Applications of Cohomology groups and Shintani descent to Broue's conjecture

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1 (1)For low rank finite Chevalley groups G of type D and F, and for p=5, we constructed a perfect isometry between the principal blocks of G and a Sylow normalizer. We also determine the number of isometries. Moreover, for blocks of cyclic defect groups, we proved that for any bijection between irreducible characters there is a derived equivalence inducing the bijecyion, provided that the local structure of the blocks are the same,(2)For finite Chevaley groups of type G, and for p=3, we establish certain Morita equivalence between principal blocks for several defining fields.(3)We proved that Broue's conjecture holds for block with elementary abelian defect groups of order 9.(4)We have obtained an easy formula for elementary divisors of the Cartan matrices of symmetric groups.(5)We gave a module theoretic proof for the Mislin's theorem on group fusions.(6)The fundamental notions of group cohomologies are extended for arbitrary blocks.2 (1)We extend Dade's conjecture, and check that it holds for many sporadic simple groups and some finte Chevaley groups.(2)For a block with dihedral defect groups of order eight, we construct functors among the derived categories of normalizers of radical chains which induce the relations between the numbers of simple modules.