Differentiating the Weyl generic dimension formula with applications to support varieties

Abstract

The authors compute the support varieties of all irreducible modules for the small quantum group uζ(g), where g is a finite-dimensional simple complex Lie algebra, and ζ is a primitive ℓ-th root of unity with ℓ larger than the Coxeter number of g. The calculation employs the prior calculations and techniques of Ostrik and of Nakano, Parshall, and 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 G1 of a reductive algebraic group scheme G defined over the prime field Fp.