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

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

2009-05-28

arxiv.org/abs/0905.4707v2

Adv. Math. 229 (2012) 2656-2668

Mathematics

Group Theory

10 pages, various typos corrected, references updated

Scientific paper

10.1016/j.aim.2012.01.007

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 $\mathbb{F}_p$.

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