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

Abstract

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 λ ∈ X ( T ) , let V be the associated irreducible kG-module. We define ν 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 ν G ( V ) for G of type A l and dim ( V ) ≤ l 3 2 , and for G of type B l , C l , or D l and dim ( V ) ≤ 4 l 3 . Moreover, we give the exact value of ν G ( V ) for G of type A l with l ≥ 15 ; for G of type B l or C l with l ≥ 14 ; and for G of type D l with l ≥ 16 .