The diagonal coinvariant ring of a complex reflection group

TL;DR

This paper constructs a large irreducible representation of the Cherednik algebra related to the diagonal coinvariant ring of a complex reflection group, proposing it as the largest such representation with minimal corrections.

Contribution

It introduces a new construction of a significant irreducible Cherednik algebra representation linked to the diagonal coinvariant ring of complex reflection groups, using a novel dot action.

Findings

Constructed a $(g+1)^n$-dimensional irreducible representation.

Proposed this as the largest representation related to the diagonal coinvariant ring.

Identified a dot action of symmetric groups on the Cherednik algebra parameter space.

Abstract

For an irreducible complex reflection group $W$ of rank $n$ containing $N$ reflections, we put $g=2N/n$ and construct a $(g+1)^n$-dimensional irreducible representation of the Cherednik algebra which is (as a vector space) a quotient of the diagonal coinvariant ring of $W$. We propose that this representation of the Cherednik algebra is the single largest representation bearing this relationship to the diagonal coinvariant ring, and that further corrections to this estimate of the dimension of the diagonal coinvariant ring by $(g+1)^n$ should be orders of magnitude smaller. A crucial ingredient in the construction is the existence of a dot action of a certain product of symmetric groups acting on the parameter space of the rational Cherednik algebra and leaving invariant both the finite Hecke algebra and the spherical subalgebra; this fact is a consequence of ideas of Berest-Chalykh on the relationship between the Cherednik algebra and quasi-invariants.