Gradings of positive rank on simple Lie algebras. (English) Zbl 1310.17017

Reeder, Mark; Levy, Paul; Yu, Jiu-Kang; Gross, Benedict H.

Transform. Groups 17, No. 4, 1123-1190 (2012).

A \(\mathbb{Z}_m\)-grading on a simple Lie algebra \(\mathfrak{g}\) is a decomposition on vector subspaces \(\mathfrak{g}=\bigoplus_{i\in \mathbb{Z}_m}\mathfrak{g}_i\) such that \([\mathfrak{g}_i,\mathfrak{g}_j]\subset \mathfrak{g}_{i+j}\) for all \(i,j\). The authors are interested in the invariant theory of the action of \(G_0\) on each summand \(\mathfrak{g}_i\), where \(G\) is a connected simple algebraic group of adjoint type over an algebraically closed field \(k\) with Lie algebra \(\mathfrak{g}\) and \(G_0\) is the connected subgroup of \(G\) with Lie algebra \(\mathfrak{g}_0\).
È. B. Vinberg proved in [Math. USSR, Izv. 10, 463–495 (1977); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 40, 488–526 (1976; Zbl 0363.20035)] for the complex field that the invariant theory of the \(G_0\)-action on \(\mathfrak{g}_1\) (for any \(m\geq0\)) has several common features with the invariant theory of the adjoint representation of \(G\) on \(\mathfrak{g }\) (the case \(m=1\)). The rank of the grading is the dimension of any Cartan subspace \(\mathfrak{c}\subset \mathfrak{g}_1\) (maximal abelian and toral). Thus the grading has positive rank if and only if \(\mathfrak{g}_1\) possesses some semisimple element of \(\mathfrak{g}\). A first objective in the paper under review is the classification of the positive-rank gradings for \(\mathfrak{g}\) of types \(E_6\), \(E_7\) and \(E_8\) when the characteristic of the field is zero or not too small (not a torsion prime for \(\mathfrak{g}\)), in order to complete the classification of the positive-rank gradings on simple Lie algebras. It also computes the little Weyl group \(W_{\mathfrak{c}}\) for each grading, that is, \(N_{G_0}(\mathfrak{c})/Z_{G_0}(\mathfrak{c})\). This group, recalling the behaviour of the Weyl group, is finite and generated by reflections of \(\mathfrak{c}\) relative hyperplanes. Furthermore, the ring of invariants \(k[\mathfrak{g}_1]^{G_0}\) is isomorphic to \(k[\mathfrak{c}]^{W_{\mathfrak{c}}}\). The paper also verifies the Popov’s conjecture, according to which \(\mathfrak{g}_1\) contains an affine subspace \(\mathfrak{v}\) of dimension equal to the rank such that the restriction \(k[\mathfrak{g}_1]^{G_0}\to k[ \mathfrak{v}]\) is an isomorphism (a so-called Kostant section).
The authors communicate that A. Elashvili, D. Panyushev and E. Vinberg had also computed these positive-rank gradings joint with the little Weyl groups in the complex case, although they never published the results. These have been checked to coincide with the obtained lists of gradings. The methods used in this paper are novel even considering arbitrary fields.
An extra motivation for classifying positive-rank gradings is its application to the representation theory of a reductive group over a \(p\)-adic field, as shown in [M. Reeder and J.-K. Yu, “Epipelagic representations and invariant theory”, Preprint (2012), https://www2.bc.edu/mark-reeder/Epipelagic.pdf].

Reviewer: Cristina Draper Fontanals (Malaga)

MSC:

17B70	Graded Lie (super)algebras
17B20	Simple, semisimple, reductive (super)algebras

Keywords:

gradings; little Weyl group; Kostant section; invariant theory.

Citations:

Zbl 0371.20041; Zbl 0363.20035

References:

