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

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.