On Rigid Tuples in Linear Groups of Odd Dimension

We construct examples of linearly rigid tuples which lead to regular Galois realizations over for linear and unitary groups GL2m + 1(q) and U2m + 1(q2), where q is odd and m > φ(q − 1) resp. m > φ(q + 1). The notion of linear rigidity was introduced by Strambach and Völklein (J. Reine Angew. Math., to appear) as an analogue of Katzʹ physically rigid systems (see N. M. Katz, “Rigid Local Systems,” Annals of Mathematics Studies, Vol. 139, Princeton Univ. Press, Princeton, NJ, 1996).