Complexity of rational $\mathrm{GL}_{n(r)}$-modules

Question

Let $k$ be an algebraically closed field of characteristic $p>0$, and let $G=\mathrm{GL}_n(k)$ for some natural number $n$. For any integer $r\ge 1$, let $G_{(r)}$ denote the $r$th Frobenius kernel of $G$, that is, the kernel of the Frobenius endomorphism which sends each entry of a matrix to its $p^r$th power. The coordinate algebra of $G_{(r)}$ is

$$k[G_{(r)}]=k[x_{ij},t]/(\det(x_{ij})t-1,x_{ij}^{p^r}-\delta_{ij})$$

Let $M$ be a finitely generated rational $G_{(r)}$-module, and let $\ldots P_2\to P_1\to P_0\to M$ be a minimal projective resolution. The complexity of $M$, denoted $c(M)$, is the smallest integer $s$ such that there is a constant $\kappa>0$ with $\dim_kP_m\le\kappa m^{s-1}$ for all $m\ge 0$.

I'm interested in $G_{(r)}$-modules of large complexity. More specifically, in terms of $r$ and $n$, do we know anything about the value of

$$\max_{M}c(M)$$

where the maximum is taken over finitely generated rational $G_{(r)}$-modules? I'd be happy with a nontrivial lower bound if one is known.

Answer 1

I don't recall seeing an explicit answer to your question in the literature, but the basic outline starts with the (restricted) Lie algebra or first Frobenius kernel: here an upper bound on the complexity of any finite dimensional module is given by the dimension of the "restricted nullcone", which for $p$ at least the Coxeter number is the full nilpotent variety in the Lie algebra.

For higher Frobenius kernels $G_r$ (a less cumbersome notation than yours), where $G$ can be any reductive group, the study of complexity seems to be much less direct than for $G_1$. But you might get some help from Theorem 2.4 in an old paper by Dan Nakano which is freely available online here. This kind of result suggests strongly that for finite dimensional $G_r$-modules the best possible upper bound should be something like $r$ times the dimension of the nilpotent variety. For a general linear group $\mathrm{GL}_n$, the latter dimension would be $n^2-n$.

I don't have time at the moment to look further into the recent literature (which extends to work on related quantum groups and Lusztig's analogue of Frobenius kernels), but there have been quite a few relevant papers ranging from the early work of Jantzen and Friedlander-Parshall to work of Carlson-Lin-Nakano, Drupieski, etc. Roughly speaking, complexity 0 corresponds to projective modules whereas the largest complexity for a simple module tends to occur for the trivial module. But along the way a lot of cohomology is involved.