Orthogonal representations of \({{\,\textrm{SL}\,}}_2(q)\) in defining characteristic

This paper determines the types of the invariant quadratic forms over their respective (finite) fields of definition for all irreducible modules of the groups \({{\,\textrm{SL}\,}}_2(q)\) ${\text{SL}}_2(q)$ in defining characteristic. We prove that for \(q>2\) $q>2$ any absolutely irreducible even dimensional orthogonal \({{\,\textrm{SL}\,}}_2(q)\) ${\text{SL}}_2(q)$ -module W in defining characteristic carries a split invariant quadratic form (i.e. it is of \(+\) $+$ type) unless \(\dim (W) \equiv 4 \pmod {8}\) $dim(W)\equiv 4(mod8)$ and the field of definition of W is the subfield of index 2 in \({\mathbb {F}}_q\) ${\mathbb{F}}_q$ ; in the latter case the type of the invariant quadratic forms is −.