A Study on Glauberman-Watanabe Correspondence and Derived Equivalence of Blocks

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In the research of this project, we studied "the abelian defect conjecture" which is one of main problems in representation theory of finite groups. We were mainly concerned with Glauberman-Watanabe correspondence.1. We have constructed a two sided complex in the principal blocks of the group U(4,q^2) which gives a stable equivalence. In our investigation. we have solved conjectures on complexes of some module categories related to Alvis-Curtis-Kawanaka duality in finite reductive groups and to self-equivalences in Hecke algebras of finite Coxeter groups.2. We studied the problem raised by Holloway-Koshitani-Kunugi which concerns with relations between blocks of SL(2,q) and its extension group by a field automorphism group. And we constructed some one sided complex for the group Sz(q) which we might expect to be a tilting complex. Blocks appearing here have non-abelian defect groups with cyclic focal subgroup.3. We also worked on a problem concerning with quasi-perfect isometries which is a generalization of notion of perfect isometries recently studied by Narasaki-Uno. And we solved their conjecture for the group U(4,q^2) and Sz(q) over the field with defining characteristic. We studied a perfect isometriy between the blocks of the groups ^2F_4(2) and ^2F_4(q).