We determine what are the fields of values of the irreducible $p$-height zero characters of all finite groups for $p=2$; we conjecture what they should be for odd primes, and reduce this statement to a problem on blocks of quasi-simple groups.

In this paper we consider the inductive Alperin--McKay condition for isolated blocks of groups of Lie type $B$ and $C$. This finishes the verification of the inductive condition for groups of this type.

The Bonnafé–Rouquier equivalence can be seen as a modular analogue of Lusztig’s Jordan decomposition for groups of Lie type. In this paper, we show that this equivalence can be lifted to include automorphisms of the finite group of Lie type. Moreover, we prove the existence of a local version of this equivalence which satisfies similar properties.

In this paper, we show that the Alperin-McKay conjecture holds for 2-blocks of maximal defect. A major step in the proof is the verification of the inductive Alperin-McKay condition for the principal 2-block of groups of Lie type in odd characteristic.

The Alperin-McKay conjecture is a longstanding open conjecture in the representation theory of finite groups. Sp\"ath showed that the Alperin-McKay conjecture holds if the so-called inductive Alperin-McKay (iAM) condition holds for all finite simple groups. In a previous paper, the author has proved that it is enough to verify the inductive condition for quasi-isolated blocks of groups of Lie type. In this paper we show that the verification of the iAM-condition can be further reduced in many cases to isolated blocks. As a consequence of this we obtain a proof of the Alperin-McKay conjecture for 2-blocks of finite groups with abelian defect.

Späth showed that the Alperin–McKay conjecture in the representation theory of finite groups holds if the so-called inductive Alperin–McKay condition holds for all finite simple groups. In a previous article, we showed that the Bonnafé–Rouquier equivalence for blocks of finite groups of Lie type can be lifted to include automorphisms of groups of Lie type. We use our results to reduce the verification of the inductive condition for groups of Lie type to quasi-isolated blocks.

In this paper we consider the inductive Alperin-McKay condition for quasi-isolated 2-blocks of exceptional groups of Lie type. Thereby, we complete the proof of the Alperin-McKay conjecture for the prime 2.

We prove that for all non-abelian finite simple groups $S$, there exists a fake mth Galois action on IBr$(X)$ with respect to $X \lhd X \rtimes $ Aut$(X)$, where $X$ is the universal covering group of $S$ and $m$ is any non-negative integer coprime to the order of $X$. This is one of the two inductive conditions needed to prove an $\ell$-modular analogue of the Glauberman-Isaacs correspondence.

In this paper we verify Navarro's refinement of the McKay conjecture for quasi-simple groups of Lie type in their defining characteristic. Navarro's refinement takes into account the action of specific Galois automorphisms on the characters present in the McKay conjecture. Our proof of this case of the conjecture relies on a character correspondence constructed by Maslowski. Building on this we verify the inductive condition for Navarro's refinement for most groups of Lie type in defining characteristic.

We prove that the cohomology group of a Deligne–Lusztig variety defines a Morita equivalence in a case which is not covered by the argument in [2], specifically we consider the situation for semisimple elements in type D whose centralizer has non-cyclic component group. Some arguments use considerations already present in an unpublished note by Bonnafé, Dat and Rouquier.