  An irreducible ordinary character of a finite reductive group is called
quadratic unipotent if it corresponds under Jordan decomposition to a
semisimple element $s$ in a dual group such that $s^2=1$. We prove that there
is a bijection between, on the one hand the set of quadratic unipotent
characters of $GL(n,q)$ or $U(n,q)$ for all $n \geq 0$ and on the other hand,
the set of quadratic unipotent characters of $Sp(2n,q)$ for all $n \geq 0$. We
then extend this correspondence to $\ell$-blocks for certain $\ell$ not
dividing $q$.
