Eigenvalue multiplicities of group elements in irreducible representations of simple linear algebraic groups

Let k be an algebraically closed field of arbitrary characteristic, let G be a simple simply connected linear algebraic group and let V be a rational irreducible tensor-indecomposable finite-dimensional kG-module. For an element g of G we denote by $V_{g}(x)$ the eigenspace corresponding to the eigenvalue x of g on V. We define N to be the minimum difference between the dimension of V and the dimension of $V_{g}(x)$, where g is a non-central element of G. In this thesis we identify pairs (G,V) with the property that $N\leq \sqrt{\dim(V)}$. This problem is an extension of the classification result obtained by Guralnick and Saxl for the condition $N\leq \max\bigg\{2,\frac{\sqrt{\dim(V)}}{2}\bigg\}$. Moreover, for all the pairs (G,V) we had to consider in our classification, we will determine the value of N.