Conjugate real classes in general linear groups

Abstract

Let F be a field with a nontrivial involution 𝑐:𝛼↦𝛼𝑐. An element 𝑔∈GL𝑛(F) is called 𝑐-real if it is conjugate to (𝑔𝑐)−1. We prove that for 𝑛≥2, 𝑔∈GL𝑛(F) is 𝑐-real if and only if it has a representation in some unitary group of degree 𝑛 over F.