Quasi-Exactly Solvable Hamiltonians related to Root Spaces

sl(2)-Quasi-Exactly-Solvable (QES) generalization of the rational A n , BC n , G 2 , F 4 , E 6,7,8 Olshanetsky-Perelomov Hamiltonians including many-body Calogero Hamiltonian is found. This generalization has a form of anharmonic perturbations and it appears naturally when the original rational Hamiltonian is written in a certain Weyl-invariant polynomial variables. It is demonstrated that for the QES Hamiltonian there exists a finite-dimensional invariant subspace in inhomogeneous polynomials. Eigenfunctions and corresponding eigenvalues which belong to this subspace are calculated algebraically.