characters for {${\rm SU}(n,q)$}},
year = {2013},
pages = {216--230},
} -
[CaEn1]
M. Cabanes and M. Enguehard, "On blocks of finite reductive groups and twisted induction," Adv. Math., vol. 145, iss. 2, pp. 189-229, 1999.
@ARTICLE{CaEn1, mrkey = {1704575},
number = {2},
issn = {0001-8708},
author = {Cabanes, Marc and Enguehard, Michel},
mrclass = {20C33 (20C20 20G40)},
doi = {10.1006/aima.1998.1814},
journal = {Adv. Math.},
zblnumber = {0954.20023},
volume = {145},
mrnumber = {1704575},
fjournal = {Advances in Mathematics},
mrreviewer = {Bhama Srinivasan},
coden = {ADMTA4},
title = {On blocks of finite reductive groups and twisted induction},
year = {1999},
pages = {189--229},
} -
[CaEnbook]
M. Cabanes and M. Enguehard, Representation Theory of Finite Reductive Groups, Cambridge: Cambridge Univ. Press, 2004, vol. 1.
@BOOK{CaEnbook, mrkey = {2057756},
author = {Cabanes, Marc and Enguehard, Michel},
mrclass = {20G05 (20G40)},
series = {New Math. Monogr.},
isbn = {0-521-82517-2},
address = {Cambridge},
publisher = {Cambridge Univ. Press},
doi = {10.1017/CBO9780511542763},
zblnumber = {1069.20032},
volume = {1},
mrnumber = {2057756},
mrreviewer = {Bhama Srinivasan},
title = {Representation Theory of Finite Reductive Groups},
year = {2004},
pages = {xviii+436},
} -
[Dat] J. -F. Dat, Equivalences of tame blocks for $p$-adic linear groups, 2016.
@MISC{Dat,
author = {Dat, J.-F.},
arxiv = {1603.07226},
title = {Equivalences of tame blocks for $p$-adic linear groups},
year = {2016},
} -
[DL]
P. Deligne and G. Lusztig, "Representations of reductive groups over finite fields," Ann. of Math., vol. 103, iss. 1, pp. 103-161, 1976.
@ARTICLE{DL, mrkey = {0393266},
number = {1},
issn = {0003-486X},
author = {Deligne, P. and Lusztig, G.},
mrclass = {20G05 (14M15)},
doi = {10.2307/1971021},
journal = {Ann. of Math.},
zblnumber = {0336.20029},
volume = {103},
mrnumber = {0393266},
fjournal = {Annals of Mathematics. Second Series},
mrreviewer = {S. I. Gelfand},
title = {Representations of reductive groups over finite fields},
year = {1976},
pages = {103--161},
} -
[dmbook]
F. Digne and J. Michel, Representations of Finite Groups of Lie Type, Cambridge: Cambridge Univ. Press, 1991, vol. 21.
@BOOK{dmbook, mrkey = {1118841},
author = {Digne, Fran{ç}ois and Michel, Jean},
mrclass = {20G05 (20-02 20C33 20G40)},
series = {London Math. Soc. Stud. Texts},
isbn = {0-521-40117-8; 0-521-40648-X},
address = {Cambridge},
publisher = {Cambridge Univ. Press},
doi = {10.1017/CBO9781139172417},
zblnumber = {0815.20014},
volume = {21},
mrnumber = {1118841},
mrreviewer = {Bhama Srinivasan},
title = {Representations of Finite Groups of {L}ie Type},
year = {1991},
pages = {iv+159},
} -
[dm-nonc]
F. Digne and J. Michel, "Groupes réductifs non connexes," Ann. Sci. École Norm. Sup., vol. 27, iss. 3, pp. 345-406, 1994.
@ARTICLE{dm-nonc, mrkey = {1272294},
number = {3},
issn = {0012-9593},
author = {Digne, Fran{ç}ois and Michel, Jean},
mrclass = {20G05},
url = {http://www.numdam.org/item?id=ASENS_1994_4_27_3_345_0},
journal = {Ann. Sci. École Norm. Sup.},
zblnumber = {0846.20040},
volume = {27},
mrnumber = {1272294},
fjournal = {Annales Scientifiques de l'École Normale Supérieure. Quatrième Série},
mrreviewer = {Bhama Srinivasan},
coden = {ASENAH},
title = {Groupes réductifs non connexes},
year = {1994},
pages = {345--406},
} -
[dm-nonc-2]
F. Digne and J. Michel, "Complements on disconnected reductive groups," Pacific J. Math., vol. 279, iss. 1-2, pp. 203-228, 2015.
