Subregular nilpotent representations of Lie algebras in prime characteristic

Jantzen, Jens

doi:10.1090/S1088-4165-99-00073-4

Subregular nilpotent representations of Lie algebras in prime characteristic
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by Jens Carsten Jantzen PDF

Represent. Theory 3 (1999), 153-222 Request permission

Abstract:

We look in this paper at representations of Lie algebras of simple reductive groups in prime characteristic. We investigate those modules that have a subregular nilpotent $p$–character. In case all roots in the corresponding root system have the same length, we determine all simple modules in generic blocks as well as the Cartan matrices of these blocks. Our results confirm conjectures by Lusztig. We determine in these cases also extension groups between non-isomorphic simple modules. There are similar, somewhat less detailed results on non-generic blocks and the cases with two root lengths.

References

N. Bourbaki, Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1337, Hermann, Paris, 1968 (French). MR 0240238
K. A. Brown and I. Gordon, The ramification of centres: Lie algebras in positive characteristic and quantised enveloping algebras, preprint, April 1999.
Roger W. Carter, Finite groups of Lie type, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1985. Conjugacy classes and complex characters; A Wiley-Interscience Publication. MR 794307
Jacques Dixmier, Algèbres enveloppantes, Cahiers Scientifiques, Fasc. XXXVII, Gauthier-Villars Éditeur, Paris-Brussels-Montreal, Que., 1974 (French). MR 0498737
James Franklin, Homomorphisms between Verma modules in characteristic $p$, J. Algebra 112 (1988), no. 1, 58–85. MR 921964, DOI 10.1016/0021-8693(88)90132-9
Eric M. Friedlander and Brian J. Parshall, Modular representation theory of Lie algebras, Amer. J. Math. 110 (1988), no. 6, 1055–1093. MR 970120, DOI 10.2307/2374686
Eric M. Friedlander and Brian J. Parshall, Deformations of Lie algebra representations, Amer. J. Math. 112 (1990), no. 3, 375–395. MR 1055649, DOI 10.2307/2374747
J. E. Humphreys, Modular representations of simple Lie algebras, Bull. Amer. Math. Soc. (N.S.) 35 (1998), no. 2, 105–122. MR 1605399, DOI 10.1090/S0273-0979-98-00749-6
Jens Carsten Jantzen, Representations of algebraic groups, Pure and Applied Mathematics, vol. 131, Academic Press, Inc., Boston, MA, 1987. MR 899071
J. C. Jantzen, Subregular nilpotent representations of $\mathfrak {sl}_{n}$ and $\mathfrak {so}_{2n+1}$, Math. Proc. Cambridge Philos. Soc. 126 (1999), 223–257.
Jens Carsten Jantzen, Representations of Lie algebras in prime characteristic, Representation theories and algebraic geometry (Montreal, PQ, 1997) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 514, Kluwer Acad. Publ., Dordrecht, 1998, pp. 185–235. Notes by Iain Gordon. MR 1649627, DOI 10.1007/978-94-015-9131-7_{5}
J. C. Jantzen, Modular representations of reductive Lie algebras, J. Pure Appl. Algebra (to appear).
V. Kac and B. Weisfeiler, Coadjoint action of a semi-simple algebraic group and the center of the enveloping algebra in characteristic $p$, Nederl. Akad. Wetensch. Proc. Ser. A 79=Indag. Math. 38 (1976), no. 2, 136–151. MR 0417308, DOI 10.1016/1385-7258(76)90059-7
G. Lusztig, Periodic $W$-graphs, Represent. Theory 1 (1997), 207–279. MR 1464171, DOI 10.1090/S1088-4165-97-00033-2
G. Lusztig, Bases in equivariant $K$–theory, Represent. Theory 2 (1998), 298–369.
G. Lusztig, Subregular elements and bases in $K$–theory, Canad. J. Math. (to appear).
G. Lusztig, Representation theory in characteristic $p$, Lecture at the Taniguchi conference, Nara 1998, preprint, December 1998.
G. Lusztig, Bases in equivariant $K$–theory II, preprint, March 1999.
Richard D. Pollack, Restricted Lie algebras of bounded type, Bull. Amer. Math. Soc. 74 (1968), 326–331. MR 219582, DOI 10.1090/S0002-9904-1968-11943-3
Peter Slodowy, Simple singularities and simple algebraic groups, Lecture Notes in Mathematics, vol. 815, Springer, Berlin, 1980. MR 584445, DOI 10.1007/BFb0090294
T. A. Springer and R. Steinberg, Conjugacy classes, Seminar on Algebraic Groups and Related Finite Groups (The Institute for Advanced Study, Princeton, N.J., 1968/69) Lecture Notes in Mathematics, Vol. 131, Springer, Berlin, 1970, pp. 167–266. MR 0268192, DOI 10.1007/BFb0081546