  In this paper, we consider the flag manifold of $p$ orthogonal subspaces of
equal dimension which carries an action of the cyclic group of order $p$. We
provide a complete calculation of the associated Fadell-Husseini index. This
may be thought of as an odd primary version of the computations of Barali\'c et
al [Forum Math., 30 (2018), pp. 1539--1572] for the Grassmann manifold
$G_n(\mathbb{R}^{2n})$. These results have geometric consequences for $p$-fold
orthogonal shadows of a convex body.
