Tensor Products as Induced Representations: The Case of Finite $\mathrm{GL}(3)$

We describe the tensor products of two irreducible linear complex representations of the group $G=\mathrm{GL}(3,\mathbb F_q)$ in terms of induced representations by linear characters of maximal tori and also in terms of Gelfand–Graev representations. Our results include MacDonald's conjectures for $G$ and are extensions to $G$ of finite counterparts to classical results on tensor products of principal series as well as holomorphic and antiholomorphic representations of the group $\mathrm{SL}(2,\mathbb R)$; besides, 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 $\mathrm{GL}(n,\mathbb F_q)$.