  In this paper we prove a new generic vanishing theorem for $X$ a complete
homogeneous variety with respect to an action of a connected algebraic group.
Let $A, B_0\subset X$ be locally closed affine subvarieties, and assume that
$B_0$ is smooth and pure dimensional. Let $\mathcal{P}$ be a perverse sheaf on
$A$ and let $B=g B_0$ be a generic translate of $B_0$. Then our theorem implies
$(-1)^{\operatorname{codim} B}\chi(A\cap B, \mathcal{P}|_{A\cap B})\geq 0$. As
an application, we prove in full generality a positivity conjecture about the
signed Euler characteristic of generic triple intersections of Schubert cells.
Such Euler characteristics are known to be the structure constants for the
multiplication of the Segre-Schwartz-MacPherson classes of these Schubert
cells.
