On rigid tuples in linear groups of odd dimension (Q1969372)

This paper is about ``rigid local systems'' (for an introduction to those systems, see \textit{N. Katz}'s book ``Rigid local systems'' [Ann. Math. Stud. 139 (1996; Zbl 0864.14013)]). One says that a tuple \((g_1,\dots,g_r)\) of elements of \(GL_n (\mathbb{C})\), such that \(g_1,\dots, g_r\in \mathbb{C}^*\), is linearly rigid if, for any other tuple \((h_1,\dots, h_r)\) in \(GL_n(\mathbb{C})\), such that \(h_i\) is conjugate to \(g_i\) \((1\leq i\leq r)\) and \(h_1\cdots h_r= g_1\cdots g_r\), then there exists an \(x\) in \(GL_n(\mathbb{C})\) such that \(h_i=xg_i x^{-1}\) for all \(i\) (this notion is introduced by \textit{K. Strambach} and \textit{H. Völklein} in J. Reine Angew. Math. 510, 57-62 (1999; Zbl 0931.12006)). The authors give new rigid tuples in \(GL_n(\mathbb{C})\). Applications to Galois relizations over \(\mathbb{Q}\) are given, for the groups \(PGL_{2m+1} (\mathbb{F}_q)\), resp. \(PU_{2m+1} (\mathbb{F}_{q^2})\) \((q=p^l\) with \(p\) an odd prime number, \(q\neq 3\), \(m>\phi (q-1)\), resp. \(m> \phi(q+1))\).