The exterior and symmetric square of the reflection representation of An(q) and Dn(q)

We decompose the exterior and symmetric square of the reflection representation of the groups GL(n) and SO+(2m) over a finite field of odd characteristic. As a byproduct of the method, we confirm a new case (r=2, m 4) of a conjecture of Aubert, Michel and Rouquier describing the unipotent part of the Howe correspondence between Sp(2r) and SO+(2m).