Schur indices of some finite Chevalley groups of rank 2. I
The author proves that the irreducible characters of \(SO_ 5(q)\), \(CSp_ 4(q)\) (the conformal symplectic group in four dimensions over \(F_ q)\), and \(G_ 2(q)\), for q odd, have Schur indices 1 over Q. One of the main tools is a lemma which states that if g is an element of prime order r of a finite group H and h is an element of H of order r-1 such that \(hgh^{-1}=g^{\nu}\) where \(\nu\) has order r-1 mod r, then the Schur index \(m_ Q(\chi)\) of an irreducible character \(\chi\) of H divides \(\chi\) (g). If \(H=G_ 2(q)\) or \(CSp_ 4(q)\) then the character table of H is known explicitly. In the case of \(SO_ 5(q)\) the author writes down a list of characters. For a constituent \(\chi\) of the Gelfand-Graev character or for an irreducible character \(\chi\) such that p does not divide \(\chi\) (1) (where q is a power of p), an earlier result of the author [Q. J. Math., Oxf. II. Ser. 32, 443-452 (1981; Zbl 0474.20022)] shows that \(m_ Q(\chi)=1\). For the other characters the theorem is checked in every case.
scientific article; zbMATH DE number 3935313