In the late 1980s, Friedlander and Parshall studied the representations of a family of algebras which were obtained as deformations of the distribution algebra of the first Frobenius kernel of an algebraic group. The representation theory of these algebras tells us much about the representation theory of Lie algebras in positive characteristic. We develop an analogue of this family of algebras for the distribution algebras of the higher Frobenius kernels, answering a 30 year old question posed by Friedlander and Parshall. We also examine their representation theory in the case of the special linear group.
References
[1] P. Berthelot, $\mathcal{D}$-modules arithmétiques. I. Opérateurs différentiels de niveau fini, Ann. Sci. École. Norm. Sup. (4), 29 (1996), 185–272.[1] P. Berthelot, $\mathcal{D}$-modules arithmétiques. I. Opérateurs différentiels de niveau fini, Ann. Sci. École. Norm. Sup. (4), 29 (1996), 185–272.
[2] R. Bezrukavnikov, I. Mirković and D. Rumynin, Localization of modules for a semisimple Lie algebra in prime characteristic, Ann. of Math. (2), 167 (2008), 945–991. 1220.17009 10.4007/annals.2008.167.945[2] R. Bezrukavnikov, I. Mirković and D. Rumynin, Localization of modules for a semisimple Lie algebra in prime characteristic, Ann. of Math. (2), 167 (2008), 945–991. 1220.17009 10.4007/annals.2008.167.945
[4] S. Chamberlin, Integral bases for the universal enveloping algebras of map algebras, J. Algebra, 377 (2013), 232–249. MR3008904 1293.17013 10.1016/j.jalgebra.2012.11.046[4] S. Chamberlin, Integral bases for the universal enveloping algebras of map algebras, J. Algebra, 377 (2013), 232–249. MR3008904 1293.17013 10.1016/j.jalgebra.2012.11.046
[5] A. Grothendieck and J. Dieudonné, Éléments de Géométrie Algébrique, Publ. Math. Inst. Hautes Études Sci., 4 (1960); 8 (1961); 11 (1961); 17 (1963); 20 (1964); 24 (1965); 28 (1966); 32 (1967).[5] A. Grothendieck and J. Dieudonné, Éléments de Géométrie Algébrique, Publ. Math. Inst. Hautes Études Sci., 4 (1960); 8 (1961); 11 (1961); 17 (1963); 20 (1964); 24 (1965); 28 (1966); 32 (1967).
[6] E. Friedlander and B. Parshall, Modular representation theory of Lie algebras, Amer. J. Math., 110 (1988), 1055–1093. 0673.17010 10.2307/2374686[6] E. Friedlander and B. Parshall, Modular representation theory of Lie algebras, Amer. J. Math., 110 (1988), 1055–1093. 0673.17010 10.2307/2374686
[7] E. Friedlander and B. Parshall, Deformations of Lie algebra representations, Amer. J. Math., 112 (1990), 375–395. 0714.17007 10.2307/2374747[7] E. Friedlander and B. Parshall, Deformations of Lie algebra representations, Amer. J. Math., 112 (1990), 375–395. 0714.17007 10.2307/2374747
[8] Y. Hashimoto, M. Kaneda and D. Rumynin, On localization of $\overline{D}$-modules, In: Representations of Algebraic Groups, Quantum Groups and Lie Algebras, Contemp. Math., 413, Amer. Math. Soc., Providence, RI, 2006, 43–62. 1121.14041[8] Y. Hashimoto, M. Kaneda and D. Rumynin, On localization of $\overline{D}$-modules, In: Representations of Algebraic Groups, Quantum Groups and Lie Algebras, Contemp. Math., 413, Amer. Math. Soc., Providence, RI, 2006, 43–62. 1121.14041
[9] J. Humphreys, Modular representations of simple Lie algebras, Bull. Amer. Math. Soc. (N.S.), 35 (1998), 105–122. 0962.17013 10.1090/S0273-0979-98-00749-6[9] J. Humphreys, Modular representations of simple Lie algebras, Bull. Amer. Math. Soc. (N.S.), 35 (1998), 105–122. 0962.17013 10.1090/S0273-0979-98-00749-6
[10] J. Jantzen, Representations of Algebraic Groups, Pure Appl. Math., 131, Academic Press, 1987. 0654.20039[10] J. Jantzen, Representations of Algebraic Groups, Pure Appl. Math., 131, Academic Press, 1987. 0654.20039
[11] J. Jantzen, Representations of Lie algebras in prime characteristic, In: Representation Theories and Algebraic Geometry, Montreal, 1997, (ed. A. Broer), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 514, Kluwer, Dordrecht, 1998, 185–235.[11] J. Jantzen, Representations of Lie algebras in prime characteristic, In: Representation Theories and Algebraic Geometry, Montreal, 1997, (ed. A. Broer), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 514, Kluwer, Dordrecht, 1998, 185–235.
