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

The authors compute the support varieties of all irreducible modules for the small quantum group $u_ζ(\mathfrak{g})$, where $\mathfrak{g}$ is a simple complex Lie algebra, and $ζ$ 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 $\mathbb{F}_p$.

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