On the quotient ring by diagonal harmonics

Abstract

For a Weyl group W and its reflection representation mathfrak{h}, we find the character and Hilbert series for a quotient ring of C[mathfrak{h} oplus mathfrak{h}^*] by an ideal containing the W--invariant polynomials without constant term. This confirms conjectures of Haiman. The proof makes use of rational Cherednik algebras, as studied by Etingof and Ginzburg, and others.