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 lambda is an element of X(T), let V be the associated irreducible kG-module. We define nu G(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 nu G(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 nu G(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.