Desingularization of Schubert varieties and orbit closures of prehomogeneous vector spaces of commutative parabolic type

Let G be a simple algebraic group, and P its parabolic subgroup whose unipotent radical U is commutative. We construct desingularizations of the closures of the P-orbits in GIP, which in the same time give desingularizations of the orbit closures of the prehomogeneous vector space (P/U, adjoint, U). Our construction is based on a formal argument concerning Coxeter systems and BN-pairs. As an application, we calculate the Frobenius traces of the intersection cohomology complexes of these orbit closures. Then using it, we study certain character sums associated to the above prehomogeneous vector spaces.