Nilpotent orbits in good characteristic and the Kempf-Rousseau theory
Let \(G\) be an algebraic group over an algebraically closed field \(k\) and \(\mathfrak g\) be the Lie algebra of \(G\). The focus of this paper is nilpotent elements and orbits in \(\mathfrak g\) with the goal of extending two results known to hold over the complex numbers to a field of prime characteristic \(p>0\) where \(p\) is good for the root system of \(G\). First, suppose that \(G\) is connected and reductive. Over the complex numbers, the Bala-Carter theorem states that any nilpotent element in \(\mathfrak g\) is \(G\)-conjugate to a Richardson element in a distinguished parabolic subalgebra of the Lie algebra of a Levi subgroup of \(G\). \textit{K. Pommerening} [J. Algebra 49, 525-536 (1977; Zbl 0367.20046), ibid. 65, 373-398 (1980; Zbl 0437.20034)] showed that the same characterization holds when \(k\) has prime characteristic \(p>0\) as long as \(p\) is good for the root system of \(G\). In this work, the author provides a new proof of this theorem in prime characteristic which avoids computations and case-checking. This is done by replacing the traditional approach of using \({\mathfrak{sl}}(2)\)-theory (which is not as effective in prime characteristic) with Kempf-Rousseau theory or the theory of optimal tori for \(G\)-unstable vectors. This new approach also leads to an improved proof of the existence theorem for good transverse slices to nilpotent orbits in \(\mathfrak g\). For previous proofs, see the work of \textit{N. Kawanaka} [J. Fac. Sci., Univ. Tokyo, Sect. I A 34, 573-597 (1987; Zbl 0651.20046)], \textit{A. Premet} [Trans. Am. Math. Soc. 347, No. 8, 2961-2988 (1995; Zbl 0865.17015)], and \textit{N. Spaltenstein} [J. Fac. Sci., Univ. Tokyo, Sect. I A 31, 283-286 (1984; Zbl 0581.17003)]. Secondly, suppose that \(G\) is a simple algebraic group of adjoint type. Over the complex numbers, \textit{E. Sommers} [Int. Math. Res. Not. 1998, No. 11, 539-562 (1998; Zbl 0909.22014)] found a bijection between \(G\)-conjugacy classes of pairs \((L,e)\) where \(L\) is a pseudo-Levi sugroup in \(G\) and \(e\) is a distinguished nilpotent element in the Lie algebra of \(L\) and \(G\)-conjugacy classes of pairs \((e,C)\) where \(e\) is a nilpotent element in \(\mathfrak g\) and \(C\) is a conjugacy class in \(A(e)\) (the component group of the centralizer of \(e\)). This bijection allows one to obtain structural information about the component group \(A(e)\). Again using the Kempf-Rousseau theory, the author shows that such a bijection also holds when the characteristic of \(k\) is a good prime and identifies the isomorphism type of \(A(e)\). Using different methods, these results have also been recently obtained by \textit{G. J. McNinch} and \textit{E. Sommers} [J. Algebra 260, No. 1, 323-337 (2003; Zbl 1026.20026)].
scientific article; zbMATH DE number 1901828
scientific article