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

Let $k$ be an algebraically closed field of characteristic $p \geq 0$, let $G$ be a simple simply connected classical linear algebraic group of rank $\ell$ and let $T$ be a maximal torus in $G$ with rational character group $X(T)$. For a nonzero $p$-restricted dominant weight $\lambda \in X(T)$, let $V$ be the associated irreducible $kG$-module. Define $\nu_{G}(V)$ to be the minimum codimension of eigenspaces corresponding to non-central elements of $G$ on $V$. In this paper, we calculate $\nu_{G}(V)$ for $G$ of type $A_{\ell}$, $\ell \geq 16$, and $dim(V) \leq \frac{\ell^{3}}{2}$; for $G$ of type $B_{\ell}$, respectively $C_{\ell}$, $\ell \geq 14$, and $dim(V) \leq 4\ell^{3}$; and for $G$ of type $D_{\ell}$, $\ell \geq 16$, and $dim(V) \leq 4\ell^{3}$. Moreover, for the groups of smaller rank and their corresponding irreducible modules with dimension satisfying the above bounds, we determine lower-bounds for $\nu_{G}(V)$.