J an 2 00 9 TENSOR PRODUCTS OF IRREDUCIBLE REPRESENTATIONS OF THE GROUP GL ( 3 , F q )

We describe the tensor products of two irreducible linear complex representations of the group G = GL(3, Fq) in terms of induced representations by linear characters of maximal torii and also in terms of classical and generalized Gelfand-Graev representations. Our results include MacDonald’s conjectures for G and at the same time they are extensions to G of finite counterparts to classical results on tensor products of holomorphic and antiholomorphic representations of the group SL(2, R). Moreover they provide an easy way to decompose these tensor products, with the help of Frobenius reciprocity. We also state some conjectures for the general case of GL(n, Fq).