We study the Galois module structure of the class groups of the Artin–Schreier extensions K over k of extension degree p, where $k:={\mathbb F}_q(T)$ is the rational function field and p is a prime number. The structure of the p-part $Cl_K(p)$ of the ideal class group of K as a finite G-module is determined by the invariant ${\lambda }_n$, where $G:=\operatorname {\mathrm {Gal}}(K/k)=\langle {\sigma } \rangle $ is the Galois group of K over k, and ${\lambda }_n = \dim _{{\mathbb F}_p}(Cl_K(p)^{({\sigma }-1)^{n-1}}/Cl_K(p)^{({\sigma }-1)^{n}})$. We find infinite families of the Artin–Schreier extensions over k whose ideal class groups have guaranteed prescribed ${\lambda }_n$-rank for $1 \leq n \leq 3$. We find an algorithm for computing ${\lambda }_3$-rank of $Cl_K(p)$. Using this algorithm, for a given integer $t \ge 2$, we get infinite families of the Artin–Schreier extensions over k whose ${\lambda }_1$-rank is t, ${\lambda }_2$-rank is $t-1$, and ${\lambda }_3$-rank is $t-2$. In particular, in the case where $p=2$, for a given positive integer $t \ge 2$, we obtain an infinite family of the Artin–Schreier quadratic extensions over k whose $2$-class group rank (resp. $2^2$-class group rank and $2^3$-class group rank) is exactly t (resp. $t-1$ and $t-2$). Furthermore, we also obtain a similar result on the $2^n$-ranks of the divisor class groups of the Artin–Schreier quadratic extensions over k.

In this paper, we study an analogue of the Bernstein–Gelfand–Gelfand category ${\mathcal {O}}$ for truncated current Lie algebras $\mathfrak {g}_n$ attached to a complex semisimple Lie algebra. This category admits Verma modules and simple modules, each parametrized by the dual space of the truncated currents on a choice of Cartan subalgebra in $\mathfrak {g}$. Our main result describes an inductive procedure for computing composition multiplicities of simples inside Vermas for $\mathfrak {g}_n$, in terms of similar composition multiplicities for ${\mathfrak {l}}_{n-1}$ where ${\mathfrak {l}}$ is a Levi subalgebra. As a consequence, these numbers are expressed as integral linear combinations of Kazhdan–Lusztig polynomials evaluated at 1. This generalizes recent work of the first author, where the case $n=1$ was treated.