Almost complex structures on quaternion-Kähler manifolds and inner symmetric spaces

We prove that compact quaternionic-Kähler manifolds of positive scalar curvature admit no almost complex structure, even in the weak sense, except for the complex Grassmannians Gr2(ℂn+2). We also prove that irreducible inner symmetric spaces M4n of compact type are not weakly complex, except for spheres and Hermitian symmetric spaces.