[12] J. Jantzen, Representations of Lie algebras in positive characteristic, In: Representation Theory of Algebraic Groups and Quantum Groups, Adv. Stud. Pure Math., 40, Math. Soc. Japan, Tokyo, 2004, 175–218. 1104.17010[12] J. Jantzen, Representations of Lie algebras in positive characteristic, In: Representation Theory of Algebraic Groups and Quantum Groups, Adv. Stud. Pure Math., 40, Math. Soc. Japan, Tokyo, 2004, 175–218. 1104.17010
[13] V. Kac and 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. 0324.17001[13] V. Kac and 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. 0324.17001
[14] M. Kaneda and J. Ye, Equivariant localization of $\overline{D}$-modules on the flag variety of the symplectic group of degree 4, J. Algebra, 309 (2007), 236–281. 1133.20038 10.1016/j.jalgebra.2006.07.023[14] M. Kaneda and J. Ye, Equivariant localization of $\overline{D}$-modules on the flag variety of the symplectic group of degree 4, J. Algebra, 309 (2007), 236–281. 1133.20038 10.1016/j.jalgebra.2006.07.023
[15] B. Kostant, Groups over $Z$, In: Algebraic Groups and Discontinuous Subgroups, Proc. Sympos. Pure Math., 9, Amer. Math. Soc., Providence, RI, 1966, 90–98.[15] B. Kostant, Groups over $Z$, In: Algebraic Groups and Discontinuous Subgroups, Proc. Sympos. Pure Math., 9, Amer. Math. Soc., Providence, RI, 1966, 90–98.
[16] A. Premet, Irreducible representations of Lie algebras of reductive groups and the Kac–Weisfeiler conjecture, Invent. Math., 121 (1995), 79–117. 0828.17008 10.1007/BF01884291[16] A. Premet, Irreducible representations of Lie algebras of reductive groups and the Kac–Weisfeiler conjecture, Invent. Math., 121 (1995), 79–117. 0828.17008 10.1007/BF01884291
[17] A. N. Rudakov, On representations of classical semisimple Lie algebras of characteristic $p$, Izv. Akad. Nauk. SSSR Ser. Mat., 34 (1970), 735–743.[17] A. N. Rudakov, On representations of classical semisimple Lie algebras of characteristic $p$, Izv. Akad. Nauk. SSSR Ser. Mat., 34 (1970), 735–743.
[18] H. Schneider, Representation theory of Hopf Galois extensions, Israel J. Math., 72 (1990), 196–231. 0751.16015 10.1007/BF02764620[18] H. Schneider, Representation theory of Hopf Galois extensions, Israel J. Math., 72 (1990), 196–231. 0751.16015 10.1007/BF02764620
[19] H. Schneider, Some remarks on exact sequences of quantum groups, Comm. Algebra, 21 (1993), 3337–3357. 0801.16040 10.1080/00927879308824733[19] H. Schneider, Some remarks on exact sequences of quantum groups, Comm. Algebra, 21 (1993), 3337–3357. 0801.16040 10.1080/00927879308824733
[20] H. Schneider, Hopf Galois extensions, crossed products, and Clifford theory, In: Advances in Hopf Algebras, Lecture Notes in Pure and Appl. Math., 158, Dekker, New York, 1994, 267–297. 0817.16017[20] H. Schneider, Hopf Galois extensions, crossed products, and Clifford theory, In: Advances in Hopf Algebras, Lecture Notes in Pure and Appl. Math., 158, Dekker, New York, 1994, 267–297. 0817.16017
[21] B. Weisfeiler and V. Kac, The irreducible representations of Lie $p$-algebras, Funktsional. Anal. i Prilozhen., 5 (1971), no. 2, 28–36. 0237.17003 10.1007/BF01076415[21] B. Weisfeiler and V. Kac, The irreducible representations of Lie $p$-algebras, Funktsional. Anal. i Prilozhen., 5 (1971), no. 2, 28–36. 0237.17003 10.1007/BF01076415
[22] M. Westaway, Higher deformations of Lie algebra representations II, arXiv:1904.10860, to appear. 1904.10860 1366.19002 10.1016/j.jalgebra.2017.03.024[22] M. Westaway, Higher deformations of Lie algebra representations II, arXiv:1904.10860, to appear. 1904.10860 1366.19002 10.1016/j.jalgebra.2017.03.024
[23] S. Witherspoon, Clifford correspondence for finite dimensional Hopf algebras, J. Algebra, 218 (1999), 608–620. 0941.16026 10.1006/jabr.1999.7866[23] S. Witherspoon, Clifford correspondence for finite dimensional Hopf algebras, J. Algebra, 218 (1999), 608–620. 0941.16026 10.1006/jabr.1999.7866
