Representations of rational Cherednik algebras of rank 1 in positive characteristic

TL;DR

This paper classifies irreducible representations of rank 1 rational Cherednik algebras in positive characteristic, distinguishing between quantum and classical cases based on the value of the parameter analogous to Planck's constant.

Contribution

It provides a complete classification of irreducible representations for these algebras in positive characteristic, highlighting differences between quantum and classical cases.

Findings

Quantum case: irreducible representations have dimension pr.

Classical case: irreducible representations have dimension r.

Distinct representation dimensions depend on the algebra's parameter.

Abstract

In this paper, we classify the irreducible representations of the rational Cherednik algebras of rank 1 in characteristic p > 0. There are two cases. One is the "quantum" case, where "Planck's constant" is nonzero and generic irreducible representations have dimension pr, where r is the order of the cyclic group contained in the algebra. The other is the "classical" case, where "Planck's constant" is zero and generic irreducible representations have dimension r.