Canonical systems of basic invariants for unitary reflection groups (Q2827415)

The authors study canonical systems for the finite unitary reflection groups. Let \(V\) be an \(n\)-dimensional unitary space and let \(W\subset U(V)\) be a finite unitary reflection group. Denote by \(S\) the symmetric algebra \(S(V^*)\) of the dual space \(V^*\). The subalgebra \(R=S^W\) of \(W\)-invariant polynomials of \(S\) is generated by \(n\) algebraically independent homogeneous polynomials. A system of such generators is called a system of basic invariants of \(R\). A system \(\{f_1,\dots,f_n\}\) of basic invariants is said to be canonical if it satisfies the following system of partial differential equations: \(f_i^*f_j=\delta_{ij}\) for \(i,j=1,\dots,n\), where \(\delta_{ij}\) is the Kronecker delta and \(f_i^*\) is a differential operator. In a previous work [J. Algebra 406, 143--153 (2014; Zbl 1328.13008)] the first and the third authors obtained an explicit formula for a canonical system for every reflection group. In the main result (Theorem 1.2) is proven the existence of the canonical systems for the finite unitary reflection groups. The proof is independent of the classification of unitary reflection groups. Moreover, the authors give an explicit formula (Theorem 4.3) for a canonical system which is also classification free.