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We give a criterion that simplifies the checking of the inductive Alperin weight condition for the remaining open cases of simple groups of Lie type (see Theorem 3.3). It is strongly related in form to the criterion of the second author for the inductive McKay conditions (see Späth, Bull. Lond. Math. Soc. 44 (2012), no. 3, 426–438, 2.12) that has proved very useful. The proof follows from a Clifford theory for weights intrinsically present in the proof of reduction theorems of the Alperin weight conjecture given by Navarro–Tiep and the second author. We also give a related criterion for the inductive blockwise Alperin weight condition (Theorem 4.5). [ABSTRACT FROM AUTHOR]

In this paper characters of the normalizer of d -split Levi subgroups in SL n (q) and SU n (q) are parametrized with a particular focus on the Clifford theory between the Levi subgroup and its normalizer. These results are applied to verify the Alperin-McKay conjecture for primes ℓ with ℓ ∤ 6 (q 2 − 1) and the Alperin weight conjecture for ℓ -blocks of those quasi-simple groups with abelian defect. The inductive Alperin-McKay condition and inductive Alperin weight condition by the second author are verified for certain blocks of SL n (q) and SU n (q). [ABSTRACT FROM AUTHOR]

We introduce H-triples and a partial order relation on them, generalizing the theory of ordering character triples developed by Navarro and Späth. This generalization takes into account the action of Galois automorphisms on characters and, together with previous results of Ladisch and Turull, allows us to reduce the Galois-McKay conjecture to a question about simple groups. [ABSTRACT FROM AUTHOR]

As a step to establish the McKay conjecture on character degrees of finite groups, we verify the inductive McKay condition introduced by Isaacs-Malle-Navarro for simple groups of Lie type An-1, split or twisted. Key to the proofs is the study of certain characters of SLn(q) and SUn(q) related to generalized Gelfand-Graev representations. As a by-product we can show that a Jordan decomposition for the characters of the latter groups is equivariant under outer automorphisms. Many ideas seem applicable to other Lie types. [ABSTRACT FROM AUTHOR]

In this paper, we propose a strengthening of Dade's Conjecture. This version, called the Character Triple Conjecture, once assumed for quasisimple groups, is shown to imply Dade's Projective Conjecture for all finite groups. In particular Dade's Projective Conjecture holds for a group whose nonabelian simple sections have only covering groups satisfying the Character Triple Conjecture. We verify the new conjecture for some classes of quasisimple groups. [ABSTRACT FROM AUTHOR]

Abstract: We prove for finite reductive groups G of classical type, that every character extends to its inertia group in N, where L is an abelian centraliser of a Sylow d-torus S of G and . This gives a precise description of the irreducible characters of N and proves according to Malle (2007) for all primes ℓ determined by S. Furthermore this enables us to verify the McKay conjecture in this situation for some primes. [Copyright &y& Elsevier]