Abstract
Let F q 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 D 4 over F q . It contains a split algebraic group G 2 over F q . In this paper we will state a conjecture on the dimensions of G 2 ( F q ) -invariant vectors on representations of G ( F q ) , and we will prove this conjecture in the cases where the representation of G ( F q ) is a principal series representation, a generic representation, or a unipotent representation.
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