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

Let $k$ be an algebraically closed field of characteristic $p\ge 0$, let $G$ be a simple simply connected classical linear algebraic group of rank $\mathrm{\ell }$ 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 $A_{\mathrm{\ell }}$, $\mathrm{\ell }\ge 16$, and $dim(V)\le \frac{{\mathrm{\ell }}^3}{2}$; for $G$ of type $B_{\mathrm{\ell }}$, respectively $C_{\mathrm{\ell }}$, $\mathrm{\ell }\ge 14$, and $dim(V)\le 4{\mathrm{\ell }}^3$; and for $G$ of type $D_{\mathrm{\ell }}$, $\mathrm{\ell }\ge 16$, and $dim(V)\le 4{\mathrm{\ell }}^3$. Moreover, for the groups of smaller rank and their corresponding irreducible modules with dimension satisfying the above bounds, we determine lower-bounds for ${\nu }_G(V)$.