@ARTICLE{dm-nonc-2, mrkey = {3437776},
number = {1-2},
issn = {0030-8730},
author = {Digne, Fran{ç}ois and Michel, Jean},
mrclass = {20G15 (20C33 20G05 20G40)},
doi = {10.2140/pjm.2015.279.203},
journal = {Pacific J. Math.},
zblnumber = {06526352},
volume = {279},
mrnumber = {3437776},
fjournal = {Pacific Journal of Mathematics},
title = {Complements on disconnected reductive groups},
year = {2015},
pages = {203--228},
} -
[DMR]
F. Digne, J. Michel, and R. Rouquier, "Cohomologie des variétés de Deligne-Lusztig," Adv. Math., vol. 209, iss. 2, pp. 749-822, 2007.
@ARTICLE{DMR, mrkey = {2296313},
number = {2},
issn = {0001-8708},
author = {Digne, Fran{ç}ois and Michel, Jean and Rouquier, Rapha{ë}l},
mrclass = {20C08 (14F43 20G40)},
doi = {10.1016/j.aim.2006.06.001},
journal = {Adv. Math.},
zblnumber = {1118.20006},
volume = {209},
mrnumber = {2296313},
fjournal = {Advances in Mathematics},
coden = {ADMTA4},
title = {Cohomologie des variétés de {D}eligne-{L}usztig},
year = {2007},
pages = {749--822},
} -
[DiDu]
R. Dipper and J. Du, "Harish-Chandra vertices," J. Reine Angew. Math., vol. 437, pp. 101-130, 1993.
@ARTICLE{DiDu, mrkey = {1212254},
issn = {0075-4102},
author = {Dipper, Richard and Du, Jie},
mrclass = {20C20},
doi = {10.1515/crll.1993.437.101},
journal = {J. Reine Angew. Math.},
zblnumber = {0768.20004},
volume = {437},
mrnumber = {1212254},
fjournal = {Journal für die Reine und Angewandte Mathematik. [Crelle's Journal]},
mrreviewer = {Bhama Srinivasan},
title = {Harish-{C}handra vertices},
year = {1993},
pages = {101--130},
} -
[DuRou]
O. Dudas and R. Rouquier, "Coxeter orbits and Brauer trees III," J. Amer. Math. Soc., vol. 27, iss. 4, pp. 1117-1145, 2014.
@ARTICLE{DuRou, mrkey = {3230819},
number = {4},
issn = {0894-0347},
author = {Dudas, Olivier and Rouquier, Rapha{ë}l},
mrclass = {20C33 (18E30 20C20)},
doi = {10.1090/S0894-0347-2014-00791-8},
journal = {J. Amer. Math. Soc.},
zblnumber = {1307.20010},
volume = {27},
mrnumber = {3230819},
fjournal = {Journal of the American Mathematical Society},
mrreviewer = {Frank Himstedt},
title = {Coxeter orbits and {B}rauer trees {III}},
year = {2014},
pages = {1117--1145},
} -
[HoLe]
R. B. Howlett and G. I. Lehrer, "On Harish-Chandra induction and restriction for modules of Levi subgroups," J. Algebra, vol. 165, iss. 1, pp. 172-183, 1994.
@ARTICLE{HoLe, mrkey = {1272585},
number = {1},
issn = {0021-8693},
author = {Howlett, R. B. and Lehrer, G. I.},
mrclass = {20C33 (20C20)},
doi = {10.1006/jabr.1994.1104},
journal = {J. Algebra},
zblnumber = {0802.20036},
volume = {165},
mrnumber = {1272585},
fjournal = {Journal of Algebra},
mrreviewer = {Peter Fleischmann},
coden = {JALGA4},
title = {On {H}arish-{C}handra induction and restriction for modules of {L}evi subgroups},
year = {1994},
pages = {172--183},
} -
[KeMa]
R. Kessar and G. Malle, "Quasi-isolated blocks and Brauer’s height zero conjecture," Ann. of Math., vol. 178, iss. 1, pp. 321-384, 2013.
