Adequate subgroups and indecomposable modules

Abstract

Abstract. The notion of adequate subgroups was introduced by Jack Thorne [59]. It is a weakening of the notion of big subgroups used by Wiles and Taylor in proving automorphy lifting theorems for certain Galois representations. Using this idea, Thorne was able to strengthen many automorphy lifting theorems. It was shown in [22] and [23] that if the dimension is smaller than the characteristic then almost all absolutely irreducible representations are adequate. We extend the results by considering all absolutely irreducible modules in characteristic p of dimension p. This relies on a modified definition of adequacy, provided by Thorne in [60], which allows p to divide the dimension of the module. We prove adequacy for almost all irreducible representations of SL2(p a ) in the natural characteristic and for finite groups of Lie type as long as the field of definition is sufficiently large. We also essentially classify indecomposable modules in characteristic p of dimension less than 2p − 2 and answer a question of Serre concerning complete reducibility of subgroups in classical groups of low dimension.

The Irreducible Subgroups of Exceptional Algebraic Groups

“…, a(G r )) otherwise, where G i are all the simple quotients of G, as proved by Jantzen, McNinch and Liebeck-Seitz (see [Ser05,Theorem 4.4]). There are also results concerning the complete reducibility of finite subgroups of G, see [Gur99,Theorem A], [GHTar,Theorem 1.9], and [Litar,Corollary 5].…”

A 2-adic automorphy lifting theorem for unitary groups over CM fields

“…The results [32,Theorem 7.1] and [32,Theorem 9.1] hold with the assumption 'ρ(G F(ζ l ) ) ⊂ GL n (k) is adequate, in the sense of [32,Definition 2.3]' replaced with the following assumption: (GL n (F l ), ad 0 ) = 0), but there are many subgroups of GL n (F l ) which are adequate in the sense of Definition 2.20, as follows from [13,Theorem 11.5].…”

“…However, in the case l|n, Corollary 7.3 is a new result, which relies upon the new definition of adequacy given in [13].…”

“…The definition of 'adequate subgroup' that we use was first written down in the case p = 2 by Guralnick et al [13]; it is pleasant to see that this turns out to be the right definition to prove automorphy lifting theorems in this case.…”

“…Acknowledgements This paper was motivated by the definition of 'adequate subgroup' given by Guralnick, Herzig and Tiep in the cases p|n and p = 2 [13]. I thank these authors for their interest in this problem.…”