Sharper character value estimates for groups of Lie type
Let \(G\) be a finite group of Lie type over the field with \(q\) elements and \(\chi\) a nonlinear irreducible complex character of \(G\). In earlier work of the same author it was shown that for an absolute constant \(C\), \(|\chi(x)/\chi(1)|\leq (C/q)^{1/2}\) if \(x\) is unipotent and \(|\chi(x)/\chi(1)|\leq C/q\) otherwise. In the present paper the author significantly sharpens these bounds by proving for unipotent \(x\): \(|\chi(x)/\chi(1)|\leq 1/(\sqrt{q} - 1)\) if \(q > 4\) and \(|\chi(x)/\chi(1)|\leq 3/4\) if \(q\leq 4\). If \(x\) is unipotent but not a transvection in a symplectic group of odd characteristic, then \(|\chi(x) /\chi(1)|\leq 5/q\). For nonunipotent elements the results is \(|\chi(x)/\chi(1)|\leq 19/20\) for all \(q\). If \(x\) is not unipotent, \(q \geq 7\) and \(G\) is not a symplectic group of odd characteristic, then \(|\chi(x)/\chi(1)|\leq 8/q - 14/q^2\). In the main situation one can assume that \(x \in P_j \setminus U_j\) where \(P_J\) is a standard maximal parabolic subgroup and \(U_J=O_p(U_j)\). Then \(\chi_{|P_j}\) is decomposed into four components \(\chi_1\), \(\chi_2\), \(\chi_3\) and \(\chi_4\). The first one collects all linear constituents, the second one all nonlinear constituents which are characters of \(P_J/ U_J\); the constituents of \(\chi_3\) lie over nonprincipal irreducible characters of \(U_J\) which are \(O^{p'}(P_J/ U_J)\)-invariant, those of \(\chi_4\) lie over irreducible characters of \(U_J\) which are not \(O^{p'}(P_J/U_J)\)-invariant. The values \(|\chi_i(x)|\) are then estimated separately. The proofs do neither use the classification of finite simple groups nor the Deligne-Lusztig theory of characters of finite groups of Lie type. Instead they use ideas from the character theory of solvable groups.
scientific article; zbMATH DE number 783864