  Let $G = GL_N$ over an algebraically closed field of odd characteristic, and
$\theta$ an involutive automorphism on $G$ such that $H = (G^{\theta})^0$ is
isomorphic to $SO_N$. Then $G^{\iota\theta} = \{ g \in G \mid \theta(g) =
g^{-1} \}$ is regarded as a symmetric space $G/G^{\theta}$. Let
$G^{\iota\theta}_{uni}$ be the set of unipotent elements in $G^{\iota\theta}$.
$H$ acts on $G^{\iota\theta}_{uni}$ by the conjugation. As an analogue of the
generalized Springer correspondence in the case of reductive groups, we
establish in this paper the generalized Springer correspondence between
$H$-orbits in $G^{\iota\theta}_{uni}$ and irreducible representations of
various symmetric groups.
