Tensor products as induced representations: The case of finite GL(3)

Abstract

We describe the tensor products of two irreducible linear complex representations of the group G = GL(3, F q) in terms of induced representations by linear characters of maximal tori and also in terms of Gelfand-Graev representations. Our results includeMacDonald'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 SL(2,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 GL(n, F q). © 2012 Pleiades Publishing, Ltd.