Orthogonal representations of \,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 \,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 Fq F q ; in the latter case the type of the invariant quadratic forms is −.