  Determining the weight distribution of a code is an old and fundamental topic
in coding theory that has been thoroughly studied. In 1977, Helleseth,
Kl{\o}ve, and Mykkeltveit presented a weight enumerator polynomial of the
lifted code over $\mathbb{F}_{q^\ell}$ of a $q$-ary linear code with
significant combinatorial properties, which can determine the support weight
distribution of this linear code. The Solomon-Stiffler codes are a family of
famous Griesmer codes, which were proposed by Solomon and Stiffler in 1965. In
this paper, we determine the weight enumerator polynomials of the lifted codes
of the projective Solomon-Stiffler codes using some combinatorial properties of
subspaces. As a result, we determine the support weight distributions of the
projective Solomon-Stiffler codes. In particular, we determine the weight
hierarchies of the projective Solomon-Stiffler codes.
