Cross-characteristic character and fixed point space ratios for groups of Lie type
Upper bounds for \(|\chi(x)/\chi(1)|\) are obtained, where \(G\) is a group of Lie type defined in characteristic \(p\), \(\chi\) is an irreducible \(r\)-Brauer character of \(G\) for a prime \(r\neq p\) and \(x\in G\) is a nonidentity unipotent element or \(x\) is a noncentral \(r'\)-element which lies in some proper parabolic of \(G\). Let \(G\) be a finite group, \(p\) a prime. We say that \(G\) is admissible if \(G=\langle y,(\overline{G_\sigma})_0\rangle\), where \(\overline G\) is a connected reductive group over \(\overline{\text{GF}(p)}\) with simply connected components, and \(y\) is a noncentral semisimple element of \(\overline{G_\sigma}\). Let \(\mathcal A\) be the class of quasisimple, simply connected groups of type \(A_2(2)\), \(A_2(3)\), \(A_3(2)\), \(^2A_3(2)\), \(^2A_3(3)\), \(^2A_4(2)\), \(^2A_5(2)\), \(B_2(3)\), \(B_3(2)\), \(B_4(2)\), \(D_4(2)\), \(^2D_4(2)\), \(^2D_5(2)\), \(^3D_4(2)\), \(F_4(2)\). Let \(\mathcal C\) be the class of quasisimple, simply connected groups of type \(A_2(3)\), \(A_3(3)\), \(B_3(3)\), \(C_3(3)\), \(G_2(3)\), \(A_2(2)\), \(A_3(2)\), \(B_3(2)\), \(B_4(2)\), \(D_4(2)\), \(F_4(2)\), \(^2A_3(2)\), \(^2A_4(2)\), \(^3D_4(2)\). Theorem 3.1. Let \(G\) be a simple admissible group of rank at least 2. Suppose \(G\) is not of type \(G_2\) or \(^2F_4(2)\). Suppose \(G_0\) does not belong to \({\mathcal A}\cup{\mathcal C}\). Let \(\chi\) be an irreducible \(r\)-Brauer character of \(G\) with \(G_0\not\leq\text{Ker }\chi\). Let \(x\in G-Z(G)\) be a unipotent element or a semisimple \(r'\)-element. Then (a) if \(x\) is unipotent, then \(|\chi(x)/\chi(1)|\leq 6/7\), \(5/6\) or \(1/2+1/2\sqrt q\) when \(q\) is respectively 2, 3, or at least 4, (b) if \(x\) is semisimple and lies in a proper parabolic of \(G\), then \(|\chi(x)/\chi(1)|\leq 5/6\) or \(1/2+3/2q\) when \(q\) is respectively at most 4 or greater than 4.
scientific article; zbMATH DE number 1218693