# A Soergel-like category for complex reflection groups of rank one

## TL;DR

This paper constructs a new category analogous to Soergel bimodules for rank one complex reflection groups, providing explicit parametrizations, presentations, and analyzing its algebraic properties.

## Contribution

It introduces a novel category for rank one complex reflection groups, with explicit indecomposable objects and a presentation of its Grothendieck ring, extending the Hecke algebra.

## Key findings

- The Grothendieck ring is an extension of the Hecke algebra.
- The algebra is generically semisimple over the complex numbers.
- The indecomposable objects are explicitly parametrized.

## Abstract

We introduce analogues of Soergel bimodules for complex reflection groups of rank one. We give an explicit parametrization of the indecomposable objects of the resulting category and give a presentation of its split Grothendieck ring by generators and relations. This ring turns out to be an extension of the Hecke algebra of the reflection group $W$ and a free module of rank $|W| (|W|-1)+1$ over the base ring. We also show that it is a generically semisimple algebra if defined over the complex numbers.