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 $\Omega =S(V^{\ast })\otimes \Lambda (V^{\ast })$ 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 $q$.

arXiv comment17 pages. Updated to correct statement of corollary 1.4(b) and to fix minor typos