Splitting subspaces of linear operators over finite fields

Let V be a vector space of dimension N over the finite field Fq and T be a linear operator on V. Given an integer m that divides N, an m-dimensional subspace W of V is T-splitting if V=W⊕TW⊕⋯⊕Td−1W where d=N/m. Let σ(m,d;T) denote the number of m-dimensional T-splitting subspaces. Determining σ(m,d;T) for an arbitrary operator T is an open problem. We prove that σ(m,d;T) depends only on the similarity class type of T and give an explicit formula in the special case where T is cyclic and nilpotent. Denote by σq(m,d;τ) the number of m-dimensional splitting subspaces for a linear operator of similarity class type τ over an Fq-vector space of dimension md. For fixed values of m,d and τ, we show that σq(m,d;τ) is a polynomial in q.