The weighted fusion category algebra and the q-Schur algebra for GL2(q)

We show that the weighted fusion category algebra of the principal 2-block b0 of GL2(q) is the quotient of the q-Schur algebra by its socle, for q an odd prime power. As a consequence, we get a canonical bijection between the set of isomorphism classes of simple kGL2(q)b0-modules and the set of conjugacy classes of b0-weights in this case.