Analogues of Weyl’s formula for reduced enveloping algebras.(English)Zbl 1162.17302

Summary: In this paper, we study simple modules for a reduced enveloping algebra $$U_\chi(\mathfrak g)$$ in the critical case when $$\chi\in\mathfrak g^*$$ is “nilpotent”. Some dimension formulas computed by Jantzen suggest modified versions of Weyl’s dimension formula, based on certain reflecting hyperplanes for the affine Weyl group which might be associated to Kazhdan-Lusztig cells.

MSC:

 17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) 17B35 Universal enveloping (super)algebras 17B45 Lie algebras of linear algebraic groups
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References:

 [1] Andersen H. H., Astérisque 220 (1994) [2] Bezrukavnikov R., ”Localization of Modules for a Semisimple Lie Algebra in Prime Characteristic.” · Zbl 1220.17009 [3] Brown K. A., Math. Z. 238 pp 733– (2001) · Zbl 1037.17011 [4] Friedlander E. M., Amer. J. Math. pp 375– (1990) · Zbl 0714.17007 [5] Humphreys J. E., Bull. Amer. Math. Soc. (N.S.) 35 pp 105– (1998) · Zbl 0962.17013 [6] Jantzen J. C., Representations of Algebraic Groups. (1987) · Zbl 0654.20039 [7] Jantzen J. C., Aarhus Univ. Preprint Series 13 (1997) [8] Jantzen J. C., Representation Theories and Algebraic Geometry (Montreal, 1997) pp 185– (1998) [9] Jantzen J. C., Math. Proc. Cambridge Philos. Soc. 126 pp 223– (1999) · Zbl 0939.17008 [10] Jantzen J. C., Represent. Theory pp 153– (1999) · Zbl 0998.17003 [11] Jantzen J. C., J. Pure Appl. Algebra 152 pp 133– (2000) · Zbl 0976.17004 [12] Lusztig G., J. Fac. Sci. Univ. Tokyo Sect. IA Math. 36 pp 297– (1989) [13] Lusztig G., Represent. Theory pp 207– (1997) · Zbl 0895.20031 [14] Lusztig G., Represent. Theory pp 298– (1998) · Zbl 0901.20034 [15] Lusztig G., Canad. J. Math. 51 pp 1194– (1999) · Zbl 0976.19002 [16] Lusztig G., Represent. Theory pp 281– (1999) · Zbl 0999.20036 [17] Lusztig, G. ”Representation Theory in Characteristic p. Taniguchi Conference On Mathematics Nara ’98. pp.167–178. Tokyo: Kinokuniya. [Lusztig 01], Adv. Stud. Pure Math. 31 · Zbl 0998.17005 [18] Lusztig G., Adv. in Math. 72 pp 284– (1988) · Zbl 0664.20028 [19] Mirković I., Transform. Groups pp 175– (2001) · Zbl 1026.17023 [20] Premet A., Invent Math. 121 pp 79– (1995) · Zbl 0828.17008 [21] Shi Jian Yi, The Kazhdan–Lusztig Cells in Certain Affine Weyl Groups (1986) · Zbl 0582.20030 [22] Steinberg R., Nagoya Math. J. 22 pp 33– (1963)
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