  The purpose of this article is to study the relationship between numerical
invariants of certain subspace arrangements coming from reflection groups and
numerical invariants arising in the representation theory of Cherednik
algebras. For instance, we observe that knowledge of the equivariant graded
Betti numbers (in the sense of commutative algebra) of any irreducible
representation in category O is equivalent to knowledge of the Kazhdan-Lusztig
character of the irreducible object (we use this observation in joint work with
Fishel-Manosalva). We then explore the extent to which Cherednik algebra
techniques may be applied to ideals of linear subspace arrangements: we
determine when the radical of the polynomial representation of the Cherednik
algebra is a radical ideal, and, for the cyclotomic rational Cherednik algebra,
determine the socle of the polynomial representation and characterize when it
is a radical ideal. The subspace arrangements that arise include various
generalizations of the k-equals arrangment. In the case of the radical, we
apply our results with Juteau together with an idea of Etingof-Gorsky-Losev to
observe that the quotient by the radical is Cohen-Macaulay for positive choices
of parameters. In the case of the socle (in cyclotomic type), we give an
explicit vector space basis in terms of certain specializations of
non-symmetric Jack polynomials, which in particular determines its minimal
generators and Hilbert series and answers a question posed by Feigin and
Shramov.

Representation Theory;Commutative Algebra