On dimension growth of modular irreducible representations of semisimple Lie algebras

TL;DR

This paper studies how the dimensions of irreducible representations of semisimple Lie algebras grow with respect to prime characteristic p, revealing polynomial behavior and connections to affine Weyl group structures.

Contribution

It demonstrates the compatibility of the canonical basis with two-sided cell filtrations and links dimension polynomial degrees to these filtrations, advancing classification conjectures.

Findings

Canonical basis is compatible with two-sided cell filtration.

Dimension of modules is polynomial in p.

Results support classification of finite-dimensional W-algebra representations.

Abstract

In this paper we investigate the growth with respect to $p$ of dimensions of irreducible representations of a semisimple Lie algebra $\mathfrak{g}$ over $\overline{\mathbb{F}}_p$. More precisely, it is known that for $p\gg 0$, the irreducibles with a regular rational central character $\lambda$ and $p$-character $\chi$ are indexed by a certain canonical basis in the $K_0$ of the Springer fiber of $\chi$. 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 $\lambda$. 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.