We propose a new reduced order fictitious domain method for the discretization of the motion of small rigid particles evolving in a viscous incompressible flow, that can be further extend for the simulation of slender elastic fibers in three-dimensional flows. The reduced model is based on the projection of Dirichlet boundary constraints on a finite dimensional approximation space of Fourier type, obtaining in this way a problem with defective interface conditions, that are imposed through a Lagrange multiplier formulation as introduced previously for the Poisson problem. We analyze the existence of the solution of a reduced Stokes problem and for arbitrarily small holes and we prove its convergence towards the original problem, the rate of which depends on the size of the inclusion and on the number of modes of the finite dimensional space. The numerical discretization of the reduced problem is addressed by the finite element method, using a computational mesh that does not fit to the holes in the framework of a fictitious domain approach. We propose a stabilized and robust formulation with respect to the hole size that is proved to be stable and convergent. One key step of the proof relies on the derivation of an inf-sup condition involving both the divergence constraint and Dirichlet boundary constraints. The properties of the discretization method are supported by numerical experiments both for rigid particles or slender elastic structures in three-dimensional flows