Splitting subspaces of linear operators over finite fields

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