Supercuspidal representations of finite reductive groups
Parabolic induction is a method to construct, in a systematic way, representations for a group \(G\) of Lie type from those of Levi subgroups. An irreducible representation of \(G\) is called cuspidal if it does not occur as a subrepresentation of a parabolically induced representation from a proper Levi subgroup. M.-F. Vignéras introduced the notion of supercuspidality for modular representations of reductive \(p\)-adic groups in non describing characteristic (i.e., non equal to \(p\)): an irreducible representation is called supercuspidal if it does not occur as a subquotient of a parabolically induced representation from a proper Levi subgroup. Thus, in particular, a supercuspidal representation is cuspidal. If the modular characteristic does not divide the order of the group \(G\), then all representations of \(G\) are semisimple. So each subquotient is also a subrepresentation. In this case any cuspidal representation is supercuspidal. In the paper under review, the author begins a systematic investigation of the supercuspidal irreducible representations of finite groups of Lie type. He shows that an irreducible representation is supercuspidal if and only if every composition factor of its projective cover is cuspidal and conjectures that a supercuspidal representation is liftable to characteristic 0. He also shows that the truth of this conjecture is implied by the truth of a conjecture of M. Geck, which is known to be true for the groups \(\text{GL}_n(q)\), \(\text{GU}_n(q)\) as well as for split classical groups and even modular characteristic (under the assumption that the order of the finite field corresponding to the group is large), and in some cases for exceptional groups. In this well written and interesting paper, the author also determines all supercuspidal unipotent representations of the classical groups (in odd characteristic) and of some exceptional groups of Lie type. The cases of Suzuki and Ree groups are also considered. The results indicate that a complete classification of all supercuspidal irreducible representations seems to be possible.
supercuspidal irreducible representations
scientific article; zbMATH DE number 941269