Minimum codimension of eigenspaces in irreducible representations of simple classical linear algebraic groups

Let k be an algebraically closed field of characteristic p≥0, let G be a simple simply connected classical linear algebraic group of rank l and let T be a maximal torus in G with rational character group X(T). For a nonzero p-restricted dominant weight λ∈X(T), let V be the associated irreducible kG-module. We define νG(V) as the minimum codimension of any eigenspace on V for any non-central element of G. In this paper, we determine lower-bounds for νG(V) for G of type Al and dim(V)≤l32, and for G of type Bl,Cl, or Dl and dim(V)≤4l3. Moreover, we give the exact value of νG(V) for G of type Al with l≥15; for G of type Bl or Cl with l≥14; and for G of type Dl with l≥16.