On infinitesimal Cherednik algebras of gl_2

TL;DR

This paper investigates the structure of infinitesimal Cherednik algebras of gl_2, establishing their centers as polynomial algebras in characteristic zero and their finite generation over the center in positive characteristic.

Contribution

It proves the polynomial nature of the center in characteristic zero and finite generation over the center in positive characteristic for these algebras.

Findings

Center of algebra is polynomial in characteristic zero.

Algebra is finitely generated over its center in positive characteristic.

Abstract

We prove that the center of an infinitesimal Cherednik algebra of $\mf{gl}_2$ is the polynomial algebra of two variables over the field of characteristic 0. In positive characteristic we show that any infinitesimal Cherednik algebra is a finitely generated module over its center.