Differentiating the Weyl generic dimension formula and support varieties for quantum 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 ${\mathbb{F}}_p$.

10 pages, various typos corrected, references updated

References in corpus (1)

• Cohomology for quantum groups via the geometry of the nullcone

Cited by in corpus (5)

• Complexity for Modules Over the Classical Lie Superalgebra gl(m|n)
• Commuting Varieties and Cohomological Complexity
• Exponentiation of commuting nilpotent varieties
• Mixed Commuting Varieties over simple Lie algebras
• Support Varieties for Frobenius Kernels of Classical Groups