On the existence of characters of the Schur index \(2\) of the simple finite Steinberg groups of type \(({^ 2E_ 6})\) (Q1318939)

Let \(\chi\) be a complex irreducible character of a finite group and \(k\) be a field of characteristic 0. Then we denote by \(m_k(\chi)\) the Schur index of \(\chi\) with respect to \(k\). In this note we show that the simple finite Steinberg group \({^2E_6(q^2)}\) has (at least) two rational-valued irreducible characters \(\chi\) such that \(m_\mathbb{Q}(\chi)=m_\mathbb{R}(\chi)=m_{\mathbb{Q}_p}(\chi)=2\) and \(m_{\mathbb{Q}_l}(\chi)=1\) for any prime number \(l\neq p\). This follows from Lusztig's classification theory of the unipotent representations of finite groups of Lie type.