Modular representations of split 𝐵𝑁 pairs

Abstract

Introduction. In 1964 C. W. Curtis classified the absolutely irreducible modular representations of a large class of groups, the so called finite groups of Lie type. This was done by finding in each irreducible module an element, called a weight element, which is an eigenvector for certain elements in the modular group algebra; proving that two irreducible modules are isomorphic if and only if the corresponding eigenvalues (the collection of such being called a weight) are equal; and finally by determining which weights are associated with irreducible modules, i.e. which weights actually occur. This paper is a continuation of Curtis' work. We construct the absolutely irreducible modular representations of a finite group of Lie type by finding weight elements in the modular group algebra which generate a full set of nonisomorphic minimal left ideals. In some work of Steinberg and Curtis (see [14] and [6]) these representations were constructed for the covering groups of the Chevalley groups using the representations of the associated modular Lie algebras. The discussion given here avoids the Lie algebras altogether. Instead we construct each irreducible submodule of certain induced modules by finding the required weight elements. We prove at the same time that these induced modules have multiplicity free socles. Some remarks are made about the related questions of degrees and block structure of these representations. The finite groups of Lie type whose representations were classified by Curtis in [7] were defined by a large number of axioms. These apparently consisted of basic properties possessed by all known examples of such groups, the Chevalley groups and variations thereon as defined by Steinberg [13], for example. A simplified axiom scheme is used here, the heart of which is the axioms for groups with BN pairs (Tits [16]). The groups discussed in [7] satisfy the simplified axioms. The disadvantage of introducing new axioms is that the classification described in the first paragraph of this introduction must be redone. The proofs in this reworking are always similar to Curtis' but rarely identical, for the new axioms are enough weaker that some of the properties Curtis used do not follow from them. The advantages of reworking the classification theorem are twofold. First, the theorem is proved in greater generality. Secondly, the commutator relation is never assumed. It is usually replaced by arguments involving lengths of words in the Weyl group. The effect is to clarify the role of the root system in the classification theorem.