The affinity of a permutation of a finite vector space

For a permutation f of an n-dimensional vector space V over a finite field of order q we let k-affinity(f) denote the number of k-flats X of V such that f(X) is also a k-flat. By k-spectrum(n, q) we mean the set of integers k-affinity(f) where f runs through all permutations of V . The problem of the complete determination of k-spectrum(n, q) seems very difficult except for small or special values of the parameters. However, we are able to establish that 0 ∈ k-spectrum(n, q) in the following cases: (i) q ≥ 3 and 1 ≤ k ≤ n− 1; (ii) q = 2, 3 ≤ k ≤ n − 1; (iii) q = 2, k = 2, n ≥ 3 odd. The maximum of k-affinity(f) is, of course, obtained when f is any semi-affine mapping. We conjecture that the next to largest value of k-affinity(f) is when f is a transposition and we are able to prove this when q = 2, k = 2, n ≥ 3 and when q ≥ 3, k = 1, n ≥ 2.