Orthogonal representations of {\, SL\, }₂ (q) in defining characteristic

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