Root-power quotients of nil-Hecke algebras and Weyl invariant theory

We study nil-Hecke quotients defined by powers of root coordinates. We identify their central ideals through the determinant components of root-orbit power ideals. Apolarity and discriminant removal reduce the graded dual centers to directional bounds on invariant polynomials. We determine their Hilbert series for every exponent and root orbit in all irreducible crystallographic types except $E_6,E_7,E_8$. Relative discriminants compare centers for reflection subgroups and transfer equivariant resolutions. Coordinate-root centers in types $B$ and $C$ are Grassmannian cohomology rings. We classify self-injectivity in type $A$, in rank two, and in type $F_4$, and determine the first nonsemisimple centers in the remaining classical families. For an explicit open set of reflection parameters, the polynomial image of the rational Cherednik algebra at $t=0$ is the full nil-Hecke algebra. This realizes our algebras as Cherednik quotients and their finite centers as closed subschemes on a section of the generalized Calogero–Moser space.