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 determine (n-1)-spectrum(n,2) and 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. For 1≤k≤n-1 and (q,k)≠(2,1), the maximum of k-affinity(f) is obtained when f is any semiaffine mapping. We conjecture that the next to largest value of k-affinity(f) occurs when f is a transposition, and we are able to prove it when q=2, k=2, n≥3 and when q≥3, k=1, n≥2.