On rigid tuples in linear groups of odd dimension

We construct examples of linearly rigid tuples which lead to regular Galois realizations over Q for linear and unitary groups GL(2m+1)(q) and U2m+1(q( 2)), where q is odd and m > phi(q - 1) resp. m > phi(q + 1). The notion of linear rigidity was introduced by Strambach and Volklein (J. Reine Angew. M ath., to appear) as an analogue of Katz' physically rigid systems (see N. M . Katz, "Rigid Local Systems," Annals of Mathematics Studies, Vol. 139, Pri nceton Univ. Press, Princeton, NJ, 1996). (C) 2000 Academic Press.