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 finite dimensional
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.Comment: 24 pages, v2 acknowledgements added; v3 references adde