Big elements in irreducible linear groups

Abstract

Let V be a linear space over a field K of dimension n > 1, and let (Formula presented.) be an irreducible linear group. In this paper we prove that the group G contains an element g such that rank (Formula presented.) for every (Formula presented.), where En is the identity operator on V. This estimate is sharp for any (Formula presented.). The existence of such an element implies that the conjugacy class of G in GL(V) intersects the big Bruhat cell (Formula presented.) 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.