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

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 $G=S{L}_n$, we study a bigraded refinement of the result.

v3, 28 pages, 2 figures. Final version, to appear in Representation Theory