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

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].

MSC:

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

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