# The center of the small quantum group II: singular blocks

## TL;DR

This paper extends the understanding of the center of small quantum groups to singular blocks, providing geometric methods to compute their dimensions and proposing a conjecture for type A.

## Contribution

It generalizes previous results to singular blocks and introduces an algebro-geometric approach for dimension calculations, including a new conjecture for type A.

## Key findings

- Describes the center of singular blocks via sheaf cohomology.
- Provides a method to compute dimensions of singular blocks.
- Formulates a conjecture for the center's dimension in type A.

## Abstract

We generalize to the case of singular blocks the result in \cite{BeLa} that describes the center of the principal block of a small quantum group in terms of sheaf cohomology over the Springer resolution. Then using the method developed in \cite{LQ1}, we present a linear algebro-geometric approach to compute the dimensions of the singular blocks and of the entire center of the small quantum group associated with a complex semisimple Lie algebra. A conjectural formula for the dimension of the center of the small quantum group at an $l$th root of unity is formulated in type A.