Globally irreducible representations of \(\text{SL}_3(q)\) and \(\text{SU}_3(q)\)
The concept of a globally irreducible representation of a finite group was introduced by \textit{B. H. Gross} [J. Am. Math. Soc. 3, No. 4, 929-960 (1990; Zbl 0745.11035)]. We will not attempt to give the precise definition of the concept, as it is somewhat technical. Briefly, it represents a generalization of the idea that the character of the representation is rational-valued and remains irreducible as a modular character for all primes. Almost all examples of globally irreducible representations have arisen either from the basic spin representations of the alternating and symmetric groups or as irreducible constituents of Weil representations of classical groups. The author considers here a finite group \(H\) which has a normal subgroup \(G\) which is a covering group of either \(\text{PSL}_3(q)\) or \(\text{PSU}_3(q)\), where \(q\) is a power of a prime. He supposes that \(H\) has a faithful globally irreducible representation. The main result of the paper may be summarized by saying that few examples other than those already discovered by Gross occur. Unusual examples are observed when \(G\) is a covering group of \(\text{PSL}_3(4)\), the phenomena being related to the exceptional Schur multiplier of this group. In particular, the Leech lattice arises from a 24-dimensional globally irreducible representation of a six-fold cover of \(\text{PSL}_3(4)\). When \(G\) is a covering group of \(\text{SU}_3(q)\) the only examples that occur are those already investigated by Gross. They involve the unipotent cuspidal character of degree \(q^2-q\), when \(q=p\) or \(p^2\), where \(p\) is a prime. Since this paper was written, more definitive results of this nature have been proved for all groups of Lie type.
scientific article; zbMATH DE number 1286707
scientific article