On the smoothness of centres of rational Cherednik algebras in positive characteristic

TL;DR

This paper investigates the structure and smoothness of centers of rational Cherednik algebras in positive characteristic, focusing on block decomposition and classification of groups with regular centers.

Contribution

It provides explicit descriptions of block decompositions for certain groups and classifies groups with regular centers at generic parameters.

Findings

Explicit block decomposition for groups G(m,d,n)

Classification of groups with regular centers

Analysis of the algebra's smoothness properties

Abstract

In this article we study rational Cherednik algebras at $t = 1$ in positive characteristic. We study a finite dimensional quotient of the rational Cherednik algebra called the restricted rational Cherednik algebra. When the corresponding pseudo-reflection group belongs to the infinite series $G(m,d,n)$, we describe explicitly the block decomposition of the restricted algebra. We also classify all pseudo-reflection groups for which the centre of the corresponding rational Cherednik algebra is regular for generic values of the deformation parameter.