Irreducible representations of Lie algebras of reductive groups and the Kac-Weisfeiler conjecture
Let \({\mathfrak g}\) be the Lie algebra of a connected reductive group \(G\) over an algebraically closed field of characteristic \(p>0\). Suppose that the commutator subgroup of \(G\) is simply connected and that \(p\) is greater than any coefficient of any positive root of the root system \(R= R(G)\) relative to a basis of simple roots in \(R\). If \(p=2\), suppose also that \({\mathfrak g}\) admits a nondegenerate \(G\)-invariant trace form. Let \(V\) be a faithful irreducible \({\mathfrak g}\)-module with \(p\)-character \(\chi\in {\mathfrak g}^*\). The author proves that \(\dim V\) is divisible by \(p^{1/2}\dim \Omega (\chi)}\) where \(\Omega (\chi)\) is the order of \(\chi\) under the coadjoint action of \(G\). A conjecture mentioned by \textit{V. Kac} and \textit{B. Weisfeiler} [Funct. Anal. Appl. 5, 111--117 (1971); translation from Funkts. Anal. Prilozh. 5, No. 2, 28--36 (1971; Zbl 0237.17003)] is thus confirmed. The necessity of \({\mathfrak g}\) admitting a nondegenerate trace form in order for the conjecture to be true is shown by an example. The paper concludes by listing some related open questions.
Lie algebra of a connected reductive group
scientific article; zbMATH DE number 784020
scientific article