Coxeter classes of unitary reflection groups
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 \(\mathcal G\) if and only if \(\text{Ker} (\overline {\sigma})=\mathbb{Z}\), or equivalently if \(\overline {\sigma} (1/n)\) is a primitive \(n\)th root of 1 for every integer \(n \geq 1\). Consider a finite dimensional \(K\)-vector space \(V\) and a finite group \(W\) of automorphisms of \(V\). Then \(W \times {\mathcal G}\) acts on \(V_{\overline{F}}=V \otimes_K \overline{F}\). Let also \(V_F=V \otimes_K F\). Let \(w \in W\). Given a topological generator \(\sigma\) of \(\mathcal G\), let \(V_{w, \sigma}=\{x \in V_{\overline{F}} \mid w (\sigma(x))=x\}\). Let \(f : V_F \to F\) be a \(W\)-invariant polynomial function. Then \(f\) extends in a natural way to a polynomial function from \(V_{\overline {F}}\) to \(\overline{F}\), which we still denote by \(f\), and which is \(W\)-invariant and \(\mathcal G\)- equivariant. Let \(f_1,\dots,f_m\) be a generating family for the \(K\)-algebra \(K[V]^W\) consisting of homogeneous functions of positive degree. Then \(\psi = (f_1, \dots, f_m) : V \to K^m\) is a finite morphism whose non-empty fibers are the \(W\)-orbits in \(V\). Let \(V^+_{w, \sigma} = V_{w, \sigma} \cap \psi^{-1} ({\mathfrak n}^{\times m})\). Using the above map \(\psi\), the author defines an order relation on the set \(W/ \sim\) of all conjugacy classes in \(W\). Definition: Let \(\sigma\) be a topological generator of \(\mathcal G\). The order on \(W/\sim\) induced by \(\sigma\) is defined by the following condition. Let \(C\), \(C'\) be conjugacy classes in \(W\) and let \(w \in C\), \(w' \in C'\). Then \(C \leq_\sigma C'\) if \(\psi(V^+_{w, \sigma}) \subset \psi(V^+_{w', \sigma})\). The author shows that this definition is independent of the choice of the family of homogeneous generators of \(K[V]^W\) used to define \(\psi\). But the order relation is affected in general by the choice of \(\sigma\). On the other hand, if \(W\) has the property that for every integer \(m\) prime to \(|W|\), every \(w \in W\) is conjugate to \(w^m\), then the order relation is independent of the choice of \(\sigma\). This holds in particular for Weyl groups. If \(W\) is generated by pseudoreflections, there is a largest conjugacy class for this order relation. In the case of Coxeter groups, this class is the usual Coxeter class. The author discusses also the order relation for some groups.
scientific article; zbMATH DE number 783990