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On the invariants of the cohomology of complements of Coxeter arrangements. (English) Zbl 1455.20003
Let \(W\) be a finite Coxeter group with a (Coxeter) generating set \(S\) of order \(\ell\), \(V_\mathbb{R}\) be an \(\ell\)-dimensional real vector space that affords the reflection representation of \(W\), \(V= \mathbb{C} \otimes_{\mathbb{R}} V_{\mathbb{R}}\) be the complexification of \(V_{\mathbb{R}}\) and let \(R\) be the set of reflections of \(W\). If \(W\) is seen as a subgroup \(\mathrm{GL}(V)\), for every \(r\in R\) let \(V^{r}\) be the hyperplane of fixed points of \(r\) in \(V\) and let \(\mathcal{A} =\{ V^{r} \mid r \in R \}\). Then \((V, \mathcal{A})\) is the complexification of a Coxeter arrangement.
The group \(W\) acts naturally on the complement \(M_{W}\) of the hyperplanes in \(\mathcal{A}\), and hence on the cohomology of \(M_{W}\) as algebra automorphisms. For \(p \geq 0\) let \(H^{p}(M_{W})\) denote the \(p^{\mathrm{th}}\) de Rahm cohomology space of \(M_{W}\).
G. Felder and A. P. Veselov [J. Eur. Math. Soc. (JEMS) 7, No. 1, 101–116 (2005; Zbl 1070.20045)] have conjectured an explicit construction of \(H^{p}(M_{W})^{W}\), the space of \(W\)-invariants in \(H^{p} (M_{W})\), in terms of so-called special involutions and they have verified their conjecture for all Coxeter groups except those with irreducible components of type \(E_{7}\), \(E_{8}\), \(F_{4}\), \(H_{3}\) or \(H_{4}\).
The purpose of the paper under review is to complete the proof of the conjecture of Felder-Veselov (Theorem 2.1). The authors get their result by refining E. Brieskorn’s [Lect. Notes Math. 317, 21–44 (1973; Zbl 0277.55003)] study of the cohomology of the complement of the reflection arrangement of the Coxeter group \(W\).
MSC:
20C08 Hecke algebras and their representations
20F55 Reflection and Coxeter groups (group-theoretic aspects)
05E10 Combinatorial aspects of representation theory
52C35 Arrangements of points, flats, hyperplanes (aspects of discrete geometry)
Software:
CHEVIE; GAP
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