On G2(Fq)-invariants of representations of D4(Fq)

Let Fq be a finite field of order q and whose characteristic is not equal to 2 or 3. Let G denote a split algebraic group of type D4 over Fq. It contains a split algebraic group G2 over Fq. In this paper we will state a conjecture on the dimensions of G2(Fq)-invariant vectors on representations of G(Fq), and we will prove this conjecture in the cases where the representation of G(Fq) is a principal series representation, a generic representation, or a unipotent representation.