[1]	M. Bhargava, A. Shankar, Binary quartic forms having bounded invariants, and the boundedness of the average rank of elliptic curves, arXiv:1006.1002 (2010). · Zbl 1307.11071
[2]	M. Bhargava, A. Shankar, Ternary cubic forms having bounded invariants and the existence of a positive proportion of elliptic curves having rank 0, arXiv:1007.0052 (2010). · Zbl 1317.11038
[3]	A. Borel, Properties and linear representations of Chevalley groups, in: Seminar in Algebraic Groups and Related Finite Groups, Lecture Notes in Mathematics, Vol. 131, Springer-Verlag, Berlin, 1970, pp. 1–55. Russian transl.: Семинар по алгебраическим группам, Мир, M., 1973.
[4]	A. Borel, Automorphic L-functions, in: Automorphic Forms, Representations, and L-Funsctions, Proc. Symp. Pure Math., Vol. 33, Part 2, Amer. Math. Soc., Providence, RI, 1979, pp. 27–61.
[5]	A. Borel, Linear Algebraic Groups, 2nd ed., Graduate Texts in Mathematics, Vol. 126, Springer-Verlag, New York, 1991. · Zbl 0726.20030
[6]	N. Bourbaki, Lie Groups and Lie Algebras, Springer-Verlag, Berlin, 2002. · Zbl 0983.17001
[7]	R. Carter, Conjugacy classes in the Weyl group, Compositio Math. 25 (1972), 1–59. · Zbl 0254.17005
[8]	W. de Graaf, Computing representatives of nilpotent orbits of {\(\theta\)}-groups, J. Symb. Comput. 46 (2011), 438–458. · Zbl 1222.17002 · doi:10.1016/j.jsc.2010.10.015
[9]	W. de Graaf, O. Yakimova, Good index behaviour of {\(\theta\)}-representations, I, Algebr. Represent. Theory 15 (2012), no. 4, 613–638. · Zbl 1259.14049 · doi:10.1007/s10468-010-9256-0
[10]	B. Gross, On Bhargava’s representations and Vinberg’s invariant theory, Frontiers of Mathematical Sciences, International Press (2011), 317–321.
[11]	B. Gross, M. Reeder, Arithmetic invariants of discrete Langlands parameters, Duke Math. J. 154 (2010), 431–508. · Zbl 1207.11111 · doi:10.1215/00127094-2010-043
[12]	V. Kac, Infinite Dimensional Lie Algebras, 3rd ed., Cambridge, 1995.
[13]	D. Kazhdan, G. Lusztig, Fixed point varieties on affine flag manifolds, Israel J. Math. 62 (1988), 129–168. · Zbl 0658.22005 · doi:10.1007/BF02787119
[14]	B. Kostant, The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie group, Amer. J. Math. 81 (1959), 973–1032. · Zbl 0099.25603 · doi:10.2307/2372999
[15]	B. Kostant, Groups over $ \(\backslash\)mathbb{Z} $ , in: Algebraic Groups and Discontinuous Subgroups, Proc. Symp. Pure Math., Vol. 9, Amer. Math. Soc., Providence, RI, 1966, pp. 71–83.
[16]	B. Kostant, S. Rallis, Orbits and representations associated with symmetric spaces, Amer. J. Math. 93 (1971), 753–809. · Zbl 0224.22013 · doi:10.2307/2373470
[17]	S. Kumar, G. Lusztig, D. Prasad, Characters of simplylaced nonconnected groups verus characters of nonsimplylaced connected groups, Contemp. Math. 478 (2009), 99–101. · Zbl 1160.20037 · doi:10.1090/conm/478/09321
[18]	P. Levy, Involutions of reductive Lie algebras in positive characteristic, Adv. Math. 210 (2007), no. 2, 505–559. · Zbl 1173.17019 · doi:10.1016/j.aim.2006.07.002
[19]	P. Levy, Vinberg’s {\(\theta\)}-groups in positive characteristic and Kostant–Weierstrass slices, Transform. Groups 14 (2009), no. 2, 417–461. · Zbl 1209.17016 · doi:10.1007/s00031-009-9056-y
[20]	P. Levy, KW-sections for exceptional type Vinberg’s {\(\theta\)}-groups, arXiv:0805.2064 (2010). · Zbl 1339.17014
[21]	I. G. Macdonald, Affine Hecke Algebras and Orthogonal Polynomials, Cambridge Tracts in Mathematics, Vol. 157, Cambridge University Press, Cambridge, 2003. · Zbl 1024.33001
[22]	D. Mumford, Stability of projective varieties, L’Enseign. Math. 23(1) (1977), 39–110. · Zbl 0363.14003
[23]	D. Panyushev, On invariant theory of {\(\theta\)}-groups, J. Algebra 283 (2005), 655–670. · Zbl 1071.17005 · doi:10.1016/j.jalgebra.2004.03.032
[24]	M. Reeder, Torsion automorphisms of simple Lie algebras, L’Enseign. Math. 56(2) (2010), 3–47. · Zbl 1223.17020
[25]	M. Reeder, Elliptic centralizers in Weyl groups and their coinvariant representations, Represent. Theory 15 (2011), 63–111. · Zbl 1251.20042 · doi:10.1090/S1088-4165-2011-00377-0
[26]	M. Reeder, J.-K. Yu, Epipelagic representations and invariant theory, preprint, 2012. · Zbl 1284.22011
[27]	J.-P. Serre, Coordonnées de Kac, Oberwolfach Reports 3 (2006), 1787–1790.
[28]	G. C. Shephard, J. A. Todd, Finite unitary reflection groups, Canad. J. Math. 6 (1954), 274–304. · Zbl 0055.14305 · doi:10.4153/CJM-1954-028-3
[29]	N. Spaltenstein, On the Kazhdan–Lusztig map for exceptional Lie algebras, Adv. Math. 83 (1990), 48–74. · Zbl 0725.17014 · doi:10.1016/0001-8708(90)90068-X
[30]	T. A. Springer, Regular elements in finite reflection groups, Invent. Math. 25 (1974), 159–198. · Zbl 0287.20043 · doi:10.1007/BF01390173
[31]	R. Steinberg, Variations on a theme of Chevalley, Pacific J. Math. 9 (1959), no. 3, 875–891. · Zbl 0092.02505 · doi:10.2140/pjm.1959.9.875
[32]	R. Steinberg, Regular elements of semisimple algebraic groups, Publ. Math. Inst. Hautes Études Sci. 25 (1965), 49–80. · Zbl 0136.30002 · doi:10.1007/BF02684397
[33]	R. Steinberg, Lectures on Chevalley groups, Yale Lecture Notes, 1967. Russian transl.: Р. Стейнберг, Лекции о группах Шевалле, Мир, M., 1975.
[34]	R. Steinberg, Endomorphisms of Linear Algebraic Groups, Mem. Amer. Math. Soc., Vol. 80, American Mathematical Society, Providence, RI, 1968. · Zbl 0164.02902
[35]	R. Steinberg, Torsion in reductive groups, Adv. in Math. 15 (1975), 63–92. · Zbl 0312.20026 · doi:10.1016/0001-8708(75)90125-5
[36]	Э. Б. Винберг, Группа Вейля градуированной алгебры Ли, Изв. АН СССР, сер. мат. 40 (1976), no. 3, 488–526. Engl. transl.: E. B. Vinberg, The Weyl group of a graded Lie algebra, Math. USSR-Izv. 10 (1977), 463–495. · Zbl 0371.20041 · doi:10.1070/IM1976v010n03ABEH001711
[37]	W. C. Waterhouse, Introduction to Affine Group Schemes, Springer-Verlag, New York, 1979. · Zbl 0442.14017