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Jordan decomposition for the Alperin-McKay conjecture / vorgelegt von Lucas Ruhstorfer. Wuppertal, [2020]
Content
Representation theory
Modular representation theory
Module categories
The Brauer functor
Brauer pairs and the Brauer category
Morita equivalences and splendid Rickard equivalences
First properties of splendid complexes
Brauer categories and splendid Rickard equivalences
Properties of splendid Rickard equivalences
Lifting Rickard equivalences
Descent of Rickard equivalences
Morita equivalences and Clifford theory of characters
Rickard equivalences for the normalizer
The Brauer functor and Clifford theory
The Harris–Knörr correspondence
Splendid Rickard equivalences and Clifford theory
Deligne–Lusztig theory and disconnected reductive groups
Disconnected reductive algebraic groups
-adic cohomology of Deligne–Lusztig varieties
Properties of Deligne–Lusztig varieties
Godement resolutions
Isogenies
Duality for connected reductive groups
Levi subgroups, isogenies and duality
Rational Lusztig series for connected reductive groups
Lusztig series for disconnected reductive groups
Lusztig series and Brauer morphism
Regular embedding and Lusztig series
The Bonnafé–Dat–Rouquier Morita equivalence
On the Bonnafé, Dat and Rouquier Morita equivalence
A remark on Clifford theory
Steinberg relation
Notation
Classifying semisimple conjugacy classes
Computations in the Weyl group
Representation theory
Proof of Morita equivalence
Equivariant Morita equivalence and local equivalences
Automorphisms of simple groups of Lie type
Equivariance of Deligne–Lusztig induction
Automorphisms and stabilizers of idempotents
Generalizations to disconnected reductive groups
Independence of Godement resolution
Comparing Rickard and Morita equivalences
Morita equivalences for local subgroups
Extending the Morita equivalence
Disconnected reductive groups and Morita equivalences
Local equivalences
Restriction of scalars for Deligne–Lusztig varieties
Duality in the context of restriction of scalars
Comparing Weyl groups
Restriction of scalars and Lusztig series
Restriction of scalars and Jordan decomposition of characters
Reduction to isolated series
Jordan decomposition for local subgroups
Application to the inductive Alperin–McKay condition
The inductive Alperin–McKay condition
A criterion for block isomorphic character triples
A condition on the stabilizer and the inductive conditions
Extension of characters
The case D4
A first reduction of the iAM-condition
Quasi-isolated blocks
Normal subgroups and character triple bijections
Application of character triples
Jordan decomposition for the Alperin–McKay conjecture