Harish-chandra Vertices

The irreducible complex representations of nite groups of Lie type are divided into series by means of the Harish-Chandra philosophy, whose starting point is the concept of cuspidal characters (see e.g..Sr]). The classiication of the irreducible characters of such a group G can be done in two steps: First nd all cuspidal irreducible characters of Levi subgroups L of G. Then apply Harish-Chandra induction and decompose the resulting character. This can be done using Howlett-Lehrer theory HL]. Recently in DF] a p-modular version of the Harish-Chandra theory was introduced by the rst author and Fleischmann, and Hiss showed subsequently in Hi], that the modular irreducible Brauer characters decompose in the same fashion into series as the ordinary irreducible characters. Here p is any prime which divides the order of G, but is diierent from the describing characteristic, that is the characteristic of the eld underlying G. The similarity with Green's vertex theory is evident (see e.g. Hi]): Here the p-modular indecomposable representations of a nite group G for primes p dividing the order of G are divided into series parametrized by G-conjugacy classes of p-subgroups of G (the vertex of a representation), and for a xed vertex containing the p-subgroup P of G we have as second parameter N G (P)-conjugacy classes of indecomposable representations of P, where N G (P) denotes the normalizer of P in G. There are two major diierences: First the vertex theory works for arbitrary nite groups, the Harish-Chandra philosophy only for nite groups with split BN-pair. In this regard, vertex theory is more general. Secondly vertex theory uses induction and restriction, that is subgroups of G (and one of the major results in the eld is, that p-subgroups suuce), The rst author was supported by the NSF under grant 9002606. The second author gratefully acknowledges support received from ARC under grant L20.24210. He also wishes to thank the University of Oklahoma for its hospitality during his visit in Spring 1991.