Application of Shintani descent to the perfect isometry problem and Dade's conjecture

1. 对于具有27阶缺陷群的G型Chevalley群的块代数，或具有27阶缺陷群的射影特殊线性群或特殊酉群的块代数。作为缺陷群的9阶初等阿贝尔群，我们证明了Broue猜想成立，即块代数与其对应的Brauer代数之间存在派生的等价。这一结果表明它们之间存在着完美的等距，并且证明了戴德猜想的正确性。在上述情况下，自同构的稳定器由denning场的稳定器导出，同样是同类型的Chevalley群。推导出的等价，从而证明了完全等距与取稳定器是相容的。换句话说，他们与新谷血统是相容的。此外，我们还证明了对于一般线性群，局部块是Morita等价的，并且与Shintani下降相容。对于小秩的辛群，我们证明了Dade猜想的不变形式是正确的。但是，我们还没有检查它是否与新谷后裔兼容。同时，艾萨克斯和纳瓦罗提出了一个猜想。我们讨论了它与完全等距和戴德猜想的关系，最后提出了一个包含它们的新猜想。对于散发性的Lyons和Thompson单群，证明了新猜想是正确的。当然有必要进一步研究这个新猜想。

1. For block algebras of Chevalley groups of type G, having an extra special group of order 27 as defect groups, or of projective special linear groups or special unitary groups, having an.elementary abelian group of order 9 as defect groups, we have shown that Broue conjecture holds, namely, that there exists a derived equivalence between the block algebra and its Brauer correspondent. This result implies that there exists a perfect isometry between them, and moreover that Dade's conjecture is true.In the situations above, the stabilizers of an automorphism induced from those of the denning fields are again Chevalley groups of the same type. The derived equivalences, and thus perfect isometries are shown to be compatible with taking the stabilizers. In other words, they are compatible with the Shintani descent. Moreover, we have proved that for general linear groups, local blocks are Morita equivalent and it is also compatible with Shintani descent.2. For syinplectic groups of small rank, we have proved that the invariant form of Dade's conjecture is true. However, we have not checked that it is compatible with Shintani descent.3. Meanwhile, Isaacs and Navarro proposed a conjecture. We have discussed the relationship between it and perfect isometries and Dade's conjecture, and finally proposed a new conjecture which includes all of them. For the sporadic simple groups of Lyons and Thompson, it is shown that the new conjecture is true. It is certainly necessary to study the new conjecture more.