  In this paper we establish a Steinberg-Lusztig tensor product theorem for
abstract Fock space. This is a generalization of the type A result of
Leclerc-Thibon and a Grothendieck group version of the Steinberg-Lusztig tensor
product theorem for representations of quantum groups at roots of unity.
Although the statement can be phrased in terms of parabolic affine
Kazhdan-Lusztig polynomials and thus has geometric content, our proof is
combinatorial, using the theory of crystals (Littelmann paths). We derive the
Casselman-Shalika formula as a consequence of the Steinberg-Lusztig tensor
product theorem for abstract Fock space.
