Strongly real classes in finite unitary groups of odd characteristic

Abstract

We classify all strongly real conjugacy classes of the finite unitary group U(n,𝔽q) when q is odd. In particular, we show that g ∈ U(n,𝔽q) is strongly real if and only if g is an element of some embedded orthogonal group O±(n,𝔽q). Equivalently, g is strongly real in U(n,𝔽q) if and only if g is real and every elementary divisor of g of the form (t ± 1)2m has even multiplicity. We apply this to obtain partial results on strongly real classes in the finite symplectic group Sp(2n,𝔽q), q odd, and a generating function for the number of strongly real classes in U(n,𝔽q), q odd, and we also give partial results on strongly real classes in U(n,𝔽q) when q is even.