Differentiating the Weyl generic dimension formula and support varieties for quantum groups

TL;DR

This paper computes support varieties for irreducible modules of small quantum groups and Frobenius kernels, revealing deep connections with Lusztig's character formula and Kazhdan-Lusztig polynomials.

Contribution

It provides explicit support variety calculations for quantum groups and Frobenius kernels, utilizing advanced representation theory techniques and deep algebraic results.

Findings

Support varieties are computed explicitly for all irreducible modules.

Results connect support varieties with Lusztig's character formula.

Calculations extend to Frobenius kernels of algebraic groups.

Abstract

The authors compute the support varieties of all irreducible modules for the small quantum group $u_\zeta(\mathfrak{g})$, where $\mathfrak{g}$ is a simple complex Lie algebra, and $\zeta$ is a primitive $\ell$-th root of unity with $\ell$ larger than the Coxeter number of $\mathfrak{g}$. The calculation employs the prior calculations and techniques of Ostrik and of Nakano--Parshall--Vella, as well as deep results involving the validity of the Lusztig character formula for quantum groups and the positivity of parabolic Kazhdan-Lusztig polynomials for the affine Weyl group. Analogous support variety calculations are provided for the first Frobenius kernel $G_1$ of a reductive algebraic group scheme $G$ defined over the prime field $\mathbb{F}_p$.