Big elements in irreducible linear groups

Abstract

Let V be a linear space over a field K of dimension n > 1, and let G ≤ GL(V ) be an irreducible linear group. In this paper we prove that the group G contains an element g such that rank(g − αEn) ≥ n 2 for every α ∈ K, where En is the identity operator on V . This estimate is sharp for any n = 2 m . The existence of such an element implies that the conjugacy class of G in GL(V ) intersects the big Bruhat cell Bẇ0B of GL(V ) non-trivially (here B is a fixed Borel subgroup of G). The latter fact is equivalent to the existence of a complete flag F such that the flags g(F), F are in general position for some g ∈ G.Introduction. Let B ≤ GL(V ) be a fixed Borel subgroup of GL(V ), let T denote a maximal split torus of B, and let N denote the normalizer of T in GL(V ). Then GL(V ) has a split BN -pair and the Bruhat decomposition GL(V ) = w∈W BẇB, where W = N/T is the Weyl group of GL(V ), T = B ∩ N is a maximal group of semisimple matrices, andẇ is any fixed representative of w ∈ W in the group N (cf.[3], 2.5). We say that an element g ∈ GL(V ) is big if the conjugacy class C g of g in GL(V ) intersects the big Bruhat cell Bẇ 0 B non-trivially (here w 0 ∈ W is the longest element in the Weyl group). A necessary and sufficient condition for an element g ∈ GL(V ) to be big is given by the set of inequalities