Products of unipotent elements of index 2 in orthogonal and symplectic groups (Q6592913)

Let \(V\) be a finite-dimensional vector space over a field \(\mathbb{F}\) of characteristic \(\neq 2\) and let \(b\) be either a symmetric or skew-symmetric nondegenerate bilinear form on \(V\). We use \(\mathrm{O}(b)\) and \(\mathrm{Sp}(b)\) to denote the corresponding orthogonal and symplectic groups on \(V\). An element \(u\in V\) is said to be unipotent of index \(2\) (briefly, a \(U_{2}\) element) if \( (u-I)^{2}=0\).\N\NA main result of the the paper is the following. \N\NTheorem. If \(b\) is symplectic and \(u\in \mathrm{Sp}(b)\) then the following are equivalent: \N\N(i) \(u\) is a product of two \(U_{2}\)-elements from \(\mathrm{GL}(V)\); \N\N(ii) \(u\) is a product of two \( U_{2}\)-elements from \(\mathrm{Sp}(b)\); \N\N(iii) \(u\) has no Jordan block of odd dimension for eigenvalue \(-1\). \N\NA consequence is that under the same hypotheses every element in \(\mathrm{Sp}(b)\) is a product of three \(U_{2}\)-elements.\N\NWhen \(b\) is symmetric there is an analogous theorem, that gives four conditions which together are necessary and sufficient for \(u\in \mathrm{O}(b)\) to be a product of two \(U_{2}\)-elements in \(\mathrm{O}(b)\). These involve Witt and Wall invariants, too complicated to describe here.