@ARTICLE{KeMa, mrkey = {3043583},
number = {1},
issn = {0003-486X},
author = {Kessar, Radha and Malle, Gunter},
mrclass = {20C20},
doi = {10.4007/annals.2013.178.1.6},
journal = {Ann. of Math.},
zblnumber = {1317.20006},
volume = {178},
mrnumber = {3043583},
fjournal = {Annals of Mathematics. Second Series},
mrreviewer = {Gabriel Navarro},
coden = {ANMAAH},
title = {Quasi-isolated blocks and {B}rauer's height zero conjecture},
year = {2013},
pages = {321--384},
} -
[KeMa2] R. Kessar and G. Malle, Brauer’s height zero conjecture for quasi-simple groups, 2015.
@MISC{KeMa2,
author = {Kessar, Radha and Malle, Gunter},
title = {Brauer's height zero conjecture for quasi-simple groups},
year = {2015},
ARXIV = {1510.07907},
} -
[L]
G. Lusztig, Representations of Finite Chevalley Groups, Providence, RI: Amer. Math. Soc., 1978, vol. 39.
@BOOK{L, mrkey = {0518617},
author = {Lusztig, George},
mrclass = {20G05},
series = {CBMS Reg. Conf. Ser. Mat.},
isbn = {0-8218-1689-6},
address = {Providence, RI},
publisher = {Amer. Math. Soc.},
volume = {39},
mrnumber = {0518617},
mrreviewer = {David B. Surowski},
title = {Representations of Finite {C}hevalley Groups},
year = {1978},
pages = {v+48},
zblnumber = {0418.20037},
doi = {10.1090/cbms/039},
} -
[N] T. H. Nguyen, Cohomologie des variétés de Coxeter pour le groupe linéaire: algèbre d’endomorphismes, compactification.
@MISC{N,
author = {Nguyen, T. H.},
note = {{Ph.D}. thesis (in preparation)},
title = {Cohomologie des variétés de {C}oxeter pour le groupe linéaire: algèbre d'endomorphismes, compactification},
} -
[Pu] L. Puig, On the Local Structure of Morita and Rickard Equivalences Between Brauer Blocks, Birkhäuser, 1999.
@BOOK{Pu, mrkey = {1707300},
number = {178},
author = {Puig, Llu{\'ı}s},
mrclass = {20C11 (16D90)},
series = {Progr. Math.},
addres = {Basel},
isbn = {3-7643-6156-5},
publisher = {Birkhäuser},
zblnumber = {0929.20012},
mrnumber = {1707300},
mrreviewer = {Jeremy Rickard},
title = {On the Local Structure of {M}orita and {R}ickard Equivalences Between {B}rauer Blocks},
year = {1999},
pages = {vi+261},
} -
[Ri]
J. Rickard, "Finite group actions and étale cohomology," Inst. Hautes Études Sci. Publ. Math., vol. 80, pp. 81-94 (1995), 1994.
@ARTICLE{Ri, mrkey = {1320604},
volume = {80},
issn = {0073-8301},
author = {Rickard, Jeremy},
mrclass = {14F20},
journal = {Inst. Hautes Études Sci. Publ. Math.},
zblnumber = {0838.14013},
mrnumber = {1320604},
fjournal = {Institut des Hautes Études Scientifiques. Publications Mathématiques},
mrreviewer = {Claudio Pedrini},
coden = {PMIHA6},
title = {Finite group actions and étale cohomology},
year = {1994},
pages = {81--94 (1995)},
doi = {10.1007/BF02698896},
} -
[Ri2]
J. Rickard, "Splendid equivalences: derived categories and permutation modules," Proc. London Math. Soc., vol. 72, iss. 2, pp. 331-358, 1996.
@ARTICLE{Ri2, mrkey = {1367082},
number = {2},
issn = {0024-6115},
author = {Rickard, Jeremy},
mrclass = {20C20 (18E30 20J05)},
doi = {10.1112/plms/s3-72.2.331},
journal = {Proc. London Math. Soc.},
zblnumber = {0862.20010},
volume = {72},
mrnumber = {1367082},
fjournal = {Proceedings of the London Mathematical Society. Third Series},
mrreviewer = {Katsuhiro Uno},
coden = {PLMTAL},
title = {Splendid equivalences: derived categories and permutation modules},
year = {1996},
pages = {331--358},
} -
[Rou1]
R. Rouquier, "Complexes de cha\^\i nes étales et courbes de Deligne-Lusztig," J. Algebra, vol. 257, iss. 2, pp. 482-508, 2002.
