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In this paper we investigate the growth with respect to p of dimensions of irreducible representations of a semisimple Lie algebra g over Fp. More precisely, it is known that for p ≫ 0, the irreducibles with a regular rational central character λ and p-character χ are indexed by a certain canonical basis in the K0 of the Springer fiber of χ. This basis is independent of p. For a basis element, the dimension of the corresponding module is a polynomial in p. We show that the canonical basis is compatible with the two-sided cell filtration for a parabolic subgroup in the affine Weyl group defined by λ. We also explain how to read the degree of the dimension polynomial from a filtration component of the basis element. We use these results to establish conjectures of the second author and Ostrik on a classification of the finitedimensional irreducible representations of W-algebras, as well as a strengthening of a result by the first author with Anno and Mirkovic on real variations of stabilities for the derived category of the Springer resolution.

On Dimension Growth of Modular Irreducible Representations of Semisimple Lie Algebras