Orbits and invariants associated with a pair of commuting involutions
Orbits and invariants associated with a pair of commuting involutions
复制标题
与一对通勤对合相关的轨道和不变量
DOI：
10.1215/s0012-7094-01-10622-4
复制
发表时间：
2001
影响因子：
2.5
通讯作者：
Gerald W. Schwarz
中科院分区：
文献类型：
--
作者：
A. Helminck;Gerald W. Schwarz
设σ，θ是连通约化代数群G的交换对合，其中σ，θ和G定义在(通常是代数闭的)域k上，char k=2.我们有不动点群H：=G和K：=G以及一个作用(H×K)×G→G，其中((h，k)，g)→hgk−1，h∈H，k∈K，g∈G.令G//(H×K)表示Spec O(G)H×K(范畴商).设A是G的子环S中的极大群，使得θ(S)=σ(S)=S−1对所有S∈S，有相应的Weyl群W：=WH×K(A).我们证明：·包含A→G诱导同构A/W∼→G//(H×K)。特别地，闭合的(H×K)轨道正好是那些与A相交的轨道。·G→G//(H×K)的纤维与某些相关的对称簇中的纤维相同。具体地说，纤维由有限多个轨道组成。我们研究：·W的结构及其与其他自然产生的Weyl群的关系，以及σθ在的A-权空间上的作用。·A/W和G//(H×K)轨道类型层结的关系。在此过程中，我们简化了Richardson关于对称情形σ=θ的一些证明，并在最后快速恢复了Berger，Flensted-Jensen，Hogenom和Matsuki[Ber57，FJ78，Hoo84，Mat97]对于k=的情形的结果。
Let σ, θ be commuting involutions of the connected reductive algebraic group G where σ, θ and G are defined over a (usually algebraically closed) field k, char k = 2. We have fixed point groups H := G and K := G and an action (H × K ) × G → G, where ((h, k), g) → hgk−1, h ∈ H, k ∈ K, g ∈ G. Let G//(H × K ) denote Spec O(G)H×K (the categorical quotient). Let A be maximal among subtori S of G such that θ(s) = σ(s) = s−1 for all s ∈ S. There is the associated Weyl group W := WH×K (A). We show: • The inclusion A → G induces an isomorphism A/W ∼ → G//(H × K ). In particular, the closed (H × K )-orbits are precisely those which intersect A. • The fibers of G → G//(H × K ) are the same as those occurring in certain associated symmetric varieties. In particular, the fibers consist of finitely many orbits. We investigate: • The structure of W and its relation to other naturally occurring Weyl groups and to the action of σθ on the A-weight spaces of . • The relation of the orbit type stratifications of A/W and G//(H × K ). Along the way we simplify some of Richardson’s proofs for the symmetric case σ = θ, and at the end we quickly recover results of Berger, Flensted-Jensen, Hoogenboom and Matsuki [Ber57, FJ78, Hoo84, Mat97] for the case k = .