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On \(m\)-quasi-invariants of a Coxeter group. (English) Zbl 1028.81027
Summary: Let \(W\) be a finite Coxeter group in a Euclidean vector space \(V\), and let \(m\) be a \(W\)-invariant \(\mathbb Z_+\)-valued function on the set of reflections in W. Chalykh and Veselov introduced an interesting algebra \(Q_m\), called the algebra of \(m\)-quasi-invariants for \(W\), such that \(\mathbb C[V]^W\subseteq Q_m\subseteq \mathbb C[V]\), \(Q_0= \mathbb C[V]\), and \(Q_m\supseteq Q_{m'}\) whenever \(m\leq m'\). Namely, \(Q_m\) is the algebra of quantum integrals of the rational Calogero-Moser system with coupling constant \(m\). The algebra \(Q_m\) was studied in [O. A. Chalykh and A. P. Veselov, Commun. Math. Phys. 126, 597–611 (1990; Zbl 0746.47025); A. P. Veselov, K. L. Styrkas and O. A. Chalykh, Teor. Mat. Fiz. 94, 253–275 (1993), English translation in Theor. Math. Phys. 94, 182–197 (1994; Zbl 0805.47070); G. Felder and A. Veselov, Action of Coxeter groups on \(m\)-harmonic polynomials and KZ equations, Mosc. Math. J. 3, No. 4, 1269–1291 (2003), see also http://arxiv.org/abs/math.QA/0108012 and M. Feigin and A. P. Veselov, Int. Math. Res. Not. 2002, No. 10, 521–545 (2003; Zbl 1009.20044)]. In particular, Feigin and Veselov propose \(Q\) a number of interesting conjectures concerning the structure of \(Q_m\) and verified them for dihedral groups and constant functions \(m\). Our objective is to prove some of these conjectures in the general case.

MSC:
81R12 Groups and algebras in quantum theory and relations with integrable systems
20F55 Reflection and Coxeter groups (group-theoretic aspects)
13A50 Actions of groups on commutative rings; invariant theory
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