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Subregular nilpotent representations of Lie algebras in prime characteristic

Author(s): Jens Carsten Jantzen
Journal: Represent. Theory 3 (1999), 153-222.
MSC (1991): Primary 17B10; Secondary 17B20, 17B45, 17B50
Posted: July 19, 1999
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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:

[Bou]
N. Bourbaki, Groupes et algèbres de Lie, Chap. 4, 5 et 6, Hermann, Paris, 1968. MR 39:1590

[BG]
K. A. Brown and I. Gordon, The ramification of centres: Lie algebras in positive characteristic and quantised enveloping algebras, preprint, April 1999.

[C]
R. W. Carter, Finite Groups of Lie Type: Conjugacy Classes and Complex Characters, Wiley, Chichester, 1985. MR 87d:20060

[D]
J. Dixmier, Algèbres enveloppantes, Gauthier-Villars, Paris, 1974. MR 58:16803a

[F]
J. Franklin, Homomorphisms between Verma modules in characteristic $p$, J. Algebra 112 (1988), 58-85. MR 89c:17015

[FP1]
E. M. Friedlander, B. J. Parshall, Modular representation theory of Lie algebras, Amer. J. Math. 110 (1988), 1055-1093. MR 89j:17015

[FP2]
E. M. Friedlander, B. J. Parshall, Deformations of Lie algebra representations, Amer. J. Math. 112 (1990), 375-395. MR 91e:17012

[H]
J. E. Humphreys, Modular representations of simple Lie algebras, Bull. Amer. Math Soc. (N.S.) 35 (1998), 105-122. MR 98k:17007

[J1]
J. C. Jantzen, Representations of Algebraic Groups, (Pure and Applied Mathematics 131), Academic, Orlando, 1987. MR 89c:20001

[J2]
J. C. Jantzen, Subregular nilpotent representations of $\mathfrak s\mathfrak l_{n}$ and $\mathfrak s\mathfrak o_{2n+1}$, Math. Proc. Cambridge Philos. Soc. 126 (1999), 223-257. CMP 99:07

[J3]
J. C. Jantzen, Representations of Lie algebras in prime characteristic (A. Broer, ed.), Representation Theories and Algebraic Geometry, Proceedings Montréal 1997 (NATO ASI series C 514), Kluwer, Dordrecht, 1998, pp. 185-235. MR 99h:17026

[J4]
J. C. Jantzen, Modular representations of reductive Lie algebras, J. Pure Appl. Algebra (to appear).

[KW]
V. Kac, B. Weisfeiler, Coadjoint action of a semi-simple algebraic group and the center of the enveloping algebra in characteristic $p$, Indag. Math. 38 (1976), 136-151. MR 54:5364

[L1]
G. Lusztig, Periodic $W$-graphs, Represent. Theory 1 (1997), 207-279. MR 99a:20042

[L2]
G. Lusztig, Bases in equivariant $K$-theory, Represent. Theory 2 (1998), 298-369. CMP 98:16

[L3]
G. Lusztig, Subregular elements and bases in $K$-theory, Canad. J. Math. (to appear).

[L4]
G. Lusztig, Representation theory in characteristic $p$, Lecture at the Taniguchi conference, Nara 1998, preprint, December 1998.

[L5]
G. Lusztig, Bases in equivariant $K$-theory II, preprint, March 1999.

[P]
R. D. Pollack, Restricted Lie algebras of bounded type, Bull. Amer. Math Soc. 74 (1968), 326-331. MR 36:2661

[Sl]
P. Slodowy, Simple Singularities and Simple Algebraic Groups, Lecture Notes in Mathematics 815, Springer, Berlin, 1980. MR 82g:14037

[SpSt]
T. A. Springer and R. Steinberg, Conjugacy classes, A. Borel et al, Seminar on Algebraic Groups and Related Finite Groups (Lecture Notes in Mathematics 131), Springer, Berlin, 1970, pp. 167-266. MR 42:3091

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Additional Information:

Jens Carsten Jantzen
Affiliation: Matematisk Institut, Aarhus Universitet, Ny Munkegade, DK-8000 Aarhus C, Denmark
Email: jantzen@imf.au.dk

DOI: 10.1090/S1088-4165-99-00073-4
PII: S 1088-4165(99)00073-4
Received by editor(s): May 3, 1999
Received by editor(s) in revised form: June 9, 1999
Posted: July 19, 1999
Copyright of article: Copyright 1999, American Mathematical Society

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