Conjugation Differential Invariants of $\mathrm{SL}_2(\mathbb{F}_q)$ on trace-free matrices

Abstract

Let $q=p^k$ be prime power, let $F= \mathbb{F}_q$ and let $V$ be the vector space of 2 by 2 matrices over $F$ with trace zero. Let $G = \mathrm{SL}_2(F)$. Then $G$ acts on $V$ via conjugation. Let $\Omega = S(V^*) \otimes \Lambda(V^*)$ be the algebra of differential forms on $V$. A minimal generating set for $\Omega^G$ when $q=3$ was computed by the author and Meyer. In this article we compute a minimal generating set for $\Omega^G$ for all $q$.