Document Zbl 1252.17011 - zbMATH Open
Differentiating the Weyl generic dimension formula with applications to support varieties. (English) Zbl 1252.17011
Summary: The authors compute the support varieties of all irreducible modules for the small quantum group \(u_\zeta (\mathfrak g)\), where \(\mathfrak g\) is a finite-dimensional simple complex Lie algebra, and \(\zeta \) is a primitive \(\ell \)-th root of unity with \(\ell \) larger than the Coxeter number of \(\mathfrak g\). The calculation employs the prior calculations and techniques of V. Ostrik [Funct. Anal. Appl. 32, 237–246 (1998); translation from Funkts. Anal. Prilozh. 32, No. 4, 22–34 (1998; Zbl 0981.17010)] and of D. K. Nakano, B. J. Parshall and D. C. Vella [J. Reine Angew. Math. 547, 15-49 (2002; Zbl 1009.17013)], as well as deep results involving the validity of the Lusztig character formula for quantum groups and the positivity of parabolic Kazhdan-Lusztig polynomials for the affine Weyl group. Analogous support variety calculations are provided for the first Frobenius kernel \(G_{1}\) of a reductive algebraic group scheme G defined over the prime field \(\mathbb F_p\).
MSC:
| 17B37 | Quantum groups (quantized enveloping algebras) and related deformations |
| 17B50 | Modular Lie (super)algebras |
| 17B56 | Cohomology of Lie (super)algebras |
References:
| [1] | Andersen, H.; Jantzen, J., Cohomology of induced representations for algebraic groups, Math. Ann., 269, 487-525 (1984) · Zbl 0529.20027 |
| [2] | Andersen, H.; Jantzen, J.; Soergel, W., Representations of quantum groups at a \(p\) th root of unity and of semisimple groups in characteristic \(p\), Astérisque, 220 (1994) · Zbl 0802.17009 |
| [3] | Bendel, C.; Nakano, D.; Pillen, C.; Parshall, B., Cohomology for quantum groups via the geometry of the nullcone · Zbl 1344.20066 |
| [4] | Bourbaki, N., Lie Groups and Lie Algebras, Elements of Mathematics (1972), Springer, Chapters 4-6 |
| [5] | Cline, E.; Parshall, B.; Scott, L., Infinitesimal Kazhdan-Lusztig theories, Contemp. Math., 139, 43-73 (1992) · Zbl 0795.20022 |
| [6] | Deodhar, V., On some geometric aspects of Burhat orderings II. The parabolic analogue of Kazhdan-Lusztig polynomials, J. Algebra, 111, 483-506 (1987) · Zbl 0656.22007 |
| [7] | Feldvoss, J.; Witherspoon, S., Support varieties and representation type of small quantum groups, Int. Math. Res. Not., 2010, 1346-1362 (2010) · Zbl 1197.16036 |
| [8] | Fiebig, P., An upper bound on the exceptional characteristics for Lusztigʼs character formula · Zbl 1266.20059 |
| [9] | Friedlander, E.; Parshall, B., Cohomology of Lie algebras and algebraic groups, Amer. J. Math., 108, 235-253 (1986) · Zbl 0601.20042 |
| [10] | Friedlander, E.; Parshall, B., Support varieties for restricted Lie algebras, Invent. Math., 86, 553-562 (1986) · Zbl 0626.17010 |
| [11] | Ginzburg, V.; Kumar, S., Cohomology of quantum groups at roots of unity, Duke Math. J., 69, 179-198 (1993) · Zbl 0774.17013 |
| [12] | I. Grojnowski, M. Haiman, Affine Hecke algebras and positivity of LLT and Macdonald polynomials, preprint, 2007.; I. Grojnowski, M. Haiman, Affine Hecke algebras and positivity of LLT and Macdonald polynomials, preprint, 2007. |
| [13] | Humphreys, J. E., Conjugacy Classes in Semisimple Algebraic Groups, Math. Surveys Monogr., vol. 43 (1995), Amer. Math. Soc. · Zbl 0834.20048 |
| [14] | Iwahori, N.; Matsumoto, H., On some Bruhat decomposition and the structure of the Hecke rings of \(p\)-adic Chevalley groups, Inst. Hautes Études Sci. Publ. Math., 25, 5-48 (1965) · Zbl 0228.20015 |
| [15] | Jantzen, J. C., Kohomologie von \(p\)-Lie-Algebren und nilpotente Elemente, Abh. Math. Semin. Univ. Hambg., 56, 191-219 (1986) · Zbl 0614.17008 |
| [16] | Jantzen, J. C., Representations of Algebraic Groups (2003), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI · Zbl 0612.20021 |
| [17] | Johnston, D. S.; Richardson, R. W., Conjugacy classes in parabolic subgroups of semisimple algebraic groups. II, Bull. Lond. Math. Soc., 9, 245-250 (1977) · Zbl 0375.22008 |
| [18] | Kashiwara, M.; Tanisaki, T., Kazhdan-Lusztig conjecture for affine Lie algebra with negative level, Duke Math. J., 77, 21-62 (1995) · Zbl 0829.17020 |
| [19] | Kashiwara, M.; Tanisaki, T., Parabolic Kazhdan-Lusztig polynomials and Schubert varieties, J. Algebra, 249, 306-325 (2002) · Zbl 1021.17015 |
| [20] | Müller, E., Some topics on Frobenius-Lusztig kernels, II, J. Algebra, 206, 659-681 (1998) · Zbl 0948.17010 |
| [21] | Nakano, D. K.; Parshall, B. J.; Vella, D. C., Support varieties for algebraic groups, J. Reine Angew. Math., 547, 15-49 (2002) · Zbl 1009.17013 |
| [22] | Ostrik, V., Support varieties for quantum groups, Funct. Anal. Appl., 32, 237-246 (1998) · Zbl 0981.17010 |
| [23] | Parshall, B.; Wang, J.-P., Cohomology of quantum groups: The quantum dimension, Canad. J. Math., 45, 1276-1298 (1993) · Zbl 0835.17009 |
| [24] | Scott, L., Some new examples in 1-cohomology, J. Algebra, 260, 416-425 (2003) · Zbl 1019.20020 |
| [25] | Suslin, A.; Friedlander, E. M.; Bendel, C. P., Infinitesimal 1-parameter subgroups and cohomology, J. Amer. Math. Soc., 10, 693-728 (1997) · Zbl 0960.14023 |
| [26] | Suslin, A.; Friedlander, E. M.; Bendel, C. P., Support varieties for infinitesimal group schemes, J. Amer. Math. Soc., 10, 729-759 (1997) · Zbl 0960.14024 |
| [27] | Tanisaki, T., Character formulas of Kazhdan-Lusztig type, (Representations of Finite Dimensional Algebras and Related Topics in Lie Theory and Geometry. Representations of Finite Dimensional Algebras and Related Topics in Lie Theory and Geometry, Fields Inst. Commun., vol. 40 (2004), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 261-276 · Zbl 1063.20052 |
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