On a question of Drinfeld on the Weil representation: The finite field case

Let F   be a finite field of odd cardinality, and let G=GL2(F)G=GL2(F). The group G×G×GG×G×G acts on F2⊗F2⊗F2F2⊗F2⊗F2 via symplectic similitudes, and has a natural Weil representation. To answer a question raised by V. Drinfeld, we decompose that representation into irreducibles. We also decompose the analogous representation of GL2(A)GL2(A), where A is a cubic algebra over F.