Coxeter classes of unitary reflection groups
type: Item
instance of: scholarly article
title: Coxeter classes of unitary reflection groups
zbMATH Open document ID: 0833.20051
DOI: 10.1007/BF01245184
published in: Inventiones Mathematicae
publication date: 1995-08-29T00:00:00Z
full work available at URL: https://eudml.org/doc/144258
review text: Let \(K\) be an algebraically closed field of characteristic 0 and let \(\overline {F}\) be the set of all functions \(f : \mathbb{Q} \to K\) whose support is contained in a subset of \(\mathbb{Q}\) of the form \((1/b)(a+\mathbb{N})\) for some integers \(a \in \mathbb{Z}\), \(b > 0\). If \(f,g \in \overline {F}\), then the functions \(f + g : r \mapsto f(r) + g(r)\) and \(fg : r \mapsto \sum_{s \in \mathbb{Q}} f(r - s) g(s)\) are also contained in \(\overline {F}\), and these operations turn \(\overline {F}\) into a field. Let \(\varepsilon \in \overline {F}\) be defined by \(\varepsilon (x)=\delta_{1x}\), and more generally, for \(r \in \mathbb{Q}\), let \(\varepsilon^r\) be defined by \(\varepsilon^r(x)=\delta_{rx}\). The ring of formal power series \(A=K\varepsilon\) embeds naturally into \(\overline {F}\), along with its field of fractions \(F=K((\varepsilon))\), and so do \(K[[\varepsilon^{1/n}]]\) and \(K((\varepsilon^{1/n}))\) for every \(n \geq 1\). For \(x \in \overline {F}^*\), let \(v(x)=\min \{r \in \mathbb{Q}\mid x_r \neq 0\}\), and let \(v(0)=\infty\). Let \({\mathfrak n}=\{x \in \overline {F} \mid v(x) > 0\}\), and \({\mathfrak m}={\mathfrak n} \cap A\). For \(n \in \mathbb{N}\) we let \({\mathfrak m}^{\times n}\) (resp. \({\mathfrak n}^{\times n})\) denote the cartesian product of \(n\) copies of \(\mathfrak m\) (resp. \(\mathfrak n\)). It is known that \(\overline {F}\) is an algebraic closure of \(F\) [\textit{J.-P. Serre}, Local fields (Graduate Texts Math. 67, 1979; Zbl 0423.12016)]. Since \(\overline {F}\) is generated over \(F\) by \((\varepsilon^{1/n})_{n \geq 1}\), the Galois group \(\mathcal G\) of \(\overline {F}\) over \(F\) is generated topologically by a single element. Let \(\sigma \in {\mathcal G}\). For \(q \in \mathbb{Q}\), let \(\overline {\sigma} (q)=\varepsilon^{-q} \sigma (\varepsilon^q) \in K\). Then \(\overline {\sigma} : \mathbb{Q} \to K^*\) is a group homomorphism, and \(\sigma\) is a topological generator of \(\mathcal G…
reviewed by: Chen Chengdong
Mathematics Subject Classification ID: 20F55
Mathematics Subject Classification ID: 51F15
Mathematics Subject Classification ID: 06A06
Mathematics Subject Classification ID: 20F05
zbMATH DE Number: 783990
zbMATH Keywords: ring of formal power series
zbMATH Keywords: Galois group
zbMATH Keywords: topological generator
zbMATH Keywords: generating family
zbMATH Keywords: homogeneous functions
zbMATH Keywords: order relation
zbMATH Keywords: conjugacy classes
zbMATH Keywords: homogeneous generators
zbMATH Keywords: Weyl groups
zbMATH Keywords: generated by pseudoreflections
zbMATH Keywords: Coxeter groups
author: Nicolas Spaltenstein
MaRDI profile type: Publication
cites work: Q5580319
cites work: Q4095018
cites work: Finite complex reflection groups
cites work: Fixed point varieties on affine flag manifolds
cites work: Lie Group Representations on Polynomial Rings
cites work: Finite Unitary Reflection Groups
cites work: Order relations on conjugacy classes and the Kazhdan-Lusztig map
cites work: Regular elements of finite reflection groups
cites work: Invariant theory
description: scientific article; zbMATH DE number 783990