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

Shintani下降在完美等距问题中的应用及Dade猜想

基本信息

批准号：
11440008
负责人：
UNO Katsuhiro
金额：
$ 7.49万
依托单位：
Osaka University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
1999
资助国家：
日本
起止时间：
1999 至 2001
项目状态：
已结题

项目摘要

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.

1. 对于具有 27 阶额外特殊群作为缺陷群的 G 型 Chevalley 群的分块代数，或者具有 9 阶基本阿贝尔群作为缺陷群的射影特殊线性群或特殊酉群，我们证明了布劳猜想成立，即分块代数与其布劳尔对应者之间存在导出的等价性。这个结果意味着它们之间存在完美的等距，而且Dade的猜想是正确的。在上述情况中，由丹宁场导出的自同构的稳定子又是同一类型的Chevalley群。推导的等价性和完美的等距性被证明与服用稳定剂是兼容的。换句话说，他们与新谷血统兼容。此外，我们还证明了对于一般线性群，局部块是森田等价的，并且也与新谷血统兼容。 2．对于小阶辛群，我们证明了戴德猜想的不变形式是正确的。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。我们讨论了它与完美等距和戴德猜想的关系，并最终提出了一个包含所有这些猜想的新猜想。对于 Lyons 和 Thompson 的零星简群，证明了新猜想是正确的。当然有必要对新猜想进行更多的研究。