On the conjugacy of nilpotent elements in the classical Lie algebras in relation to their representations (Q1089431)

Let \(G\) be an algebraic group over an algebraically closed field \(K\), and \({\mathfrak g}\) the Lie algebra of \(G\). Let \((\sigma,V)\) be a finite dimensional rational representation of \(G\) or also the corresponding representation of \({\mathfrak g}\). The author investigates the following problems. (I) For nilpotent elements \(X, X'\), if \(X\) and \(X'\) are conjugate under the adjoint group \(\mathrm{Ad}(G)\), then are \(\sigma(X)\) and \(\sigma(X')\) conjugate under \(\mathrm{GL}(V)\) ? (II) Find all the conjugacy classes which are glued up in \(\sigma\)-conjugacy. For simple Lie algebras of types \(E_ 6\), \(E_ 7\), \(E_ 8\), \textit{T. Hirai} announced the affirmative answer to (I) with \(\sigma\) the adjoint representation when the characteristic of \(K\) is good for \({\mathfrak g}\) [Proc. Japan Acad., Ser. A 60, 225--228 (1984; Zbl 0583.17006)]. The author shows that for classical groups \(G=\mathrm{GL}_ n(K)\), \(\mathrm{Sp}_ n(K)\), \(O_ n(K)\), if \(G\) is not \(O_{2^{2r+1}}(K)\) \((r=1,2,...)\), then (I) holds for the adjoint representation, and there exists a counter-example for \(O_{2^{2r+1}}(K)\) when \(r\geq 1\) except \(r=2,4.\) As for the problem (II), when \(G=O_ n(K)\) and \(\sigma\) is the adjoint representation, the author reduces the problem to looking for certain polynomials, and giving explicit examples of such polynomials, the author approaches to the problem yet without completely solving it.