On the Schur indices of \(\text{SU}_{l+1}(\mathbb{F}_ q)\) and \(\text{Spin}_{2l}^ -(\mathbb{F}_ q)\) (Q1903369)

Let \(\mathbb{F}_q\) be a finite field with \(q\) elements of characteristic \(p>0\). Let \(G^F\) be the group \(\text{SU}_{l+1}(\mathbb{F}_q)\) or \(\text{Spin}^-_{2l}(\mathbb{F}_q)\). Let \(U^F\) be a Sylow \(p\)-subgroup of \(G^F\) and let \(\lambda\) be a linear character of \(U^F\). The author considers the Schur indices of an irreducible character of \(G^F\) occurring in the induced character \(\lambda^{G^F}\). He gives some sufficient conditions for that all the \(\lambda^{G^F}\) are realizable in the rational number field \(\mathbb{Q}\), where \(\lambda\) ranges over the linear characters of \(U^F\). He shows that in some cases all the \(\lambda^{G^F}\) are realizable in the \(r\)-adic number field \(\mathbb{Q}_r\) for any prime number \(r\neq p\). He also gives some sufficient conditions for the existence of characters of Schur index equal to two (note that the Schur indices of many irreducible characters of \(G^F\) are not greater than two). Notice that the main results of the present paper were announced in the author's previous paper [Proc. Japan Acad., Ser. A 64, No. 7, 253-255 (1988; Zbl 0667.20034)].