  It is known that the Dirac index of a $Spin^c$ structure is localized to the
characteristic submanifold. We introduce the notion of $G^{\pm}(n,s^+,s^-)$
structure on a manifold as a common generalization of the $Spin^c$ structure
and the $H_n(s)$ structure defined by D.~Freed--M.~Hopkins, and formulate a
version of characteristic submanifold for the $G^{\pm}(n,s^+,s^-)$ structure.
We show that the $KO^*(pt)$-valued index associated with the
$G^{\pm}(n,s^+,s^-)$ structure is localized to the characteristic submanifold.
As an application, we give a topological sufficient condition for the moduli
space of $Pin^-(2)$ monopoles to be orientable.
