Abstract:
Let k be an algebraically closed field of characteristic p ≥ 0 and let G be a linear algebraic group of rank l ≥ 1 over k. Let V be a rational kG-module and let V_g(µ) denote the eigenspace corresponding to the eigenvalue µ of a group element g on V. We set ν_G(V ) = min{ dim(V) − dim(V_g(µ)) | g in G, µ in k* with µg =/= 1}. In this talk we will identify pairs (G, V) of simple simply connected classical linear algebraic groups of rank l ≥ 10, respectively of exceptional linear algebraic groups, and of rational irreducible tensor-indecomposable kG-modules with the property that ν_G(V) ≤ √dim(V). This problem is an extension of the classification result obtained by Guralnick and Saxl for ν_G(V) ≤ max{2,√dim(V)/2}. One motivation for studying such problems is to identify subgroups of linear algebraic groups based on element behaviour.