On the affine Springer fibers inside the invariant center of the small quantum group

TL;DR

This paper explores the geometric structure of the invariant center of the small quantum group associated with a simple Lie algebra, providing dimension calculations and a refined analysis for special cases.

Contribution

It computes the dimension of the geometric subalgebra of the center and offers a detailed study for the case of G=SL_n, advancing understanding of quantum group centers.

Findings

Dimension of the geometric subalgebra of the center is computed.

A bigraded refinement of the center's structure is developed for G=SL_n.

Provides new insights into the geometric realization of quantum group centers.

Abstract

Let $\mathfrak{u}_\zeta^\vee$ denote the small quantum group associated with a simple Lie algebra $\mathfrak{g}^\vee$ and a root of unity $\zeta$. In [9], a geometric realization of $Z(\mathfrak{u}_\zeta^\vee)^{G^\vee}$, the $G^\vee$-invariant part of the center of $\mathfrak{u}_\zeta^\vee$, was proposed. We compute the dimension of the geometric subalgebra of the center and in the case where $G=SL_n$, we study a bigraded refinement of the result.