@ARTICLE{Rou1, mrkey = {1947973},
number = {2},
issn = {0021-8693},
author = {Rouquier, Rapha{ë}l},
mrclass = {20C33 (14F05 18E30 20G40)},
doi = {10.1016/S0021-8693(02)00530-6},
journal = {J. Algebra},
zblnumber = {1030.18008},
volume = {257},
mrnumber = {1947973},
fjournal = {Journal of Algebra},
mrreviewer = {Jean Michel},
coden = {JALGA4},
title = {Complexes de cha\^\i nes étales et courbes de {D}eligne-{L}usztig},
year = {2002},
pages = {482--508},
} -
[Rou2] R. Rouquier, "Block theory via stable and Rickard equivalences," in Modular Representation Theory of Finite Groups, de Gruyter, Berlin, 2001, pp. 101-146.
@INCOLLECTION{Rou2, mrkey = {1889341},
author = {Rouquier, Rapha{ë}l},
mrclass = {20C20 (18E30)},
publisher = {de Gruyter, Berlin},
zblnumber = {0998.20006},
mrnumber = {1889341},
booktitle = {Modular Representation Theory of Finite Groups},
venue = {{C}harlottesville, {VA},
1998},
mrreviewer = {Thorsten Holm},
title = {Block theory via stable and {R}ickard equivalences},
pages = {101--146},
year = {2001},
} -
[Rou3] R. Rouquier, Finite generation of cohomology of finite groups, 2014.
@MISC{Rou3,
author = {Rouquier, Rapha{ë}l},
note = {preprint},
title = {Finite generation of cohomology of finite groups},
year = {2014},
arxiv = {1608.03579},
} -
@BOOK{Springer, mrkey = {1642713},
author = {Springer, T. A.},
mrclass = {20G15 (14L10)},
series = {Progr. Math.},
edition = {Second},
isbn = {0-8176-4021-5},
address = {Boston},
publisher = {Birkhäuser},
doi = {10.1007/978-0-8176-4840-4},
volume = {9},
mrnumber = {1642713},
mrreviewer = {Andy R. Magid},
title = {Linear Algebraic Groups},
year = {1998},
pages = {xiv+334},
zblnumber = {0927.20024},
} -
[steinberg] R. Steinberg, Endomorphisms of Linear Algebraic Groups, Providence, RI: Amer. Math. Soc., 1968, vol. 80.
@BOOK{steinberg, mrkey = {0230728},
author = {Steinberg, Robert},
mrclass = {14.50 (22.00)},
series = {Mem. Amer. Math. Soc.},
address = {Providence, RI},
publisher = {Amer. Math. Soc.},
zblnumber = {0164.02902},
volume = {80},
mrnumber = {0230728},
mrreviewer = {E. Abe},
title = {Endomorphisms of Linear Algebraic Groups},
year = {1968},
pages = {108},
} -
[Th] J. Thévenaz, $G$-Algebras and Modular Representation Theory, New York: The Clarendon Press, Oxford Univ. Press, 1995.
@BOOK{Th, mrkey = {1365077},
author = {Th{é}venaz, Jacques},
mrclass = {20C20 (20-02 20C05)},
series = {Oxford Math. Monogr.},
isbn = {0-19-853587-2},
address = {New York},
publisher = {The Clarendon Press, Oxford Univ. Press},
zblnumber = {0837.20015},
mrnumber = {1365077},
mrreviewer = {Burkhard K{ü}lshammer},
title = {{$G$}-Algebras and Modular Representation Theory},
year = {1995},
pages = {xxviii+470},
} -
[Tho]
R. W. Thomason, "The classification of triangulated subcategories," Compositio Math., vol. 105, iss. 1, pp. 1-27, 1997.
@ARTICLE{Tho, mrkey = {1436741},
number = {1},
issn = {0010-437X},
author = {Thomason, R. W.},
mrclass = {18E30 (18F30)},
doi = {10.1023/A:1017932514274},
journal = {Compositio Math.},
zblnumber = {0873.18003},
volume = {105},
mrnumber = {1436741},
fjournal = {Compositio Mathematica},
mrreviewer = {Steven E. Landsburg},
coden = {CMPMAF},
title = {The classification of triangulated subcategories},
year = {1997},
pages = {1--27},
} -
[W] H. Wang, L’espace symétrique de Drinfeld et correspondance de Langlands locale II, 2014.
@misc{W,
author = {Wang, Haoran},
title = {L'espace symétrique de {D}rinfeld et correspondance de {L}anglands locale {II}},
year = {2014},
arxiv={1402.1265},
}