Normalizers of Parabolic Subgroups of Reflection Groups
Normalizers of Parabolic Subgroups of Reflection Groups
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DOI：
10.1112/jlms/s2-21.1.62
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发表时间：
1980-02
影响因子：
1.2
通讯作者：
R. Howlett
中科院分区：
文献类型：
--
作者：
R. Howlett
设X是真实的域U上的有限维向量空间，且X具有正定内积。对于x ∈ X，设wv是在与x正交的超平面中的反射，设H^是由反射生成的有限群。因此W是0（X）的子群。在X中定义一个对应于Win的根系Z（例如见[2]），设I+，I”和n分别是相对于某个固定序的正根，负根和基本根。若Kczfl令WK为抛物子群（wa\aeK）.本文的目的是描述在所有情况下WK在W中的正规化子Nw（WK）的结构。G. Lusztig（在§ 5的”Coxeter轨道和Frobenius的本征空间”发明数学，38（1977），101-159），并且裁判员告诉我，关于下面定义的群W的大部分信息已经由D. Deriziotis（Ph. D.论文，沃里克大学（1977））。对于Kczn，设ZK是WK的根系，由K中包含的所有支撑的元素组成。对于每个W，令N（W）={a e I+| w（a）eS”}并且令f（w）=\N（w）\（N（w）的基数）。下面的引理包含了一个集合的杂项众所周知的事实需要以后。
Let X be a finite dimensional vector space over the real field U, and suppose that X is equipped with a positive definite inner product. For x e X let wv be the reflection in the hyperplane orthogonal to x, and let H^ be a finite group generated by reflections. Thus Wis a subgroup of 0 {X). Define a root system Z in X corresponding to Win the usual way (see [2] for example), and let I+, I" and n be respectively the positive, negative and fundamental roots relative to some fixed ordering. If K cz fl let WK be the parabolic subgroup (wa\a e K). The purpose of this paper is to describe the structure of Nw {WK), the normalizer of WK in W, in all cases. Some similar results have been obtained by G. Lusztig (in § 5 of" Coxeter Orbits and Eigenspaces of Frobenius" Inventiones Math., 38 (1977), 101-159), and the referee informs me that much of the information about the group W defined below has been obtained by D. Deriziotis (Ph. D. thesis, Warwick University (1977)). For K cz n let ZK be the root system of WK consisting of all elements of£ with support contained in K. For each weW let N (w)={a e I+| w (a) eS"} and let/(w)=\N (w)\(the cardinality of N (w)). The following lemma contains a collection of miscellaneous well known facts required later.