A Study on Derived Equivalences of Blocks and Shintani Descent in Group Algebras

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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 concerned with Shintani descent and Glauberman correspondent.1.We have constructed one sided tilting complexes for finite Chevalley groups of small rank, for example, U(3,q^2)、Sp(4,q)、G_2(q). We could construct Splendid tilting complexes which coincide with these complexes as one side complexes and studied perfect isometries between character rings induced by them.2.The situations described in 1 can be considered problems of derived category version of "Glauberman correspondent". We have studied solvable group case and obtain some generalization of results by Harris and Linckelmann. And we clarify the strategy raised by Rouquier in this setting.3.We also worked on a problem giving quiver presentations for various types of 5-Sylow normalizers of rank 2 to apply for constructing tilting complexes of some finite Chevalley groups, for example, ^2F_4(q) of which 5-Sylow normalizer is of 4S_4-type.