  We compute the irreducible constitutents of the product of the Weil character
and the Steinberg character in those finite classical groups for which a Weil
character is defined, namely the symplectic, unitary and general linear groups.
It turns out that this product is multiplicity free for the symplectic and
general unitary groups, but not for the general linear groups.
  As an application we show that the restriction of the Steinberg character of
such a group to the subgroup stabilizing a vector in the natural module is
multiplicity free. The proof of this result for the unitary groups uses an
observation of Brunat, published as an appendix to our paper.
  As our "Weil character" for the symplectic groups in even characteristic we
use the 2-modular Brauer character of the generalized spinor representation.
Its product with the Steinberg character is the Brauer character of a
projective module. We also determine its indecomposable direct summands.
