The polynomial representation of the type $A_{n - 1}$ rational Cherednik algebra in characteristic $p \mid n$

TL;DR

This paper investigates the polynomial representation of the type A_{n-1} rational Cherednik algebra in characteristic p dividing n, providing explicit formulas and computing the Hilbert series of the irreducible quotient.

Contribution

It offers explicit formulas for generators of the maximal proper graded submodule and shows they form a complete intersection in characteristic p dividing n.

Findings

Generators for the maximal proper graded submodule are explicitly described.

The submodule cuts out a complete intersection.

Hilbert series of the irreducible quotient is computed.

Abstract

We study the polynomial representation of the rational Cherednik algebra of type $A_{n-1}$ with generic parameter in characteristic $p$ for $p \mid n$. We give explicit formulas for generators for the maximal proper graded submodule, show that they cut out a complete intersection, and thus compute the Hilbert series of the irreducible quotient. Our methods are motivated by taking characteristic $p$ analogues of existing characteristic $0$ results.