Modular representations of profinite groups

Abstract

a b s t r a c tOur aim is to transfer several foundational results from the modular representation theory of finite groups to the wider context of profinite groups. We are thus interested in profinite modules over the completed group algebra k [[G]] of a profinite group G, where k is a finite field of characteristic p.We define the concept of relative projectivity for a profinite k[[G]]-module. We prove a characterization of finitely generated relatively projective modules analogous to the finite case with additions of interest to the profinite theory. We introduce vertices and sources for indecomposable finitely generated k [[G]]-modules and show that the expected conjugacy properties hold-for sources this requires additional assumptions. Finally we prove a direct analogue of Green's Indecomposability Theorem for finitely generated modules over a virtually pro-p group.

Block theory and Brauer's first main theorem for profinite groups

“…Remark 2.11. Lemma 2.10 may be compared with [15,Lemma 3.4], which says that when k is finite and U, W are finitely generated, then if U N is isomorphic to a direct summand of W N for each N , then U is isomorphic to a direct summand of W . On one hand Lemma 2.10 is stronger, as it does not require that W be finitely generated.…”

“…This suggests that the block theoretic approach to the representation theory of profinite groups will be as central to the theory as it is with finite groups. For comparison, we note that with modules, conditions on the group are frequently required to prove the most powerful results -see for instance [14,15].…”

Green correspondence for virtually pro-p groups

“…By the Mackey decomposition formula [5, 2.2] we have [3, 6.2] the module x(V ) is projective relative to xQ x −1 so by [3,4.7] we can choose a kJxQ…”

“…The main concepts mentioned in this paragraph are introduced and discussed in [3]. Let G be a virtually pro-p group and k a finite field of characteristic p. All modules are assumed to be profinite left modules.…”

Blocks of profinite groups with cyclic defect group

“…The study of the modular representation theory of profinite groups was begun in [13,12], while the study of blocks and defect groups has been initiated recently in [5]. In this article we classify, in Theorem 4.3, the blocks of an arbitrary profinite group whose defect groups are cyclic (meaning either a finite cyclic p-group or the p-adic integers Z p ): they are Brauer tree algebras in strict analogy with the finite case.…”