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

1.对于G型Chevalley群的块代数，如果亏群是27阶的额外特殊群，或者对于射影特殊线性群或特殊酉群的块代数，如果亏群是9阶的初等交换群，我们证明了Broue猜想成立，即块代数与它的Brauer对应群之间存在一个导出等价。这个结果意味着它们之间存在完美等距，而且Dade猜想是正确的。在上述情况下，由denning域的自同构导出的自同构的稳定子又是同一类型的Chevalley群。导出的等价，从而完美的等距被证明是兼容的稳定剂。换句话说，他们与Shintani血统兼容。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.对于小秩辛群，我们证明了Dade猜想的不变式成立。然而，我们还没有检查它是否与Shintani血统兼容。3.与此同时，艾萨克斯和纳瓦罗提出了一个猜想。讨论了它与完全等距和Dade猜想的关系，并提出了一个包含这两个猜想的新猜想。对于里昂和汤普森的零星单群，证明了新猜想成立。当然有必要对这个新猜想进行更多的研究。

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.