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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.

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