Galois group of Looijenga-Lyashko mapping (Q1818770)

Let \(\Pi= \{y^k+ a_1y^{k-1}+\cdots+ a_n: a_1,\dots, a_n\in \mathbb{C}\}\) be the space of complex polynomials of leading coefficient \(1\). By \(\Delta\subset\Pi\) we denote the discriminent hypersurface consisting of polynomials having multiple roots. The Looijenga-Lyashko mapping is defined by the function \(L: T-B\to \Pi-\Delta\) \[ L(t)= (y- w_1(t))\cdots(y- w_k(t)),\quad t\in\mathbb{C}^k, \] where \(w_i(t)\) are critical values of the semiuniversal deformation \(\phi(x,t)\) of a quasihomogeneous polynomial \(f(x)\), and \(B= L^{-1}(\Delta)\). The author studies the Galois group -- the group of covering transformations of the Looijenga-Lyashko mapping which is a ramified covering from the parameter space of a semiuniversal deformation of the singularity to the polynomial space. The author proves that if the simple singularity \(f\) is of type \(A_n\), \(D_n\), \(E_n\) and \(G\) is a Galois group of the corresponding Looijenga-Lyashko mapping then \(G\cong Z_A(c)/\{\pm 1\}\), where \(Z_A(c)\) is the centralizer of \(c\) in the \(A\)-automorphism group of a root system of the singularity and \(c\) is a Coxeter element in the corresponding Weyl group \(W\subset A\).